My understanding is that large cardinals are ordered by "consistency strength", but how does this correlate with their size (cardinality)?


More specifically, are there any systematic results on the lines of:
If A and B are two types of large cardinals such that 
Cons(ZFC + Type A exists) => Cons( ZFC + Type B exists)
THEN
Cardinality of smallest Type A cardinal >= Cardinality of smallest Type B cardinal



Large cardinal consistency strength and size
I may note that a cardinal of type $A$ may has more consistency strength of a cardinal of type $B$, while the smallest cardinal of type $A$ is smaller than the least cardinal of type $B$ (assuming cardinals of both types exist). For example:



> 
> The consistency of a huge cardinal implies the consistency of a supercompact cardinal, nevertheless, the least huge cardinal is smaller than the least supercompact cardinal (assuming both exist).
> 
> 
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