A Prince, his vassals, and his Court are not a faction, since they're tied together by common geography and Prince fealty. However, Factions are merely creatures whose agendas and motivations are connected by common political interest, who band together to further those common agendas, who usually have some kind of leader or leaders, and who have at least two other members in the faction. Because the only thing linking together members of a Faction is political interest and motives, many factions form and dissolve so quickly that they're never noticed. There can be many factions in Hell at any one time. However, some factions take root in the rich political compost underlying most of Hell's society, and don't die, at least not quickly.
Some factions are legendary, whispered things.
All are, ultimately, political and are therefore subject to the vagaries and whimsies of Hell-wide politics. Council of Imps Tolerated. Most of the time. Endorsed by many powerful Fallen. Alchemical Society Legal.