1-automorphism-invariant subgroup | |

AEP-subgroup | |

Abelian subgroup | |

Ascendant subgroup | |

Automorph-conjugate subgroup | |

Base of a wreath product | |

C-closed subgroup | |

CEP-subgroup | |

Central factor | factor in central product product with centralizer is whole group quotient action by outer automorphisms is trivial every inner automorphism restricts to an inner automorphism |

Central subgroup | |

Centrally closed subgroup | |

Characteristic central factor | |

Characteristic direct factor | |

Characteristic subgroup | invariant under all automorphisms automorphism-invariant strongly normal normal under outer automorphisms |

Characteristic subgroup of group of prime power order | |

Characteristic transitively normal subgroup | |

Characteristically complemented characteristic subgroup | |

Characteristically complemented normal subgroup | |

Characteristically complemented subgroup | |

Cocentral subgroup | |

Commutator-verbal subgroup | |

Completely distinguished subgroup | |

Completely divisibility-closed subgroup | |

Component | |

Composition subgroup | |

Conjugacy-closed characteristic subgroup | |

Conjugacy-closed normal subgroup | |

Conjugacy-closed subgroup | |

Conjugate-dense subgroup | |

Conjugate-large subgroup | |

Contranormal subgroup | |

Cyclonormal subgroup | |

Descendant subgroup | |

Direct factor | factor in internal direct product normal with normal complement has centralizing complement |

Divisibility-closed subgroup | |

Finite direct power-closed characteristic subgroup | |

Free factor | |

Fully invariant subgroup | invariant under all endomorphisms endomorphism-invariant |

Fully normalized subgroup | |

Hall direct factor | |

Hall subgroup | |

Hereditarily normal subgroup | |

IA-automorphism-balanced subgroup | |

Image-closed characteristic subgroup | |

Image-closed fully invariant subgroup | |

Improper subgroup | |

Inert subgroup | |

Injective endomorphism-invariant subgroup | invariant under all injective endomorphisms injective endomorphism-invariant |

Isomorph-containing subgroup | contains all isomorphic subgroups weakly closed in any ambient group |

Isomorph-free subgroup | no other isomorphic subgroups no isomorphic copies only subgroup of its isomorphism type |