Group in which every nontrivial normal subgroup contains a cyclic normal subgroup

Definition
A group in which every nontrivial normal subgroup contains a cyclic normal subgroup is a group with the property that any nontrivial normal subgroup of the group, contains a cyclic normal subgroup.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Nilpotent group:
 * Weaker than::Group whose center is normality-large:
 * Weaker than::Supersolvable group:

Metaproperties
It can happen that a group $$G$$ satisfies the property that every nontrivial normal subgroup contains a cyclic normal subgroup, but the quotient of $$G$$ doesn't satisfy that property. For instance, if we take $$G$$ to be a quasisimple group that is not simple, then its center is normality-large, so $$G$$ satisfies the property, but $$G/Z(G)$$, being simple, doesn't. In a similar vein, if we take $$G = GL(2,3)$$, then the center of $$G$$ is normality-large, and $$G/Z(G)$$, which is isomorphic to the symmetric group on four elements, does not satisfy the property. (in fact, it has no cyclic normal subgroups).

Facts
If $$G$$ has a subgroup $$H$$ that is normality-large, transitively normal, and satisfies this property (namely, every nontrivial normal subgroup of $$H$$ contains a cyclic normal subgroup), then $$G$$ also satisfies the property. This generalizes the observation that if the center is normality-large, then the group satisfies the property.