Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
644 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 125 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 | C17×#397 | C2 |
| hypercenter | #397 | D17 |
| hypocenter | C17 | #2815 |
| LCS | G ▷ C23×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ #397 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#10587581 | C2 | semidirect |
| 2. C2×#10587570 | C2 | semidirect |
| 3. #11762738 | C2 | semidirect |
| 4. #11762735 | C2 | semidirect |
| 5. #11762719 | C2 | semidirect |
| 6. #11762719 | C2 | semidirect |
| 7. #11762718 | C2 | semidirect |
| 8. #11762718 | C2 | semidirect |
| 9. #11762718 | C2 | semidirect |
| 10. #11762671 | C2 | semidirect |
| 11. C17×#397 | C2 | semidirect |
| 66. C2×C4×D17 | D4 | semidirect |
| 164. #397 | D17 | semidirect |
| 205. C17 | #2815 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587581 | 2 |
| C2×#10587570 | 2 |
| #11762738 | 2 |
| #11762735 | 2 |
| #11762719 | 2 |
| #11762718 | 2 |
| #11762671 | 2 |
| C17×#397 | 2 |
| #2815 | 17 |