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1-automorphism-invariant subgroup, Abelian-extensible automorphism-invariant subgroup, Abelian-extensible endomorphism-invariant subgroup, Abelian-quotient-pullbackable automorphism-invariant subgroup, Center-fixing automorphism-balanced subgroup, Center-fixing automorphism-invariant subgroup, Central factor, Central subgroup, Characteristic subgroup, Cofactorial automorphism-invariant subgroup, Conjugacy-closed normal subgroup, Direct projection-invariant subgroup, Fully invariant subgroup, Hereditarily normal subgroup, IA-automorphism-balanced subgroup, IA-automorphism-invariant subgroup, Injective endomorphism-invariant subgroup, Locally inner automorphism-balanced subgroup, Normal subgroup, Normal subgroup whose center is contained in the center of the whole group, Normal-extensible automorphism-invariant subgroup, Normality-preserving endomorphism-balanced subgroup, Projectivity-invariant subgroup, Quasiautomorphism-invariant subgroup, Retraction-invariant subgroup, SCAB-subgroup, Strictly characteristic subgroup, Surjective endomorphism-balanced subgroup, Transitively normal subgroup, Unipotent automorphism-invariant subgroup