Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
760 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 185 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D17 |
| commutator, socle, cosocle | C23×C17 | C24 |
| Fitting | C4×C4⋊C4×C17 | C2 |
| hypercenter | C4×C4⋊C4 | D17 |
| hypocenter | C17 | #2540 |
| LCS | G ▷ C23×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#10587561 | C2 | semidirect |
| 2. C2×#10587583 | C2 | semidirect |
| 3. C2×#10587581 | C2 | semidirect |
| 4. C2×C4×D68 | C2 | semidirect |
| 5. #11762746 | C2 | semidirect |
| 6. #11762719 | C2 | semidirect |
| 7. #11762711 | C2 | semidirect |
| 8. C4×C4⋊C4×C17 | C2 | semidirect |
| 49. C4⋊C4×C17 | D4 | semidirect |
| 51. C42×C17 | D4 | semidirect |
| 123. C4×C4⋊C4 | D17 | semidirect |
| 151. C17 | #2540 | semidirect |
| 174. C4⋊C4 | D68 | semidirect |
| 181. C42 | D68 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587561 | 2 |
| C2×#10587583 | 2 |
| C2×#10587581 | 2 |
| C2×C4×D68 | 2 |
| #11762746 | 2 |
| #11762719 | 2 |
| #11762711 | 2 |
| C4×C4⋊C4×C17 | 2 |
| #2540 | 17 |