Although the role has has no formal powers, it is normally been held by top ranking cabinet ministers and confers some seniority over other secretaries of state.
Theresa May has sought to bolster her position following last week's disastrous showing at the General Election, effectively making one of her closest Cabinet allies deputy prime minister.
Patel holds on to her position as the senior-most Indian-origin Cabinet minister and will remain in charge as the secretary of state for global development in the new government, which May said will focus on delivering a successful Brexit.
Green was previously secretary of state for work and pensions and is now expected to have a key role in supporting the prime minister and particularly in coordinating the negotiations for the UK's exit from the European Union.
Mrs May's new chief of staff, Gavin Barwell, was also a passionate Remainer, having branded the Leave campaign as the "politics of hate and division".
May, who chairs a meeting of her new Cabinet on Monday morning, will hear first-hand the anger of Conservative members of Parliament who blame her for the election campaign that saw the Tories unexpectedly lose their parliamentary majority.
But there is a demotion for Liz Truss, who has gone from Justice Secretary to fill in Mr Gauke's position - which is the number two role at the Treasury underneath the Chancellor.
Negotiations are set to start in a week's time, but it is still not clear what form of Brexit David Davis will present. The Oxford graduate, who was elected as an MP in 2005, has been unpopular with many National Health workers, with his decision to impose new contracts on junior doctors attracting particularly heavy criticism.
Mr Gove, who clashed bitterly with Mrs May over tackling extremism when they were in government under David Cameron, said that he had been taken by surprise when the invitation came through to visit No 10.
Speaking to Lord Falconer and Lord Howard, Morgan demanded to know why May hasn't yet resigned, four days after she lost her Commons majority in the snap election. She included a link to the They Work For You website detailing how Gove voted with the government in favour of culling badgers, selling England's state-owned forests, and building high speed rail infrastructure. Sunday's media reports showed that James Brokenshire was likely to keep the post of Northern Ireland Secretary. Andrea Leadsom takes over from Lidington as Leader of the House of Commons.
Greg Clark is remaining in his post of Business, Energy and Industrial Strategy Secretary.