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×#123 | C2 |
| hypercenter | #123 | D29 |
| hypocenter | C29 | #560 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×#123 | C2 | semidirect |
| 2. #11745302 | C2 | semidirect |
| 3. #11745299 | C2 | semidirect |
| 4. #11745292 | C2 | semidirect |
| 5. #11745260 | C2 | semidirect |
| 6. #11745257 | C2 | semidirect |
| 7. #11745246 | C2 | semidirect |
| 8. #11745245 | C2 | semidirect |
| 9. #11745244 | C2 | semidirect |
| 12. C22×C4×C29 | C22 | semidirect |
| 15. C2×C22⋊D29 | C22 | semidirect |
| 18. C2×D116 | C22 | semidirect |
| 19. C2×C4×D29 | C22 | semidirect |
| 20. C4⋊C4×C29 | C22 | semidirect |
| 22. C22⋊C4×C29 | C22 | semidirect |
| 27. #90781 | C22 | semidirect |
| 28. C4⋊Dic29 | C22 | semidirect |
| 29. #90779 | C22 | semidirect |
| 30. C4×Dic29 | C22 | semidirect |
| 78. #123 | D29 | semidirect |
| 84. C29 | #560 | semidirect |
| 94. C22×C4 | C2×D29 | semidirect |
| 97. C4⋊C4 | C2×D29 | semidirect |
| 104. C22⋊C4 | C2×D29 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#123 | 2 |
| #11745302 | 2 |
| #11745299 | 2 |
| #11745292 | 2 |
| #11745260 | 2 |
| #11745257 | 2 |
| #11745246 | 2 |
| #11745245 | 2 |
| #11745244 | 2 |
| #560 | 29 |