# Nilpotent subnormal subgroup

This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): nilpotent group
View a complete list of such conjunctions

## Definition

A subgroup of a group is termed a nilpotent subnormal subgroup if it is nilpotent as a group and subnormal as a subgroup.

## Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Group partSubgroup partQuotient part
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Non-normal subgroups of M16M16Cyclic group:Z2
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian subnormal subgroup the subgroup is an abelian group and is a subnormal subgroup of the whole group |FULL LIST, MORE INFO
abelian normal subgroup the subgroup is an abelian group and is a normal subgroup of the whole group Abelian subnormal subgroup|FULL LIST, MORE INFO
abelian characteristic subgroup the subgroup is an abelian group and is a characteristic subgroup of the whole group Abelian subnormal subgroup|FULL LIST, MORE INFO
nilpotent normal subgroup nilpotent and a normal subgroup |FULL LIST, MORE INFO
hereditarily normal subgroup every subgroup of it is normal |FULL LIST, MORE INFO
nilpotent characteristic subgroup nilpotent and a characteristic subgroup |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subnormal subgroup Hereditarily subnormal subgroup, Right-transitively fixed-depth subnormal subgroup|FULL LIST, MORE INFO