Assume $G$ is a finite group and $H$ a subgroup. Is it true that the number of irreducible representations of $G$ is always larger than (or equal to) the number of irreducible representations of $H$?



Is the number of irreducible representations of G always bigger (or equal) to the number of irreducible representations of H, with H a subgroup of G?
Here is a simple example. As mentionned by Gjergji, this is a question about the number of conjugacy classes. Take $G=\frak A\_4$, which has $3$ classes (the identity, the double transpositions and the $4$-cycles). Now take $H$ the subgroup spanned by a $4$-cycle. Because $|H|=4$ and $H$ is abelian, it has $4$ classes.


**Edit**. This is incorrect: $H$ is not a subgroup of $\frak A\_4$, because a $4$-cycle is an odd permutation. I apologize. Note that I an accepted answer can not be deleted.


