Recognizability of symmetric groups by spectrum

The spectrum of a finite group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group whose spectrum coincides with the spectrum of $G$ is isomorphic to $G$. It is proved the symmetric group $S_n$ is recognizable by spectrum for $n\not\in\{2,3,4,5,6,8,10,15,16,18,21,27,33,35,39,45\}$.