Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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 | C22×C17 | C23×C4 |
| Fitting | C17×#384 | C2 |
| socle, cosocle | C23×C17 | C24 |
| prime radical | C22×D68 | C22 |
| hypercenter | #384 | D17 |
| hypocenter | C17 | #2326 |
| LCS | G ▷ C22×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#10587573 | C2 | semidirect |
| 4. #11762714 | C2 | semidirect |
| 6. #11762699 | C2 | semidirect |
| 7. #11762680 | C2 | semidirect |
| 8. #11762655 | C2 | semidirect |
| 9. C17×#384 | C2 | semidirect |
| 26. #10587561 | C4 | semidirect |
| 46. C2×Dic34 | C2×C4 | semidirect |
| 50. C42×C17 | C2×C4 | semidirect |
| 53. #63227 | C2×C4 | semidirect |
| 113. #384 | D17 | semidirect |
| 141. C17 | #2326 | semidirect |
| 169. C42 | C4×D17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587561 | 2 |
| C2×#10587573 | 2 |
| C2×#10587579 | 2 |
| #11762714 | 2 |
| #11762710 | 2 |
| #11762699 | 2 |
| #11762680 | 2 |
| #11762655 | 2 |
| C17×#384 | 2 |
| #2326 | 17 |