# Subgroups of $$p$$-groups

Algebra Level pending

Let $$G$$ be a group of order $$p^5,$$ where $$p$$ is prime. Consider the following statements:

I. $$G$$ has a normal subgroup of order $$p.$$
II. $$G$$ has a normal subgroup of order $$p^2.$$
III. $$G$$ has a normal subgroup of order $$p^3.$$
IV. $$G$$ has a normal subgroup of order $$p^4.$$

How many of the statements I-IV is/are always true for any $$G$$ of order $$p^5$$?

Terminology: The order of a finite group is the number of elements in the group.

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