# GAP:ConjugacyClassesSubgroups

This article is about a GAP function.

This GAP function takes as input a group. See more functions like this.

## Definition

### Function type

`ConjugacyClassesSubgroups` is a GAP command that takes as input a group and outputs a list of groups.

### Behavior

The command inputs a group and outputs a list of conjugacy classes of subgroups *of* that group. Each member of the list is thus a list of subgroups, all conjugate to each other and form a full conjugacy class.

## =Typical use

The typical use of the function is in the form:

ConjugacyClassesSubgroups(group);

(for more, see the #Examples of usage section).

## Related functions

- GAP:Subgroups lists all the subgroups of a group.
- GAP:NormalSubgroups lists only the normal subgroups of the group.
- GAP:CharacteristicSubgroups lists only the characteristic subgroups of the group.
- GAP:ConjugateSubgroups takes as input a group and a subgroup and outputs the list of subgroups conjugate to that subgroup in the whole group.

## Examples of usage

Here is an example showing the listing of the conjugacy classes of subgroups of a group constructed using the SmallGroup command.

gap> ConjugacyClassesSubgroups(SmallGroup(32,15)); [ Group( <identity> of ... )^G, Group( [ f5 ] )^G, Group( [ f3*f4 ] )^G, Group( [ f5, f4 ] )^G, Group( [ f5, f3 ] )^G, Group( [ f5, f3*f4 ] )^G, Group( [ f5, f3, f4 ] )^G, Group( [ f5, f2*f4, f3 ] )^G, Group( [ f5, f1, f4 ] )^G, Group( [ f5, f1*f2, f4 ] )^G, Group( [ f5, f2, f3 ] )^G, Group( [ f5, f3, f4, f1 ] )^G, Group( [ f5, f3, f4, f1*f2 ] )^G, Group( [ f5, f3, f4, f2 ] )^G, Group( [ f5, f3, f4, f1, f2 ] )^G ]

Here is another example that combines the command with the List and Representative commands to output a list of subgroups with one representative of each conjugacy class of subgroups.

gap> List(ConjugacyClassesSubgroups(SymmetricGroup(4)), Representative); [ Group(()), Group([ (1,3)(2,4) ]), Group([ (3,4) ]), Group([ (2,4,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,2)(3,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ]), Group([ (3,4), (2,4,3) ]), Group([ (1,3)(2,4), (1,4)(2,3), (1,2) ]), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ]