American LHDs never operate on their own. The smallest formation is the Expeditionary Strike Group which consists of 1 LHD and 2 LPDs carrying a single MEU and two-three escorts - originally a cruiser with command facilities, a destroyer and a frigate - as well as an attack sub.
On an operational scale however en ESG never moves into position without a CSG present in the area.
LHDs are not command ships. Their role is to deploy land forces and the only C3 involves staging the assault or retrieving the forces. The rest is managed by the ground forces. This is why - at least in the American military - the USN provides the ships and the USMC provides everything else.
Consider that even in the CSG the carrier is not the capital ship for the naval force - that's the cruiser. The only reason why a carrier would carry an admiral is because the entire task force consisting of the CAW and the ships has to be managed jointly and it's easier to manage air operation which are the primary offensive and defensive capability from the carrier while the escorts do their job separately.
In short I don't know where the idea that an LHD is some capital ship would come from. Only from misunderstanding of fleet operations.
