Finite Group DatabasebetaFinite Group DatabasebetaOrder 1856.
260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 260 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×#124 | C2 |
| hypercenter | #124 | D29 |
| hypocenter | C29 | H(3,𝔽4) |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×#124 | C2 | semidirect |
| 2. #11745309 | C2 | semidirect |
| 3. #11745299 | C2 | semidirect |
| 4. #11745247 | C2 | semidirect |
| 5. #11745244 | C2 | semidirect |
| 6. #11745237 | C2 | semidirect |
| 7. C23×D29 | C22 | semidirect |
| 8. C2×Q8×C29 | C22 | semidirect |
| 13. C2×Dic58 | C22 | semidirect |
| 14. C22⋊C4×C29 | C22 | semidirect |
| 15. C42×C29 | C22 | semidirect |
| 16. C22⋊Dic29 | C22 | semidirect |
| 17. #90781 | C22 | semidirect |
| 18. #90781 | C22 | semidirect |
| 19. #90781 | C22 | semidirect |
| 20. C4×Dic29 | C22 | semidirect |
| 54. #124 | D29 | semidirect |
| 57. C29 | H(3,𝔽4) | semidirect |
| 60. C2×Q8 | C2×D29 | semidirect |
| 69. C22⋊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×#124 | 2 |
| #11745309 | 2 |
| #11745299 | 2 |
| #11745247 | 2 |
| #11745244 | 2 |
| #11745237 | 2 |
| H(3,𝔽4) | 29 |