It's not clear whether an Admiral is both fleet commander and direct commander of their destroyer. We know for example that Petrarch is a rear admiral, and that he commands 3rd Fleet. However it's never mentioned whether he commands the Aquitaine directly or not, and the existence of anyone else who does is also never mentioned. The practice of giving a flag-rank officer direct command of a major warship is historically very unusual, but not unheard of. On the other hand, a fleet is a permanent institution, not ad-hoc like a task force where the most senior officer present would be given command, and common sense dictates giving overall command to an officer who is not burdened by the concerns of fighting his own ship. We're also never really directly informed of whether lesser vessels have literal Captains or merely figurative ones (article of faith in most navies; the CO is The Captain, regardless of rank, when aboard ship).
I tend to believe that cruisers commanded by Lieutenant Commanders, sometimes Lieutenants for older ones or under unusual circumstances, sometimes full Commanders for more advanced models. Corvettes are commanded by Commanders, occasionally by very junior Captains. A destroyer has a Captain in direct overall command, probably another commanding the aerospace group, and perhaps even one or two more (most likely, engineering and weapons). Installations like the Arcadia, and planetside facilities, may be Captain's or Admiral's postings, depending on their importance. Admirals command fleets, systems, or regional commands.
Of course, I can't prove a word of that.
