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 |