Finite Group DatabasebetaFinite Group DatabasebetaOrder 1856.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 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 |
| prime radical | C2×D4×C29 | C22 |
| hypercenter | #123 | D29 |
| hypocenter | C29 | #563 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #11745286 | C2 | semidirect |
| 5. #11745251 | C2 | semidirect |
| 6. #11745250 | C2 | semidirect |
| 7. #11745241 | C2 | semidirect |
| 8. #11745241 | C2 | semidirect |
| 9. #11745240 | C2 | semidirect |
| 71. C29 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#123 | 2 |
| #11745298 | 2 |
| #11745297 | 2 |
| #11745286 | 2 |
| #11745251 | 2 |
| #11745250 | 2 |
| #11745241 | 2 |
| #11745240 | 2 |
| #563 | 29 |