Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#393 | C2 |
| hypercenter | #393 | D17 |
| hypocenter | C17 | #2388 |
| LCS | G ▷ C23×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#63256 | C2 | semidirect |
| 2. C2×#10587567 | C2 | semidirect |
| 3. #11762728 | C2 | semidirect |
| 4. #11762727 | C2 | semidirect |
| 5. #11762718 | C2 | semidirect |
| 6. #11762656 | C2 | semidirect |
| 7. C17×#393 | C2 | semidirect |
| 38. C2×C4×D17 | D4 | semidirect |
| 46. C23×C17 | #41 | semidirect |
| 64. C2×Dic17 | #41 | semidirect |
| 88. #393 | D17 | semidirect |
| 110. C17 | #2388 | semidirect |
| 140. C23 | #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×#10587567 | 2 |
| #11762728 | 2 |
| #11762727 | 2 |
| #11762718 | 2 |
| #11762656 | 2 |
| C17×#393 | 2 |
| #2388 | 17 |