Finite Group DatabasebetaFinite Group DatabasebetaOrder 1856.
312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#127 | C2 |
| hypercenter | #127 | D29 |
| hypocenter | C29 | #550 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#90807 | C2 | semidirect |
| 2. C29×#127 | C2 | semidirect |
| 3. #11745302 | C2 | semidirect |
| 4. #11745301 | C2 | semidirect |
| 5. #11745300 | C2 | semidirect |
| 6. D4×Dic29 | C2 | semidirect |
| 7. C4×D116 | C2 | semidirect |
| 8. C4×Dic58 | C2 | semidirect |
| 42. Dic58 | D4 | semidirect |
| 52. C4×C29 | #41 | semidirect |
| 64. #127 | D29 | semidirect |
| 70. C29 | #550 | semidirect |
| 117. C4 | #90807 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#90807 | 2 |
| C29×#127 | 2 |
| #11745302 | 2 |
| #11745301 | 2 |
| #11745300 | 2 |
| D4×Dic29 | 2 |
| C4×D116 | 2 |
| C4×Dic58 | 2 |
| #550 | 29 |