2-subnormal subgroup | | Normal subgroup Normal closure Normalizer of a subgroup Conjugate subgroups Normal core of normalizer Normal core Subnormal subgroup Subnormal depth Commutator of two subgroups |
3-subnormal subgroup | | Normal subgroup 2-subnormal subgroup |
4-subnormal subgroup | | Subnormal subgroup Subnormal depth 2-subnormal subgroup Normal subgroup 3-subnormal subgroup |
Abelian group | | Center Derived subgroup Normal subgroup Diagonal-in-square operator |
Abelian normal subgroup | | Normal subgroup Abelian group |
Abelian normal subgroup of maximum rank | | Abelian normal subgroup Abelian group Normal subgroup Rank of a p-group Normal rank of a p-group |
Almost normal subgroup | | Normal subgroup Subgroup of finite index Normalizer of a subgroup Index of a subgroup Conjugate subgroups |
Almost subnormal subgroup | | Normal subgroup Subgroup of finite index |
Automorph-conjugate subgroup | | Automorphic subgroups Conjugate subgroups Normal subgroup Normalizer of a subgroup |
Automorphism-faithful normal subgroup | | Automorphism-faithful subgroup Normal subgroup |
C-closed normal subgroup | | C-closed subgroup Normal 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 | Inner automorphism Centralizer Normal subgroup Quotient group maps to outer automorphism group of normal subgroup Internal central product Image-potentially operator Direct factor Upper join-closure operator Cocentral subgroup |
Central factor of normal subgroup | | Central factor Normal subgroup Normal closure |
Centralizer-free normal subgroup | | Centralizer-free subgroup Normal subgroup |
Class two normal subgroup | | Normal subgroup Group of nilpotency class two |
Closed normal subgroup | | Closed subgroup of semitopological group Normal subgroup |
Commutator of a normal subgroup and a subset | | Normal subgroup Commutator of two subsets Commutator |
Commutator-in-center subgroup | | Normal subgroup Commutator-in-centralizer subgroup |
Complete direct factor | | Direct factor Complete group Normal subgroup Complemented normal subgroup |
Complete group | | Normal subgroup Direct factor |
Completely divisibility-closed normal subgroup | | Completely divisibility-closed subgroup Normal subgroup |
Conjugacy-closed normal subgroup | | Conjugacy-closed subgroup Normal subgroup |
Coprime automorphism-faithful normal subgroup | | Coprime automorphism-faithful subgroup Normal subgroup |
Coprime automorphism-invariant normal subgroup | | Coprime automorphism-invariant subgroup Normal subgroup |
Coprime automorphism-invariant normal subgroup of group of prime power order | | Coprime automorphism-invariant normal subgroup Group of prime power order Normal subgroup Normal subgroup of group of prime power order Coprime automorphism-invariant subgroup Coprime automorphism-invariant subgroup of group of prime power order |
Cyclic normal subgroup | | Normal subgroup Cyclic group |
Dedekind group | every subgroup is normal every cyclic subgroup is normal normal closure of element is cyclic | Normal subgroup Inner automorphism Power automorphism Normal closure Levi operator Cyclic group Hamiltonian operator |
Dedekind normal subgroup | | Normal subgroup Dedekind group |
Direct factor | factor in internal direct product normal with normal complement has centralizing complement | Internal direct product Normal subgroup Normal complement Retract |
Direct factor of normal subgroup | | Direct factor Normal subgroup Normal closure |
Divisible normal subgroup | | Normal subgroup Divisible group |
Elementary abelian normal subgroup | | Normal subgroup Elementary abelian group |
Finite normal subgroup | | Normal subgroup Finite group |
Finitely generated normal subgroup | | Normal subgroup Finitely generated group |
Group in which every automorphism is inner | | Normal subgroup Central factor Automorphism of a group Inner automorphism |
Group in which every nontrivial normal subgroup has finite index | | Normal subgroup Subgroup of finite index |
Group in which every normal subgroup is a central factor | | Normal subgroup Central factor |
Group in which every normal subgroup is a direct factor | | Normal subgroup Direct factor |
Group in which every normal subgroup is characteristic | | Normal subgroup Characteristic subgroup Transitively normal subgroup Automorphism of a group Normal automorphism |
Group in which every normal subgroup is fully invariant | | Normal subgroup Fully invariant subgroup |
Group in which every normal subgroup is powering-invariant | | Normal subgroup Powering-invariant subgroup |
Group in which every permutable subgroup is normal | | Permutable subgroup Normal subgroup |
Group in which every pronormal subgroup is normal | | Pronormal subgroup Normal subgroup |
Group in which every retract is a direct factor | | Normal subgroup Retract Direct factor |
Group in which every subgroup is pronormal | | Pronormal subgroup Normal subgroup Hamiltonian operator |
Group whose automorphism group is abelian | abelian automorphism group any two automorphisms commute | Automorphism group of a group Abelian group Normal subgroup Normal subgroup contained in centralizer of derived subgroup |
Group whose inner automorphism group is central in automorphism group | | Inner automorphism group Central subgroup Automorphism group of a group Inner automorphism Central automorphism Normal subgroup Commutator-in-center subgroup |
Groups embeddable as normal subgroups in a finite group with a common complement | | Normal subgroup Permutable complements Normal complement |
Hall subgroup of normal subgroup | | Hall subgroup Normal subgroup |
Hereditarily normal subgroup | | Transitively normal subgroup Dedekind group Normal subgroup Hereditarily operator |