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

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., group having subgroups of all orders dividing the group order)notsatisfying a group metaproperty (i.e., quotient-closed group property).

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## Statement

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

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

## Related facts

- Every finite solvable group is a subgroup of a finite group having subgroups of all orders dividing the group order: The proof of this uses an external direct product construction that can be used to construct an infinitude of examples, for any isomorphism class of quotient group that is finite and solvable and fails to have the property.

## Proof

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