TITLE: Groups whose normal subgroups form a chain with respect to inclusion
QUESTION [7 upvotes]: Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume that G has such a property, that is, the normal subgroups of G constitute a chain with respect to inclusion.  For example, simple groups, cyclic groups and symmetric groups satisfy this property. Certainly, if G has that property, then the normal subgroups are necessarily characteristic. Furthermore, the center of G must be cyclic. Indeed, every abelian group with this property must be a cyclic p-group (and vice-versa). This also shows that G/G' is cyclic, for the property is hereditary under quotients. 
I would like to know if these groups have been studied before. If so, can you please provide some references?

REPLY [6 votes]: For solvable groups without Frattini chief factors, this is equivalent to each of the following (individually):

having a unique chief series,
every quotient group having a faithful primitive permutation action,
the upper Fitting series being a chief series
the lower Fitting series being a chief series

This is shown in:

Hawkes, Trevor O.
  "Two applications of twisted wreath products to finite soluble groups."
  Trans. Amer. Math. Soc. 214 (1975), 325–335.
  MR379657
  DOI:10.2307/1997110

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However, there are solvable groups with Frattini factors whose normal subgroups form a chain: SL(2,3) for example.