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 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 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 p-group (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.
For solvable groups without Frattini chief factors, this is equivalent to each of the following (individually):
This is shown in:
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However, there are solvable groups with Frattini factors whose normal subgroups form a chain: SL(2,3) for example.