A few days ago I asked a question (http://mathoverflow.net/questions/158302) 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.