Let \(G\) be a finite group and \(H\) a subnormal subgroup of \(G\). Suppose the index \([G : N_G (H)]\) of the normalizer of \(H\) is the product of \(r\) primes. The authors’ main result is that the subnormal depth of \(H\) in \(G\) is at most \(r+1\). The proof is basically elementary but does use the following two theorems of Wielandt. Minimal normal subgroups of G normalize every subnormal subgroup of \(G\) and any subgroup of \(G\) generated by two subnormal subgroups of \(G\) is itself subnormal in \(G\). The paper is written in a style suitable for readers with a limited knowledge of group theory.