Finite Group DatabasebetaFinite Group DatabasebetaOrder 26944.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D421 |
| commutator, socle, cosocle | C22×C421 | C24 |
| Fitting | #123×C421 | C2 |
| prime radical | #27212338 | C2 |
| hypercenter | #123 | D421 |
| hypocenter | C421 | #529 |
| LCS | G ▷ C22×C421 ▷ C421 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27212361 | C2 | semidirect |
| 2. #27212359 | C2 | semidirect |
| 3. #27212356 | C2 | semidirect |
| 4. #27212352 | C2 | semidirect |
| 5. #27212349 | C2 | semidirect |
| 6. #27212343 | C2 | semidirect |
| 8. #27212331 | C2 | semidirect |
| 9. D4×Dic421 | C2 | semidirect |
| 10. #27212326 | C2 | semidirect |
| 11. #27212326 | C2 | semidirect |
| 12. #27212324 | C2 | semidirect |
| 13. #27212322 | C2 | semidirect |
| 14. C2×C4×Dic421 | C2 | semidirect |
| 15. #123×C421 | C2 | semidirect |
| 103. C421 | #529 | semidirect |
| 110. #123 | D421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27212361 | 2 |
| #27212359 | 2 |
| #27212356 | 2 |
| #27212352 | 2 |
| #27212349 | 2 |
| #27212343 | 2 |
| #27212338 | 2 |
| #27212331 | 2 |
| D4×Dic421 | 2 |
| #27212326 | 2 |
| #27212324 | 2 |
| #27212322 | 2 |
| C2×C4×Dic421 | 2 |
| #123×C421 | 2 |
| #529 | 421 |