Ok. This is what i think the solution should be an d i think i have made a mistake in the last part. Just see whats wrong and recitfy.
We have . Clearly it has unique 5-sylow subgroup and hence its normal in Now if we work out the number of 3 sylow subgroups one can see that its either 1 or 25 such subgroups are possible. If its one then your question is trivial since and since they are normal so .
Now if the number of 3-sylow subgroups are 25 then by a lemma given in hersteins "topics in algebra book" we see that cannot be simple as . But i am unable to deduce this part.