Group has Subgroups of All Prime Power Factors

Theorem
Let $$G$$ be a finite group of order $$n$$ such that $$p \backslash n$$.

If $$p^k \backslash n$$ then $$G$$ has at least one subgroup order $$p^k$$.

Proof
From Composition Series, a $p$-group has subgroups corresponding to every divisor of its order.

Thus, taken with the First Sylow Theorem, a finite group has a subgroup corresponding to every prime power divisor of its order.