Let G be a finite group, p a prime number and \(H\trianglelefteq G\). The author defines \(m_ p(G-H)\) as the least integer n for which every maximal chain of subgroups whose orders are multiples of p and which do not contain H (a chain consisting of n such subgroups) such a chain necessarily contains a proper modular subgroup of G. If there is not any such subgroup, then \(m_ p(G-H)=1\). If the word ”modular” in this definition is substituted by ”p-subnormal” then it defines a function \(h_ p(G-H)\). The generalized p-length \(L_ p(G)\) of an arbitrary finite group G was introduced and studied by L. A. Shemetkov [in Mat. Sb., Nov. Ser. 72(114), 97-107 (1967; Zbl 0179.324)] (for a p- solvable group G \(L_ p(G)\) coincides with the usual p-length \(l_ p(G)\) of Hall and Higman). Theorem 1 states that always \(L_ p(H)\leq m_ p(G-H)\) and theorem 2 that \(L_ p(H)\leq h_ p(G-H)\).