On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = `(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound `(m)](\frak F)\overline m(\frak F) in terms of the `(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that `(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.