Was Frankie Carbo a Genovese or Lucchese member?

I say Lucchese, but others say Genovese. So what Family did Frankie Carbo actually belong to?

For many years he was listed erroneously on Lucchese charts as a family member. But in fact Frankie Carbo was believed to have been a member under the Luciano family. Hence, thats why he was a prime suspect in the murder of Benjamin (Bugsy) Siegel, a associate directly connected to the Luciano family.

Each borgata typically handles their own "work," its not usually handed out for outsiders to accomplish.

I always thought Frankie Carbo was kinda out of place in the Lucchese Family. He is more in line with the old Luciano Family style but then again, each family has outliers.

I also think because Carbo was active in the prize-fight rackets in conjunction and partnership on occasion with Lucchese bigwigs Jimmy "Doyle" Plumeri and Eddie Coco authorities may have erroneously suspected him to be in that crew.

This leads to a question. Who was Carbo associated to in the Genovese Family? Because the only thing I ever heard in reference to him being with the West Side was the association to the Bugsy Siegel hit.