Functional subgroups

Given a group, there are some subgroups of interest that can be defined uniquely and functionally in terms of ${\displaystyle G}$, such as the center, commutator, etc. These should be invariant under isomorphisms, that is, any isomorphism on the groups, must induce a corresponding isomorphism on the subgroups. Hence, such subgroups are invariant under automorphisms, and are always characteristic subgroups.