P-normal not implies p-solvable

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., p-normal group) need not satisfy the second subgroup property (i.e., p-solvable group)
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Statement

We can have a finite group $G$ and a prime number $p$ such that $G$ is a p-normal group but is not a p-solvable group.

Facts used

1. A-group implies p-normal for all p

Proof

Example of the alternating group of degree five

Further information: alternating group:A5

The alternating group of degree five is a $p$-normal group for all primes $p$, because all its Sylow subgroups are abelian. On the other hand, since the group is simple non-abelian, it is not $p$-solvable for $p=2,3,5$.