Let $p$ be a prime number and let $G$ be a $p$-group. Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$

Question

Let $p$ be a prime number, and let $G$ be a $p$-group: $|G|=p^r$ . Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$

But are there any normal subgroup of order $p^n$ such that $n>r$ ?

How can we show that, I couldn't show.

Answer 1

No, since Lagrange's Theorem ensures that the order of any subgroup of a group divides the order of the group; $p^n\nmid p^r$ for $n>r$.