P-normal not implies p-solvable

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) uses::A-group implies p-normal for all p

Example of the alternating group of degree five
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 $$p = 2,3,5$$.