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 got a lot of useful information about it. Thanks for the nice and very helpful answers. Now I have a question:

Is it possible that we can conclude that any group of order $(p^2+1)/2$, where $p>13$ and $p\ne 239$ 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 enough that the subgroup of that order is normal and abelian. Of course this is not always true.