Non-subnormal subgroups of groups

According to Mohres's Theorem an arbitrary group whose proper subgroups are all subnormal (or a group without non-subnormal proper subgroups) is solvable. In this paper we generalize Mohres's Theorem, by proving that every group with at most 56 non-subnormal subgroups is solvable. Also we show that the derived length of a solvable group with a finite number k of non-n-subnormal subgroups is bounded in terms of n and k. (c) 2012 Elsevier B.V. All rights reserved.