I'm still drowning in Lominsan lore over here, but should point out that the Syndicate is not a secret society. They're the ruling elite of Ul'dah in name, title, and public recognition. Their motivations and deals may at times be shady, but its membership is plain as day - the six wealthiest people in town.

In the Immortal Flames story arc, Raubahn (leader of the Flames) takes part in a high-stakes duel and the winnings end up pushing him over the cusp of being one of the six wealthiest men in the city. This deposes the least-wealthy of the Syndicate members and puts him in his place (this is possibly Rycharde, who says that Raubahn ruined him that day).

Now, usually, the Syndicate is at odds with the Sultana and her advisors - they have direct control of the economy and fear anything that could assert more sway over it than they could. However, the sultana is ecstatic that Raubahn is now on the board and they seem very close friends - he even refers to the fact that his ascension to the Syndicate was a promise he made to her, with her end of the bargain to come (though, perhaps it already has - this is a flashback after all, maybe her end of the bargain was that he would lead the company. He seems to value prestige and wealth little if it didn't come with victory).