A technical clarification on subgraph isomorhism
The subgraph isomorphism problem is "Is $G$ isomorphic to some subgraph of&nbsp;$H$?" so, yes, if $H$ contains a $k$-clique, that clique on its own has subgraphs isomorphic to any graph on $k$&nbsp;or fewer vertices. Your "exact presence" version is the induced subgraph isomorphism problem: is $G$ isomorphic to some induced subgraph of&nbsp;$H$? Both versions are NP-complete.

You could have found this trivially by Googling for "subgraph isomorphism problem".