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Characteristic subgroup, Complemented normal subgroup, Direct factor, Endomorphism kernel, Finite direct power-closed characteristic subgroup, Fully invariant subgroup, Homomorph-containing subgroup, Injective endomorphism-quotient-balanced subgroup, Intermediately characteristic subgroup, Intermediately endomorphism kernel, Periodic normal subgroup, Purely definable subgroup, Quotient-powering-invariant subgroup, Strictly characteristic subgroup, Verbal subgroup