The Orion Syndicate can be considered a subset of Orion Culture.
Not all Orions are members of the syndicate. Most genetically Orion people aren't members of the Syndicate.
Not all Syndicate members are Orion.
It is pretty safe to assume that the leader ship of the Orion Syndicate is all Orions. They have so far defeated every attempt by police and intelligence agencies of known space to penetrate their organization and onbtain information.
The leaders of the syndicate are exceptionally intelligent, paranoid and ruthless. Enough so that the Tal Shiar, the Obsidian Order, and the most insane of Federation intelligence teams have been stymied so far.
The Orion Syndicate is an organized crime syndicate. They deal in illegal weapons, drugs, goods and services. They excel at espionage and frequently corrupt high officials in the goveernments of known space.
While murder, mercenaries, sex-slavery, blackmail and smuggling are the Syndicates stock in trade - everyone captured so far has turned out to be an associate member, a sub contractor to a higher boss. trying to follow this line to the top has resulted in failure so far.
The Orion Syndicate has an extreme code of Omerta or "Silence". Members are expected to die before revealing their allegiance or providing useful information. They have been unfortunately consistent in this area.
The Orion Syndicate does not have their hands in every crime in known space. They are not the only source for smugglers, assassins, drug dealers and pimps. But they are, by far the nastiest and most liable to infiltrate and murder rivals, witnesses or inconvenient people.
As suspected the Orion Syndicate is in a deadly rivalry with The Committee based out of Nar's Star It is expected to result in a compromise organization composed of the surviving members of both parties. This has not happened yet.