The maximal subgroups of S n fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form S k × S n – k for 1 ≤ k < n /2. The imprimitive maximal subgroups are exactly those of the form S k wr S n / k, where 2 ≤ k ≤ n /2 is a proper divisor of n and "wr" denotes the wreath product. The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem and the classification of finite simple groups, (Liebeck, Praeger & Saxl 1988) gave a fairly satisfactory description of the maximal subgroups of this type, according to (Dixon & Mortimer 1996, p. 268).