Joe Bonanno had strong connections with the Vatican.

On the Gambino family. If I remember right, when Stefano Bontade was killed, Salvatore Inzerillo took over those connections for a short time before he was killed. The Gambinos took in the Inzerillos to protect them after Joe Inzerillo was murdered, with that the Gambino family gained more connections with the Vatican. The Vatican has connections with many mafia families, who relatives who works for the church.