OD-characterization of all finite nonabelian simple groups with orders having prime divisors at most 13

The main purpose of this article is to determine $h_{OD}(M)$ for every finite nonabelian simple group $M$ with order having prime divisors at most 13. This result is an analog of the result by A. V. Vasil'ev [1] about the recognizability of these simple groups by spectrum (the set of element orders). By the available results, we need only consider the groups $L_6(3)$, $U_4(5)$, $G_2(4)$, $L_5(3)$, $S_4(8)$, $U_6(2)$ and $O^+_8(3)$.