The index of every proper subgroup of a simple group containing an order 22 element

Question

The complete question is the following:

For $G$ a simple group containing an element of order $22$. Show that every proper subgroup of $G$ has index at least $13$.

I think I am supposed to use Sylow's Theorems to show this is true, but I don't know exactly what to do. Is it easier to show contradiction?

Answer 1

Hint: Show that if $G$ is a simple group with a subgroup of index $n>1$, then $G$ injects into the symmetric group $S_n$.