Let $G$ be a subgroup of $S_n$. What can be said on the number of conjugacy classes of $G$, in terms of $n$?

I will review many results from the literature and give examples. I will then present an upper bound for the case where $G$ is primitive with nonabelian socle. This states that either $G$ belongs to explicit families of examples, or the number of conjugacy classes is smaller than $n/2$, and in fact, it is $o(n)$. I will finish with a few questions. Joint work with Nick Gill.