A subgroup is characteristic when every automorphism of the ambient group maps it to itself. A uniquely determined subgroup of a given order or structural type is characteristic.
A nontrivial group with no proper nontrivial characteristic subgroup. A minimal nontrivial normal subgroup of a finite group has this property: every characteristic subgroup of it is normal in the ambient group.

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Characteristic subgroup by Wikipedia Bot 1
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A **characteristic subgroup** of a group \( G \) is a subgroup \( H \) that is invariant under all automorphisms of the group \( G \). This means that for any automorphism \( \phi \) of \( G \), the image \( \phi(H) \) is still a subgroup of \( G \) and is equal to \( H \) itself.
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