# Category:Subgroup properties

From Groupprops

This is a category of properties for the following kind of objects: subgroups. In other words, this category lists subgroup properties

View a complete list of property categories

Important group properties, sorted by importance rank:

Extremely important (importance rank one):## Related categories

### Subcategory for pivotal elements

To see only the more important, or pivotal, among subgroup properties, refer Category:Pivotal subgroup properties.

### Properties/metaproperties

Related categories include:

- Category:Properties of subsets of groups
- Category:Subgroup pair properties
- Category:Properties of subgroup series
- Category:Properties of subgroup families
- Category:Subgroup metaproperties

### Testing/computational problems

- Category:Subgroup property implications is a list of articles, each article describing an implication between two subgroup properties. For instance, characteristic implies normal meaning that every characteristic subgroup is a normal subgroup.
- Category:Subgroup property non-implications is a list of articles, each article describing why a particular subgroup property does
*not*imply another one. Usually, the proof is constructive, giving an example of a subgroup satisfying the second property but not the first. For instance, normal not implies characteristic - Category:Subgroup metaproperty satisfactions is a list of articles, each article describing how/why a given subgroup property satisfies a given subgroup metaproperty.
- Category:Subgroup metaproperty dissatisfactions is a list of articles, each article describing how/why a given subgroup property does
*not*satisfy a given subgroup metaproperty.

## Notations page

The page giving notations for important subgroup properties is:

Notation for subgroup properties

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### C

## Pages in category "Subgroup properties"

The following 200 pages are in this category, out of 729 total. The count *includes* redirect pages that have been included in the category. Redirect pages are shown in italics.

### A

- Abelian characteristic subgroup
- Abelian conjugacy-closed subgroup
- Abelian critical subgroup
- Abelian direct factor
- Abelian fully invariant subgroup
- Abelian hereditarily normal subgroup
- Abelian marginal subgroup
- Abelian normal subgroup
- Abelian normal subgroup of group of prime power order
- Abelian normal subgroup of maximum order
- Abelian pronormal subgroup
- Abelian subgroup
- Abelian subgroup of maximum order
- Abelian subgroup of maximum order which is normal
- Abelian subgroup of maximum rank
- Abelian subnormal subgroup
- Abelian Sylow subgroup
- Abelian verbal subgroup
- Abelian-completed subgroup
- Abelian-extensible automorphism-invariant subgroup
- Abelian-extensible endomorphism-invariant subgroup
- Abelian-potentially characteristic subgroup
- Abelian-potentially verbal subgroup
- Abelian-quotient subgroup
- Abelian-quotient-pullbackable automorphism-invariant subgroup
- Abnormal subgroup
- Action-isomorph-free subgroup
- AEP-subgroup
- Algebraic subgroup
- Almost normal subgroup
- Almost subnormal subgroup
- Amalgam-characteristic subgroup
- Amalgam-normal-subhomomorph-containing subgroup
- Amalgam-strictly characteristic subgroup
- Amalgamated free factor
- Ascendant subgroup
- Asymptotically fixed-depth join-transitively subnormal subgroup
- Autoclinism-invariant subgroup
- Automorph-commensurable subgroup
- Automorph-conjugate subgroup
- Automorph-dominating subgroup
- Automorph-join-closed subnormal subgroup
- Automorph-permutable subgroup
- Automorphically endomorph-dominating subgroup
- Automorphism-faithful AEP-subgroup
- Automorphism-faithful normal subgroup
- Automorphism-faithful subgroup

