TITLE: What are the solvable subgroups of $S_n$?
QUESTION [7 upvotes]: What are the solvable subgroup of $S_n$?
I know that when $n \geq 5$, both $S_n$ and $A_n$ are not solvable. But, how "large" can a solvable permutation group be when $n$ is given. 
Many thanks~

REPLY [6 votes]: If $n = p$ is prime, then any transitive solvable subgroup of $S_p$ is contained 
in the group of affine transformations $x \mapsto ax + b$ (here $a \in \mathbb F_p^{\times}$ and $b \in \mathbb F_p$), thought of as a group of transformations from $\mathbb F_p$ (a set of order $p$) to itself.  (This result goes back to Galois himself, and was one of the motivations for his invention of finite fields.)