Suppose $G$ is a group of order $n$ . Let $p$ and $q$ be distinct primes which divide $n$. Can we say that $G$ has a subgroup of order $p\cdot q$?

Question

Suppose $G$ is a group of order $n$ . Let $p$ and $q$ be distinct primes both of which divide $n$. Can we say that $G$ has a subgroup of order $p\cdot q$?

Answer 1

No. For example $A_5$ is a group of order $60=2^2\times3\times5$ but if you know this group is simple then you can easily check that it can't have any subgroups of order $15$.