Let G be a finite group. In general case, given two normal subgroups N and K of G, we need not to have N < K or K< N. The easiest example is the Klein 4group 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 such a property, then the normal subgroups are necessarily characteristic. Also we may find out that the center of G must be cyclic. Indeed, every abelian group with this property must be a cyclic pgroup (and vice versa). This also shows that G/G' is cyclic, for, the property in quotient hereditary. Please let me know, if these groups are studied before.
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For solvable groups without Frattini chief factors, this is equivalent to each of the following (individually):
This is shown in:
You might also be interested in the safari for zebra groups. However, there are solvable groups with Frattini factors whose normal subgroups form a chain: SL(2,3) for example. 

