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Normalizer of subgroup of p-group

Here's the whole question:

H is a proper subgroup of p-group G. Show that normalizer of H, N(H) is strictly larger than H, and that H is contained in a normal subgroup index p.

With some help I already managed to prove that N(H) is strictly larger than H. But I'm having some trouble with the later half.

I'm thinking we could do something with the quotient group G/N(H) and its order, but then it wouldn't cover the case where H is normal and N(H) is all of G.

Any help appreciated.

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