Here's the way I understood it to be, although it's still fairly murky to me. The late fifties and early sixties were fairly difficult to decipher, organizationally and structurally:

Battaglia: Boss of the western suburbs. Pranno, Amabile, DeGrazia, etc. reported to him. Pranno definitely ran Stone Park and reported to Battaglia.

Alderisio: Boss level by a certain point in the mid to late sixties, mainly focused on the Grand Avenue area (a sort of buffer zone between Buccieri's and Cerone's traditional territory), the Loop, and certain parts of the North Side.

Nicoletti: Kind of the same boat as Alderisio, he was headquartered in the Garfield Park area, a little west of where Alderisio operated, and later shared Melrose Park with Aiuppa once Battaglia and his top guys went to prison.

Daddano: Boss of Lake County. Sent Joe Amato to McHenry County to run that territory (or maybe he sent him to Lake County from McHenry - can't recall off the top of my head). I don't think he reported directly to Battaglia.

Caifano was a guy who had some spots on the North Side but I'm not 100% sure if he was under Alderisio or not. Considering his role in the Grand Avenue crew when he got out of prison the first time he may have been, but I'm honestly not sure.

Willie Aloisio had a chunk of territory west of the Chicago River between Belmont and Bloomingdale. I am not sure who he reported to but it very well could have been Battaglia.