Suppose that is a finite group, is the set of prime divisors of the order , and is the class of finite groups such that for any proper subgroup of . Groups from the class will be called prime spectrum minimal. Many but not all finite simple groups are prime spectrum minimal. For finite simple groups not from the class , the question whether they are isomorphic to nonabelian composition factors of groups from the class is interesting. We describe some finite simple groups that are not isomorphic to nonabelian composition factors of groups from the class and construct an example of a finite group from that has as its composition factor a finite simple sporadic McLaughlin group not from the class .