  The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its
elements. The following sufficient criterion of nonsolvability is proved: if
among the prime divisors of the order of a group $G$, there are four different
primes such that $\omega(G)$ contains all their pairwise products but not a
product of any three of these numbers, then $G$ is nonsolvable. Using this
result, we show that for $q\geqslant 8$ and $q\neq 32$, the direct square
$Sz(q)\times Sz(q)$ of the simple exceptional Suzuki group $Sz(q)$ is uniquely
characterized by its spectrum in the class of finite groups, while for
$Sz(32)\times Sz(32)$, there are exactly four finite groups with the same
spectrum.
