The Syrian National Council, consisting mainly of exiles, is failing to gain diplomatic traction.
Officials from the Syrian National Council said the suspension indicated it was time for world leaders to take more severe steps.
If the Syrian National Council ultimately does take charge, it may struggle to contain the worst excesses of the militia groups, he said.
CNN: Syria: As al-Assad's grip loosens, what could come next?
The Syrian National Council, the Syrian opposition, congratulated Pope Francis and called on him to pay special consideration to the suffering of the Syrian people.
The opposition Syrian National Council has accused the Syrian government of planning Monday's explosion, but has not offered any evidence to back up that claim.
The Syrian National Council (SNC), which was formerly recognised as the main opposition, had been concerned it might be sidelined by the new opposition body.
The Syrian National Council, a group mainly of exiles, includes several.
Dozens of countries have recognized the Syrian National Council as a legitimate representative of the Syrian opposition, though many members of the group's leadership are expatriates.
"Our Foreign Minister rejected meeting with the Syrian National Council until they widened their base to include different religions and sects, " a Foreign Ministry spokesman said.
Russia has recently been pivoting in an anti-Assad direction, saying that it would suspend arms sales and holding talks with the Syrian National Council.
Adib al Shishakly, a member of the Syrian National Council, an umbrella group of exiles, said Damascus needs to abide by all six points of Annan's peace plan.
"We cannot overlook the fact that Kofi Annan's proposals have not been agreed to by some of the opposition groups, including the well-known Syrian National Council, " Lavrov said.
Burhan Ghalioun, the professorial leader of the Syrian National Council, the main opposition group in exile, has pledged to oversee and organise the rebels, but not to equip them.
Speaking on Al Arabiya TV, the former head of the Syrian National Council, which preceded the coalition, said that he and other coalition members were surprised by the resignation.
The Syrian National Council, an opposition coalition that includes many exiles, is set to make a presentation at the Istanbul meeting to summarize its efforts to unite the opposition.
Consequently, Turkey rapidly emerged as one of the boldest supporters of the Syrian opposition, and the Syrian National Council, formed as the dissidents' umbrella group, found a base in Turkey.
Burhan Ghalioun, of the Syrian National Council - the divided body which says it represents the political opposition to Mr Assad - said Russia was standing in the way of progress.
The opposition Syrian National Council warned that 40, 000 civilians' lives were in danger amid the violence and called for the Arab League to hold an emergency meeting to protect the besieged city.
Shaikh now plans to split off and form his own group in coming days amid disagreements over the rebel army's relationship to the Syrian National Council, if rebel and opposition leaders can't come to a power-sharing compromise, rebels familiar with his plans said Friday.
Such a zone would reflect the separate coastal Alawite state created on and off between 1920 and 1936 when France was the colonial power in Syria, according to Osama Klaib, a Sunni lawyer from the Homs area and now a Tripoli-based member of the opposition Syrian National Council.
In early July, Mr Barzani himself brokered a deal whereby the PYD agreed to share power with a coalition of smaller Syrian Kurdish parties, the Kurdish National Council.
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