A few days ago I asked a question (Groups of order $p(p^2+1)/2$) about a finite group of order $p(p^2+1)/2$ and I get many useful information about it. Thanks for the nice and very helpful answers.

Now I have a question: is it possible we conclude that any group of order $(p^2+1)/2$, where $p>5$ is a prime, has an abelian and normal Sylow subgroup?

For small $p$, i.e., $p<1000$ we can see that most of the time there exists an odd prime $p'$ which is large enought such that the subgroup of that order is normal and abelian. Of course this is not always true.