On the subgroup index problem in finite groups

We address the subgroup index problem in a given finite subgroup lattice L = l(G) which is P-indecomposable and determine out of the structure of L the existence in G of a subgroup (D) over tilde invariant for all automorphisms of L, with a cyclic complement R in G and where for any pair X <= Y of subgroups of (D) over tilde the index vertical bar Y : X vertical bar can be computed using only structural properties of L. As a consequence, we show that in such an L all the terms of the Fitting series of G can be determined, as well as an upper bound of the order of G can be computed out of L as long as G has no cyclic Hall direct factor.