Finite Group DatabasebetaFinite Group DatabasebetaOrder 27584.
248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×D431 |
| commutator, socle, cosocle | C22×C431 | C24 |
| Fitting | #117×C431 | C2 |
| prime radical | #28352159 | C2 |
| hypercenter | #117 | D431 |
| hypocenter | C431 | #551 |
| LCS | G ▷ C22×C431 ▷ C431 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28352184 | C2 | semidirect |
| 2. #28352183 | C2 | semidirect |
| 3. #28352177 | C2 | semidirect |
| 4. #28352174 | C2 | semidirect |
| 5. #28352174 | C2 | semidirect |
| 6. #28352172 | C2 | semidirect |
| 7. #28352167 | C2 | semidirect |
| 8. #28352166 | C2 | semidirect |
| 10. C4×D1724 | C2 | semidirect |
| 11. (C22⋊D431)⋊C4 | C2 | semidirect |
| 12. #28352146 | C2 | semidirect |
| 13. C2×(C2×D431)⋊C4 | C2 | semidirect |
| 14. C22⋊C4×D431 | C2 | semidirect |
| 15. #117×C431 | C2 | semidirect |
| 17. C2×Dic862 | C22 | semidirect |
| 24. C2×C4×D431 | C22 | semidirect |
| 25. C4⋊Dic431 | C22 | semidirect |
| 26. Dic431⋊C4 | C22 | semidirect |
| 27. Dic431⋊C4 | C22 | semidirect |
| 28. Dic431⋊C4 | C22 | semidirect |
| 29. Dic431⋊C4 | C22 | semidirect |
| 30. Dic431⋊C4 | C22 | semidirect |
| 40. (C2×D431)⋊C4 | C22 | semidirect |
| 41. (C2×D431)⋊C4 | C22 | semidirect |
| 42. (C2×D431)⋊C4 | C22 | semidirect |
| 43. C4×Dic431 | C22 | semidirect |
| 45. C4⋊C4×C431 | C22 | semidirect |
| 46. C4⋊C4×C431 | C22 | semidirect |
| 49. C42×C431 | C22 | semidirect |
| 50. C42×C431 | C22 | semidirect |
| 110. C431 | #551 | semidirect |
| 124. #117 | D431 | semidirect |
| 143. C4⋊C4 | C2×D431 | semidirect |
| 144. C4⋊C4 | C2×D431 | semidirect |
| 160. C42 | C2×D431 | semidirect |
| 161. C42 | C2×D431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28352184 | 2 |
| #28352183 | 2 |
| #28352177 | 2 |
| #28352174 | 2 |
| #28352172 | 2 |
| #28352167 | 2 |
| #28352166 | 2 |
| #28352159 | 2 |
| C4×D1724 | 2 |
| (C22⋊D431)⋊C4 | 2 |
| #28352146 | 2 |
| C2×(C2×D431)⋊C4 | 2 |
| C22⋊C4×D431 | 2 |
| #117×C431 | 2 |
| #551 | 431 |