# Number of conjugacy classes in a subgroup may be more than in the whole group

## Statement

It is possible to have a finite group $G$ and a subgroup $H$ of $G$ such that the Number of conjugacy classes (?) in $H$ is more than in $G$.

## Proof

### Example of the dihedral group of degree five

Further information: dihedral group:D10, cyclic group:Z5

The smallest pair of examples is where the group $G$ is $D_{10}$, the dihedral group of degree five and order ten, and $H$ is the subgroup is cyclic group:Z5. The group $D_{10}$ has four conjugacy classes: one of involutions, one identity element, and two conjugacy classes in the cyclic subgroup of order five. On the other hand, $H$ is an abelian group of order five hence has five conjugacy classes.

### Other dihedral examples

Further information: element structure of dihedral groups

More generally, for odd $n$, the dihedral group of order $2n$ and degree $n$ has $(n + 3)/2$ conjugacy classes, and the cyclic subgroup of order $n$ has $n$ conjugacy classes. For $n \ge 5$, the cyclic subgroup has more conjugacy classes than the whole group.

For even $n$, the dihedral group of order $2n$ has $(n + 6)/2$ conjugacy classes and the cyclic subgroup of order $n$ has $n$ conjugacy classes. For $n \ge 8$, the cyclic subgroup has more conjugacy classes than the whole group. The first example of this is cyclic group:Z8 in dihedral group:D16.