# Number of conjugacy classes in extension group is bounded by product of number of conjugacy classes in normal subgroup and quotient group

## Statement

### Statement in terms of conjugacy classes

Suppose $G$ is a finite group and $H$ is a normal subgroup of $G$ with quotient group $G/H$. Denote by $c(G),c(H),c(G/H)$ respectively the number of conjugacy classes in $G,H,G/H$ respectively. Then, we have the relation:

$c(G) \le c(H)c(G/H)$

### Statement in terms of conjugacy classes

Suppose $G$ is a finite group and $H$ is a normal subgroup of $G$ with quotient group $G/H$. Denote by $CF(G),CF(H),CF(G/H)$ respectively the commuting fractions of $G,H,G/H$ respectively. Then, we have the relation:

$CF(G) \le CF(H)CF(G/H)$