Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×D271 |
| commutator, socle, cosocle | C22×C271 | C24 |
| Fitting | #123×C271 | C2 |
| hypercenter | #123 | D271 |
| hypocenter | C271 | #546 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16864654 | C2 | semidirect |
| 2. C2×D4×D271 | C2 | semidirect |
| 3. #22576146 | C2 | semidirect |
| 4. #22576145 | C2 | semidirect |
| 5. #22576138 | C2 | semidirect |
| 6. #22576135 | C2 | semidirect |
| 7. #22576133 | C2 | semidirect |
| 8. #22576132 | C2 | semidirect |
| 9. #22576130 | C2 | semidirect |
| 10. #22576130 | C2 | semidirect |
| 11. #22576122 | C2 | semidirect |
| 12. (C22⋊D271)⋊C4 | C2 | semidirect |
| 13. C2×(C2×D271)⋊C4 | C2 | semidirect |
| 14. C22⋊C4×D271 | C2 | semidirect |
| 15. #123×C271 | C2 | semidirect |
| 19. C2×Dic542 | C22 | semidirect |
| 22. C2×D1084 | C22 | semidirect |
| 23. C2×D1084 | C22 | semidirect |
| 26. C2×C22⋊D271 | C22 | semidirect |
| 27. C2×C22⋊D271 | C22 | semidirect |
| 29. C22×Dic271 | C22 | semidirect |
| 31. C2×C4×D271 | C22 | semidirect |
| 32. C2×C4×D271 | C22 | semidirect |
| 33. C2×C4×D271 | C22 | semidirect |
| 34. Dic271⋊C4 | C22 | semidirect |
| 35. Dic271⋊C4 | C22 | semidirect |
| 36. C22⋊Dic271 | C22 | semidirect |
| 40. (C2×D271)⋊C4 | C22 | semidirect |
| 41. (C2×D271)⋊C4 | C22 | semidirect |
| 42. (C2×D271)⋊C4 | C22 | semidirect |
| 43. (C2×D271)⋊C4 | C22 | semidirect |
| 44. (C2×D271)⋊C4 | C22 | semidirect |
| 45. C4×Dic271 | C22 | semidirect |
| 47. C22×C4×C271 | C22 | semidirect |
| 48. C4⋊C4×C271 | C22 | semidirect |
| 49. C4⋊C4×C271 | C22 | semidirect |
| 50. C22⋊C4×C271 | C22 | semidirect |
| 51. C22⋊C4×C271 | C22 | semidirect |
| 52. C22⋊C4×C271 | C22 | semidirect |
| 71. C22⋊D271 | D4 | semidirect |
| 110. C2×D271 | #41 | semidirect |
| 129. C271 | #546 | semidirect |
| 135. #123 | D271 | semidirect |
| 165. C22×C4 | C2×D271 | semidirect |
| 168. C4⋊C4 | C2×D271 | semidirect |
| 169. C4⋊C4 | C2×D271 | semidirect |
| 179. C22⋊C4 | C2×D271 | semidirect |
| 180. C22⋊C4 | C2×D271 | semidirect |
| 181. C22⋊C4 | C2×D271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864654 | 2 |
| C2×D4×D271 | 2 |
| #22576146 | 2 |
| #22576145 | 2 |
| #22576138 | 2 |
| #22576135 | 2 |
| #22576133 | 2 |
| #22576132 | 2 |
| #22576130 | 2 |
| #22576122 | 2 |
| (C22⋊D271)⋊C4 | 2 |
| C2×(C2×D271)⋊C4 | 2 |
| C22⋊C4×D271 | 2 |
| #123×C271 | 2 |
| #546 | 271 |