The order of a subgroup divides the order of the group. Groups of prime order are cyclic.
For a slightly more descriptive answer, Lagrange's theorem says that for finite groups, a subgroup's order divides that of the group. Since the group's order is a product of two primes, a nontrivial subgroup should have order p, q or 1. Groups of prime order are cyclic, because in particular, the subgroup generated by any non-identity element must be the whole thing, as it would have order dividing a prime number.