Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 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×#391 | C2 |
| hypercenter | #391 | D17 |
| hypocenter | C17 | #2431 |
| LCS | G ▷ C23×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#63256 | C2 | semidirect |
| 2. C2×#10587560 | C2 | semidirect |
| 3. C2×#10587633 | C2 | semidirect |
| 4. C2×#10587581 | C2 | semidirect |
| 5. #11762719 | C2 | semidirect |
| 6. #11762710 | C2 | semidirect |
| 7. #11762687 | C2 | semidirect |
| 8. C4×#63227 | C2 | semidirect |
| 9. C17×#391 | C2 | semidirect |
| 38. C2×Q8×C17 | D4 | semidirect |
| 59. C2×D68 | D4 | semidirect |
| 65. C2×C4×D17 | D4 | semidirect |
| 100. C2×C4×C17 | #41 | semidirect |
| 146. #391 | D17 | semidirect |
| 181. C17 | #2431 | semidirect |
| 188. C2×Q8 | D68 | semidirect |
| 262. C2×C4 | #63256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#63256 | 2 |
| C2×#10587560 | 2 |
| C2×#10587633 | 2 |
| C2×#10587581 | 2 |
| #11762719 | 2 |
| #11762710 | 2 |
| #11762687 | 2 |
| C4×#63227 | 2 |
| C17×#391 | 2 |
| #2431 | 17 |