I think it was in the late 1960s that they adopted a power-sharing arrangement. Fish Cafaro confirmed Benny Squint was the real boss after Vito Genovese died in 1969, but used puppet bosses like Tommy Ryan and Funzi to shield himself. It worked out for him, as he died a free man in 1987.

Not sure if this was the Genovese ruling administration in 1981:

Official Boss: Philip "Benny Squint" Lombardo
Street Boss: Vincent "Chin" Gigante
Front Boss: Anthony "Fat Tony" Salerno
Underboss: Saverio "Sammy" Santora
Consigliere: Louis "Bobby" Manna