The paper An Elementary Proof of the Simplicity of the Mathieu Groups $M_{11}$ and $M_{23}$ by Robin J. Chapman proves the following theorem: Let $G$ be a transitive subgroup of $S_p$, and suppose $|G| = pmr$ where $m > 1$, $m \equiv 1 \pmod p$, $r < p$, and $r$ is prime. Then $G$ is simple.
As stated in the papers name, this allows one to immediately deduce the simplicity of the Mathieu groups $M_{11}$ and $M_{23}$ from only their order and basic facts about their action on $11$ and $23$ points; I found the simplicity of the argument quite refreshing.
I tried applying this lemma to other groups, starting with the alternating groups $A_n$; amusingly, using Wilson's theorem, it's not difficult to show this theorem allows you to conclude $A_p$ is simple for $p$ a safe prime, that is, a prime $p$ such that $(p-1)/2$ is also prime. I also know that no sporadic groups besides $M_{11}$ and $M_{23}$ can be proved sporadic via this method, as no others act on a small enough prime number of points. One can also prove $PSL_2(11)$ simple using its action on $11$ points (not a typo!).
My questions is: which other finite groups can be proved simple using this theorem? I'm not very familiar with most of the families of finite simple groups, but am still curious as to the applicability of this lemma.