Characteristic subgroups are in particular invariant under inner automorphisms, so they are normal subgroups. However, the converse is not true; for example, consider the Klein group *V*_{4}. Every subgroup of this group is normal; but there is an automorphism which essentially "swaps" the various subgroups of order 2, so these subgroups are not characteristic.

On the other hand, if *H* is a normal subgroup of *G*, and there are no other subgroups of the same order, then *H* must be characteristic; since automorphisms are order-preserving.

A related concept is that of a **strictly characteristic subgroup**. In this case the subgroup *H* is invariant under the applications of surjective endomorphisms. (Recall that for an infinite group, a surjective endomorphism is not necessarily an automorphism).

For an even stronger constraint, a **fully characteristic subgroup** (also called a *fully invariant subgroup*) *H* of a group *G* is a group remaining invariant under every endomorphism of *G*; in other words, if *f* : *G* → *G* is any homomorphism, then *f*(*H*) is a subgroup of *H*.

Every fully characteristic subgroup is, perforce, a characteristic subgroup; but a characteristic subgroup need not be fully characteristic. The center of a group is always a strictly characteristic subgroup, but not always fully characteristic; for example, consider the group *D*_{6} × *C*_{2} (the direct product of a dihedral group and a cyclic group of order 2).

The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group.

The property of being characteristic or fully characteristic is transitive; if *H* is a (fully) characteristic subgroup of *K*, and *K* is a (fully) characteristic subgroup of *G*, then *H* is a (fully) characteristic subgroup of *G*.

The relationship amongst these types of subgroups can be expressed as:

subgroup ← normal subgroup ← characteristic subgroup ← strictly characteristic subgroup ← fully characteristic subgroup