Group Isomorphism regarding Sylow Subgroups

Question

Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of Syllow subgroups of these are also same then "are $G_1,G_2$ isomorphic"? I have always find this statement as true considering lower order groups but can't proved it. Is it true or there are some counterexamples too! Thanks for reading.

Answer 1

This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.

Answer 2

Groups with Identical Subgroup Lattices in All Powers shows there are many, many examples even when the Sylow subgroups are required to be cyclic.