### B

### C

- C-closed normal subgroup
- C-closed self-centralizing subgroup
- C-closed subgroup
- C-normal subgroup
- C-subnormal subgroup
- Carter subgroup
- CDIN-subgroup
- Center of pronormal subgroup
- Center-fixing automorphism-balanced subgroup
- Center-fixing automorphism-invariant subgroup
- Central factor
- Central factor of normal subgroup
- Central factor of normalizer
- Central subgroup
- Central subgroup of finite group
- Central subgroup of normalizer
- Centralizer of involution
- Centralizer-annihilating endomorphism-invariant subgroup
- Centralizer-dense subgroup
- Centralizer-free normal subgroup
- Centralizer-free subgroup
- Centralizer-large subgroup
- Centrally closed subgroup
- Centrally large subgroup
- Characteristic AEP-subgroup
- Characteristic central factor
- Characteristic central subgroup
- Characteristic core of subgroup of finite index
- Characteristic direct factor
- Characteristic direct factor of abelian group
- Characteristic direct factor of nilpotent group
- Characteristic maximal subgroup of group of prime power order
- Characteristic subgroup
- Characteristic subgroup of abelian group
- Characteristic subgroup of center
- Characteristic subgroup of direct factor
- Characteristic subgroup of finite group
- Characteristic subgroup of finite index
- Characteristic subgroup of group of prime power order
- Characteristic subgroup of nilpotent group
- Characteristic subgroup of pronormal subgroup
- Characteristic subgroup of solvable group
- Characteristic subgroup of Sylow subgroup
- Characteristic transitively normal subgroup
- Characteristic-isomorph-free subgroup
- Characteristic-potentially characteristic subgroup
- Characteristically complemented characteristic subgroup
- Characteristically complemented normal subgroup
- Characteristically complemented subgroup
- Characteristically DP-decomposable subgroup
- Class two characteristic subgroup
- Class two normal subgroup
- Closure-characteristic subgroup
- Cocentral subgroup
- Cocentral subgroup of normalizer
- Cofactorial automorphism-invariant subgroup
- Commutator of a normal subgroup and a subset
- Commutator-in-center subgroup
- Commutator-in-centralizer subgroup
- Commutator-verbal subgroup
- Complemented central factor
- Complemented characteristic subgroup
- Complemented fully invariant subgroup
- Complemented homomorph-containing subgroup
- Complemented isomorph-containing subgroup
- Complemented normal subgroup
- Complemented transitively normal subgroup
- Complete characteristic subgroup
- Complete direct factor
- Completely divisibility-closed normal subgroup
- Completely divisibility-closed subgroup
- Component
- Composition subgroup
- Compressed subgroup
- Conjugacy-closed characteristic subgroup
- Conjugacy-closed Hall subgroup
- Conjugacy-closed normal subgroup
- Conjugacy-closed subgroup
- Conjugate-commensurable subgroup
- Conjugate-comparable subgroup
- Conjugate-dense subgroup
- Conjugate-join-closed subnormal subgroup
- Conjugate-large subgroup
- Conjugate-permutable subgroup
- Conjugation-invariantly permutably complemented subgroup
- Contracharacteristic subgroup
- Contranormal subgroup
- Coprime automorphism-faithful characteristic subgroup
- Coprime automorphism-faithful normal subgroup
- Coprime automorphism-faithful subgroup
- Coprime automorphism-invariant normal subgroup
- Coprime automorphism-invariant normal subgroup of group of prime power order
- Coprime automorphism-invariant subgroup
- Coprime automorphism-invariant subgroup of group of prime power order
- Core-characteristic subgroup
- Core-free 2-subnormal subgroup
- Core-free permutable subgroup
- Core-free subgroup
- Core-free subnormal subgroup
- Corollary of Timmesfeld's replacement theorem for abelian subgroups
- Critical subgroup
- Cutting subgroup
- Cyclic characteristic subgroup
- Cyclic isomorph-containing subgroup
- Cyclic normal subgroup
- Cyclic normal subgroup of finite group

### D

- Dedekind normal subgroup
- Descendant subgroup
- Diagonal subgroup of a direct power
- Direct factor
- Direct factor of characteristic subgroup
- Direct factor of complemented normal subgroup
- Direct factor of fully characteristic subgroup
- Direct factor of normal subgroup
- Direct factor over central subgroup
- Direct power-closed characteristic subgroup
- Direct product of subgroups of factors in direct product decomposition into cyclic groups
- Direct projection-invariant subgroup
- Distinguished subgroup
- Divisibility-closed subgroup
- Divisibility-closed subgroup of abelian group
- Divisibility-closed subgroup of nilpotent group
- Divisible normal subgroup
- Double coset-ordering subgroup
- Double coset-separated subgroup
- DRC-subgroup

### E

- EEP-subgroup
- Elementarily characteristic subgroup
- Elementary abelian normal subgroup
- Elementary abelian subgroup of maximum rank
- Elliptic subgroup
- Endoclinism-invariant subgroup
- Endomorph-dominating subgroup
- Endomorphism image
- Endomorphism kernel
- Epimarginal subgroup
- Existence-bound-word subgroup
- Existentially bound-word subgroup