of the distinct conjugacy classes of maximal abelian subgroups of |G|. This pattern of partitioning the prime divisors of |G| according to conjugacy classes of certain Hall subgroups (a Hall subgroup is one whose order and index are relatively prime) which correspond to the maximal subgroups of G (up to conjugacy) is repeated in both the proof of the Feit–Hall–Thompson CN-theorem and in the proof of the Feit–Thompson odd-order theorem. Each maximal subgroup M has a certain nilpotent Hall subgroup Mσ with normalizer contained in M, whose order is divisible by certain primes forming a set σ(M). Two maximal subgroups are conjugate if and only if the sets σ(M) are the same, and if they are not conjugate then the sets σ(M) are disjoint. Every prime dividing the order of G occurs in some set σ(M). So the primes dividing the order of G are partitioned into equivalence classes corresponding to the conjugacy classes of maximal subgroups. The proof of the CN-case is already considerably more difficult than the CA-case: the main extra problem is to prove that two different Sylow subgroups intersect in the identity. This part of the proof of the odd-order theorem takes over 100 journal pages. A key step is the proof of the Thompson uniqueness theorem, stating that abelian subgroups of normal rank at least 3 are contained in a unique maximal subgroup, which means that the primes p for which the Sylow p-subgroups have normal rank at most 2 need to be considered separately. Bender later simplified the proof of the uniqueness theorem using Bender's method. Whereas in the CN-case, the resulting maximal subgroups M are still Frobenius groups, the maximal subgroups that occur in the proof of the odd-order theorem need no longer have this structure, and the analysis of their structure and interplay produces 5 possible types of maximal subgroups, called types I, II, III, IV, V. Type I subgroups are of "Frobenius type", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups. They have the structure MF⋊U where MF is the largest normal nilpotent Hall subgroup, and U has a subgroup U0 with the same exponent such that MF⋊U0 is a Frobenius group with kernel MF. Types II, III, IV, V are all 3-step groups with structure MF⋊U⋊W1, where MF⋊U is the derived subgroup of M. The subdivision into types