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?



Groups whose normal subgroups form a chain with respect to inclusion
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.
>  MR[379657](http://www.ams.org/mathscinet-getitem?mr=379657)
>  DOI:[10.2307/1997110](http://dx.doi.org/10.2307/1997110)
> 
> 
> 


You might also be interested in the safari for [zebra groups](https://math.stackexchange.com/questions/14919/zebra-groups-and-counting-stripes).


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


