TITLE: Is there a name for a group having a normal subgroup for every divisor of the order?
QUESTION [6 upvotes]: Suppose, G is a group of order $n$. 

Is there a name (or an easy criterion) for the property that for every divisor $d|n$, there is a normal subgroup of order $d$ ?

The abelian groups and the p-groups have this property, but other groups satisfy the property as well. The dihedral groups (excluding the 2-groups) do not have this property.

REPLY [2 votes]: If you drop the condition on the normality of the subgroups, then this class of groups is called CLT groups (satifiying the Converse of Lagrange's Theorem). There is a classic paper of Henry Bray that provides the basic properties of these groups. We have the following proper inclusions of classes $$\{Nilpotent \text{ } Groups\} \subsetneq \{Supersolvable \text{ } Groups\} \subsetneq\{CLT \text{ } Groups\} \subsetneq \{Solvable \text{ } Groups\}.$$
CLT groups are neither subgroup- nor quotient-closed, but a finite direct product of CLT groups is again CLT.