Module-theoretic characterizations of a general radical class and a special radical class are known [see F. A. Szász, Radicals of rings (1981; Zbl 0461.16009)]. The authors, by using two methods in the same way, characterize supernilpotent radical classes defined, in the first by the semiprime module property, a weakly special class of modules and, in the other, a class of modules of another kind.