As many as 1,000 boys from strictly Orthodox Jewish families may be pupils at a network of between 12 and 20 illegal private schools in east London.
These schools are not registered with the authorities, which makes them illegal, and they offer a narrow, religious syllabus.
The Department for Education is working with Ofsted to find and shut them.
Some of the illegal schools, however, are registered as charities, gaining them an advantaged tax status.
These private schools serve the small so-called Charedi community - a grouping that contains within it a wide variety of strictly Orthodox Jewish traditions. Hackney council estimates there are around 30,000 Charedi Jews in the borough.
Charedi parents are more likely to want a relatively mainstream education for their girls. There is demand, however, from parts of the community for a narrow education for boys, one that is largely focused on religious education and delivered in the Yiddish language.
While there are registered Charedi schools, many are unregistered because they fear being shut or made to follow a broader syllabus.
Officials from schools watchdog Ofsted and the DfE believe many of these schools offer an insufficiently broad education. Former pupils have complained of not being equipped by them to leave the Charedi community.
The education authorities have similar concerns about stricter Christian and Muslim groups. The fundamental issue is that English education has long held that parents should be able to educate their children within their faith. But that conflicts with another principle: children should be equipped to do as they wish as adults.
One ex-student of illegal Charedi schools, now in his 20s and outside the community, told Newsnight: "I'm starting to study for my GCSEs. I'm maybe like an eight-year-old, nine-year-old. That's my level of education."
Men who stayed in the community, however, told Newsnight that they supported the schools - even if they wish the schools were registered with the authorities. Eli Spitzer, now a head teacher at a registered Charedi primary school, says that his education at an unregistered school was "overwhelmingly positive".
The lowest estimate for the number of illegal schools in Hackney heard by Newsnight came from a senior member of the community, who estimated that there were a dozen such institutions, teaching 600 boys.
The highest estimate, from a someone who recently left the community, was that there were around 20. A senior local government official in Hackney estimated there are 1,000 boys in the illegal schools.
For boys under the age of 13, some parents want only a few hours of secular education each week.
From interviews conducted by Newsnight with current and former community members, an hour a day of such study seems the typical amount. A former pupil told us they received "two hours a week for English and maths". The rest of the time was spent studying scripture and was taught in Yiddish.
Around the age of 13, the boys move to "yeshivas". These schools often have very long school days - lasting more than 14 hours. Pupils are exclusively taught in Yiddish and only study scripture. They are not entered for GCSEs or for other qualifications.
Former pupils who left the community described struggling with English and maths, in particular, because of this education and one said he was "very unhappy with the education I received. I'm struggling to find a job because of that".
Some of the schools and yeshivas are run in contravention of the 2008 Education and Skills Act, which stipulates that "a person must not conduct an independent educational institution unless it is registered".
An individual convicted of running an unregistered school could face up to a year in prison. Ofsted and the DfE began a crackdown on illegal schools in January.
The DfE says: "We have announced an escalation of Ofsted investigations into unregistered schools, with additional inspectors dedicated to rooting them out, a new tougher approach to prosecuting them and a call to local authorities to help identify any settings of concern."
Ofsted says it has inspected and closed seven illegal schools across the country since last November - including a Charedi school.
Even though the institutions are currently on high alert, however, Newsnight established the location of four unregistered Charedi schools with relative ease.
We also established that whistleblowers had alerted the DfE to all four. Newsnight has been asked not to publish the locations of the schools by anti-semitism experts.
In one case, we learned that the authorities had investigated and concluded that no children were being taught in the building. So we asked an actor, posing as the parent of a prospective 13-year-old pupil, to telephone the yeshiva and ask in Yiddish what services it offered.
The school employee who answered the phone stated that the normal day ran from 8am to around 9 or 10pm at night. There was, however, a "dawn framework" for boys who wished to start the day even earlier. When asked what the school taught, the employee listed no secular subjects. When asked how many boys were there, they believed there were 80 or 90 boys in the "junior yeshiva".
The building it is based in also failed a Fire Brigade inspection last year, which chastised the building managers for a "failure to take general fire precautions to ensure the safety of persons on the premises".
The school was registered in one place, however - with the Charity Commission.
Research by the British Humanist Association identified a further seven cases where suspected illegal Charedi schools have registered as charities with the Charity Commission in order to benefit from charity status. One of those schools was the institution recently ordered to close by the DfE.
Andrew Copson, chief executive of the BHA, said: "By being allowed to register as charities, these schools are being given access to all sorts of tax and other benefits that supplements the hundreds of thousands of pounds being poured into them. This means that this is not just a matter for the DfE in some regulatory sense to sort out. It's also a serious matter for the Charity Commission to investigate."
The Charity Commission said it would "have concerns" where charities were not meeting the requirements imposed on it by other bodies and will "liaise with the Department for Education if necessary to determine what regulatory action may be required".
Dr Yaakov Wise, an expert on the Charedi community based at Huddersfield University, said: "Their parents and teachers gave them a highly intellectual, very sophisticated education for the life that they planned for them to lead. Now they've rejected all that and they've gone off into the world, so they have to start again. They have to get an education that suits the world outside."
Mr Spitzer said that he could have retrained for other activities if he so wished: "I attended yeshiva from the age of 14 and i spent the majority of the day studying Biblical and Talmudical texts... [which are] very challenging and highly intellectual."
Referring to the case of the young men who told Newsnight they were let down by these schools, Mr Spitzer argues that these young men "were let down in a primary system where the provision of secular studies was not good enough". Rather than having the authorities dismantle the school system, he argues, the focus should be on improving their secular studies provision.