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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.

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Yes, these are precisely the finite nilpotent groups.

This is because a finite group is nilpotent if and only if it has a unique - that is, normal - Sylow $p$-subgroup for each prime divisor $p$ of its order. And thus, the finite nilpotent groups are exactly the direct products of groups of prime-power order.

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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.

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