Maria Caulfield's appointment as Vice-chair for Women caused anger because of her views on abortion.
"Justine was offered DWP but declined to take it".
Greening's decision to quit came at the end of a long day of Cabinet meetings for the Prime Minister.
Following the ministerial changes, May said the reshuffle makes the government look "more like the country it serves" with a "new generation" of "fresh talent". "I'll continue to do everything I can to create a country that has equality of opportunity for young people and I'll keep working hard as MP for Putney". One minister said: "I feel sorry for Steve".
One Tory former minister told the Press Association: "She ended the year in not a bad place, to the point where she was making a virtue of her ability to walk through fires". Although he has a low profile in the party and media but, according to his government biography, he previously "spent 18 years working in the pubs, brewing and hotel industries, in Britain and overseas".
May's office flagged Monday's events as "a refresh" of her top team.
Health Secretary Jeremy Hunt has been tipped as a possible replacement. Even Greg Clark, who is opposed to her trade policy, remains unscathed.
Tory HQ tweeted congratulations to "new party chairman" Chris Grayling.
He replaced PM's close ally Damian Green, who was dismissed as the Cabinet Office minister and de facto Prime Minister last month for misleading statements over pornography found on his office computer.
Ms Greening has been mounting what many have seen as a campaign to remain in post, posting a series of messages on Twitter listing her achievements as education secretary.
Everybody needs to be reminded of Esther McVey vs Glenda Jackson.Esther McVey has been reappointed to the Department of Work and Pensions despite being forever associated with some of its cruellest measures against disabled people.
Digital minister Matt Hancock took over from his old boss as Culture Secretary.
In a reshuffle that unraveled over the course of several hours and lasted into the night, it became clear that the key takeaway was how little wriggle room May seemed to have to enact change as minister after minister were confirmed in their posts.
Karen Bradley has been appointed as the new Northern Secretary.
Hunt's title has been tweaked to Secretary of State For Health and Social Care.
Claire Perry will also be able to attend the Cabinet as a new Business Minister.
It follows the resignation of James Brokenshire on health grounds. Lidington will, however, likely take over Green's role in Brexit negotiations with the Scottish and Welsh governments.
Britain's Foreign Secretary Boris Johnson has kept his portfolio despite a series of high-profile gaffes.
Chancellor Philip Hammond stays at the Treasury.
One week later, as global development secretary, after a row over unauthorised meetings with Israeli officials. Home Secretary Amber Rudd was also appointed by May as Minister of Women and Equalities.