The other groups are the Haqqani network and the faction led by Hafiz Gul Bahadur.
BBC: Taliban's Mullah Nazir death spells trouble for Pakistan
And the faction leaders have ordered their fighters not to join the government's new forces.
The faction is part of the Shiite-led United Iraqi Alliance, the political bloc that holds sway in Iraq.
The faction headed by Mr Mitsuzuka wants to retain control of the job.
The ex-prime minister said the faction, Rally for the UMP (R-UMP), would be dissolved if a new vote was announced.
According to the statement, the faction is headed by Alghabass Ag Intalla, an important Ansar Dine leader and influential figure in Kidal.
Some analysts said they believe Mr. Rehman could have cut a deal with Islamabad, leaving the faction around the more extreme Mr. Hakimullah isolated.
The task before Kesri is to reunite the faction-ridden party.
But in the quarter-century after her death, the family became divided over a possible sale with William Cox III, Mr. Cox's son, helping lead the faction favoring a sale.
Insiders expect the present chief cabinet secretary, Mikio Aoki, aged 65, an upper-house member from Mr Takeshita's district in western Japan, to take over the faction for the time being.
Fatah, the faction they favour, is fractious and disorganised.
ECONOMIST: Democracy is flagging in both the Palestinian territories
The meeting also highlighted the enormous power of the faction leaders from the mainly Tajik former Northern Alliance, who orchestrated both Mr Karzai's election and the selection of his government from behind the scenes.
For the past year, Livni has led the negotiating team with the Fatah faction of the Palestinian Authority.
He became, for Tony Blair, an important bridge between the traditional Labour left and the modernising faction of the party.
The dominant faction of the GOP now seems to be positioning itself to show responsibility in facing various impending fiscal crises.
They cut their political teeth together in the 1980s in the same faction of the Liberal Party in Antioquia, their home region.
His group has been the dissenting faction in the negotiations since Thursday.
Reports in Japan say Ichiro Ozawa, who leads the biggest faction despite the suspension of his party membership over a political funding scandal, is unlikely to support the former foreign minister.
BBC: Japan PM Naoto Kan announces resignation amid criticism
Many fear that Mullah Nazir's death may upset this balance, especially if the fourth faction in the region, the Tehreek-e-Taliban Pakistan group (TTP), is able to force an advantage in Wana region.
BBC: Taliban's Mullah Nazir death spells trouble for Pakistan
However, the talks are being boycotted by Abdel-Wahid al-Nur, the leader of the largest faction of the SLM, as well as by the leader of the other main Darfur rebel group, the Justice and Equality Movement (JEM), Khalil Ibrahim.
And even if the GOP wise guys figured out how to lure libertarians and moderates back, it runs the risk of disaffecting the largest faction of the party, the social conservatives, and leaving them sitting on their hands rather than marking ballots on Election Day.
Riina, known as The Beast, is currently serving multiple life-sentences after leading the Corleonesi faction of the criminal network to the top of the Sicilian mafia and waging a ruthless campaign of violence against rival mobsters and the Italian state in the 1980s and early 1990s.
Mr Sadr's 30-strong block of followers in parliament, which has 275 members, was the largest faction in the ruling Shia alliance and had backed Mr Maliki as prime minister largely to stop the post going to the Supreme Council for Islamic Revolution in Iraq (SCIRI), their bitter rivals in the Shia alliance.
ECONOMIST: The prime minister, Nuri al-Maliki, is ever more embattled
Viktor Alksnis, one of the most prominent spokesmen for the Soviet old guard and leader of the Soyuz faction of the Supreme Soviet Parliament, became the latest in a series of prominent authorities on Kremlin intrigues to speculate openly that Mikhail Gorbachev may have been involved in orchestrating the recent abortive coup.
The Hagelin faction is challenging the FEC decision in the courts.
His faction, the Supreme Council - which is an allied Shiite faction - the Kurds, and a few other factions, they want the United States there because in their view, the United States is defending them.
Rand Paul, a leader of the tea party faction of the Republican party, told CNN's Candy Crowley.
Abit is with the mainstream Palestinian faction in the camp.
But the Misrata faction is also lobbying for one of its own to have the job.
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