Finite Group DatabasebetaFinite Group DatabasebetaOrder 1856.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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×D29 |
| commutator, socle, cosocle | C22×C29 | C24 |
| Fitting | C29×#126 | C2 |
| hypercenter | #126 | D29 |
| hypocenter | C29 | #560 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×#126 | C2 | semidirect |
| 2. #11745257 | C2 | semidirect |
| 3. #11745256 | C2 | semidirect |
| 4. #11745246 | C2 | semidirect |
| 5. #11745244 | C2 | semidirect |
| 6. #11745238 | C2 | semidirect |
| 7. #11745236 | C2 | semidirect |
| 12. C2×D116 | C22 | semidirect |
| 13. C2×C4×D29 | C22 | semidirect |
| 14. C4⋊C4×C29 | C22 | semidirect |
| 16. C42×C29 | C22 | semidirect |
| 19. #90781 | C22 | semidirect |
| 20. #90779 | C22 | semidirect |
| 53. #126 | D29 | semidirect |
| 58. C29 | #560 | semidirect |
| 66. C4⋊C4 | C2×D29 | semidirect |
| 71. C42 | C2×D29 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#126 | 2 |
| #11745257 | 2 |
| #11745256 | 2 |
| #11745246 | 2 |
| #11745244 | 2 |
| #11745238 | 2 |
| #11745236 | 2 |
| #560 | 29 |