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 | #125×C421 | C2 |
| prime radical | #27212341 | C2 |
| hypercenter | #125 | D421 |
| hypocenter | C421 | #529 |
| LCS | G ▷ C22×C421 ▷ C421 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27212361 | C2 | semidirect |
| 2. #27212354 | C2 | semidirect |
| 3. #27212351 | C2 | semidirect |
| 4. #27212346 | C2 | semidirect |
| 6. #27212331 | C2 | semidirect |
| 7. #27212328 | C2 | semidirect |
| 8. #27212328 | C2 | semidirect |
| 9. #27212324 | C2 | semidirect |
| 10. C42×D421 | C2 | semidirect |
| 11. #125×C421 | C2 | semidirect |
| 74. C421 | #529 | semidirect |
| 77. #125 | D421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27212361 | 2 |
| #27212354 | 2 |
| #27212351 | 2 |
| #27212346 | 2 |
| #27212341 | 2 |
| #27212331 | 2 |
| #27212328 | 2 |
| #27212324 | 2 |
| C42×D421 | 2 |
| #125×C421 | 2 |
| #529 | 421 |