Number of conjugacy classes in a subgroup of finite index is bounded by index times number of conjugacy classes in the whole group

Statement in terms of number of conjugacy classes
Suppose $$G$$ is a group and $$H$$ is a subgroup of finite index. Suppose the number of conjugacy classes in $$G$$ is a finite number $$c$$ and the index of $$H$$ in $$G$$ is $$d$$. Then, the number of conjugacy classes in $$H$$ is at most $$cd$$.

Statement in terms of commuting fraction
Suppose $$G$$ is a finite group and $$H$$ is a subgroup of finite index. Suppose the commuting fraction of $$G$$ is a finite number $$\mu$$ and the index of $$H$$ in $$G$$ is $$d$$. Then, the commuting fraction of $$H$$ is at most $$d^2\mu$$.

Related facts

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