## Statement

It is possible to have a group , a lattice automorphism of the lattice of subgroups, and subgroups of with , such that:

### Property-theoretic implication

No subgroup property of any of the following kinds is a conditionally lattice-determined subgroup property:

[SHOW MORE]
This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about subnormal subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about normal subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about characteristic subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about fully invariant subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal Hall subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about normal Hall subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal Sylow subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about normal Sylow subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., complemented normal subgroup) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about complemented normal subgroup|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow retract) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about Sylow retract|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall retract) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about Hall retract|Get more facts about conditionally lattice-determined subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., retract) **not** satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about retract|Get more facts about conditionally lattice-determined subgroup property|

## Related facts

## Proof

### General example

Choose primes such that divides . Let be the semidirect product of the group of order by the subgroup of order in its automorphism group. is a group of order . Its lattice has size , including the trivial subgroup, whole group, and intermediate mutually incomparable subgroups, one of order and of order .

Let be the subgroup of order and be any subgroup of order . The map from to itself interchanging and is a lattice automorphism, and it interchanges the two subgroups. Also, and satisfy the stated conditions, completing the proof.

### Smallest example

The smallest example is obtained by setting , giving:

For more on the subgroup structure, see subgroup structure of symmetric group:S3.

The lattice of subgroups is also shown below: