| 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 |
