Finite Group DatabasebetaFinite Group DatabasebetaOrder 20032.
238 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 95 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×D313 |
| commutator, socle, cosocle | C22×C313 | C24 |
| Fitting | #126×C313 | C2 |
| hypercenter | #126 | D313 |
| hypocenter | C313 | #553 |
| LCS | G ▷ C22×C313 ▷ C313 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23756492 | C2 | semidirect |
| 2. #23756487 | C2 | semidirect |
| 3. #23756482 | C2 | semidirect |
| 4. #23756481 | C2 | semidirect |
| 5. #23756472 | C2 | semidirect |
| 6. #23756470 | C2 | semidirect |
| 7. #23756470 | C2 | semidirect |
| 8. #23756470 | C2 | semidirect |
| 9. C4×D1252 | C2 | semidirect |
| 10. #23756459 | C2 | semidirect |
| 11. #23756457 | C2 | semidirect |
| 12. #23756457 | C2 | semidirect |
| 13. #23756454 | C2 | semidirect |
| 14. C4⋊C4×D313 | C2 | semidirect |
| 15. #126×C313 | C2 | semidirect |
| 109. C313 | #553 | semidirect |
| 112. #126 | D313 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23756492 | 2 |
| #23756487 | 2 |
| #23756482 | 2 |
| #23756481 | 2 |
| #23756472 | 2 |
| #23756470 | 2 |
| C4×D1252 | 2 |
| #23756459 | 2 |
| #23756457 | 2 |
| #23756454 | 2 |
| C4⋊C4×D313 | 2 |
| #126×C313 | 2 |
| #553 | 313 |