# Having subgroups of all orders dividing the group order is not quotient-closed

This article gives the statement, and possibly proof, of a group property (i.e., group having subgroups of all orders dividing the group order) not satisfying a group metaproperty (i.e., quotient-closed group property).
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## Statement

It is possible to have a finite group $G$ and a normal subgroup $H$ such that:

• $G$ satisfies the property that it has subgroups of every order dividing the order of $G$.
• The quotient group $G/H$ does not have this property, i.e., there exists a positive divisor of the order of $G/H$ such that $G/H$ has no subgroup of that order.

## Proof

We can take $G$ to be the direct product of A4 and Z2 and $H$ to be the direct factor subgroup cyclic group:Z2.