Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 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×D271 |
| commutator, socle, cosocle | C22×C271 | C24 |
| Fitting | C4×D4×C271 | C2 |
| hypercenter | C4×D4 | D271 |
| hypocenter | C271 | #538 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22576155 | C2 | semidirect |
| 2. #22576154 | C2 | semidirect |
| 3. #22576148 | C2 | semidirect |
| 4. #22576146 | C2 | semidirect |
| 5. #22576145 | C2 | semidirect |
| 6. #22576143 | C2 | semidirect |
| 7. #22576137 | C2 | semidirect |
| 8. #22576134 | C2 | semidirect |
| 9. #22576133 | C2 | semidirect |
| 10. #22576132 | C2 | semidirect |
| 11. #22576130 | C2 | semidirect |
| 12. #22576128 | C2 | semidirect |
| 13. C4×C22⋊D271 | C2 | semidirect |
| 14. C2×(C2×D271)⋊C4 | C2 | semidirect |
| 15. C4×D4×C271 | C2 | semidirect |
| 17. C2×Dic542 | C22 | semidirect |
| 23. C2×C22⋊D271 | C22 | semidirect |
| 25. C2×C4×D271 | C22 | semidirect |
| 26. C4⋊Dic271 | C22 | semidirect |
| 27. Dic271⋊C4 | C22 | semidirect |
| 28. Dic271⋊C4 | C22 | semidirect |
| 29. Dic271⋊C4 | C22 | semidirect |
| 30. Dic271⋊C4 | C22 | semidirect |
| 32. C22⋊Dic271 | C22 | semidirect |
| 41. (C2×D271)⋊C4 | C22 | semidirect |
| 42. (C2×D271)⋊C4 | C22 | semidirect |
| 43. C4×Dic271 | C22 | semidirect |
| 46. C22×C4×C271 | C22 | semidirect |
| 47. C4⋊C4×C271 | C22 | semidirect |
| 49. C22⋊C4×C271 | C22 | semidirect |
| 50. C42×C271 | C22 | semidirect |
| 116. C271 | #538 | semidirect |
| 131. C4×D4 | D271 | semidirect |
| 145. C22×C4 | C2×D271 | semidirect |
| 151. C4⋊C4 | C2×D271 | semidirect |
| 165. C22⋊C4 | C2×D271 | semidirect |
| 167. C42 | C2×D271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22576155 | 2 |
| #22576154 | 2 |
| #22576148 | 2 |
| #22576146 | 2 |
| #22576145 | 2 |
| #22576143 | 2 |
| #22576137 | 2 |
| #22576134 | 2 |
| #22576133 | 2 |
| #22576132 | 2 |
| #22576130 | 2 |
| #22576128 | 2 |
| C4×C22⋊D271 | 2 |
| C2×(C2×D271)⋊C4 | 2 |
| C4×D4×C271 | 2 |
| #538 | 271 |