As $PSL(2,7)$ has a subgroup of index $7$, and $PSL(2,7)$ is simple, hence it can be embedded in $A_7$. How many copies of $PSL(2,7)$ are in $A_7$?
(There should be at-least 15 copies: If $H\leq A_7$, $H\cong PSL(2,7)$, then $|A_7\colon H|=15$. If we count number of distinct conjugates of $H$ in $A_7$ then we get (some) copies of $PSL(2,7)$ in $A_7$, and this is equal to the index of normalizer of $H$ in $A_7$. As $A_7$ is simple, there must be more than one copies of $PSL(2,7)$ in $A_7$. If $|A_7\colon H|\in \{3,5\}$ then we will have injective homomorphism from $A_7$ into $S_3$ or $S_5$, contradiction. Hence $|N_{A_7}(H)\colon H|=15$, so we have at-least 15 copies of $PSL(2,7)$ in $A_7$.)