Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 580 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×#393 | C2 |
| prime radical | C24×D17 | C22 |
| hypercenter | #393 | D17 |
| hypocenter | C17 | #2840 |
| 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. #11762722 | C2 | semidirect |
| 2. #11762721 | C2 | semidirect |
| 3. #11762720 | C2 | semidirect |
| 5. C17×#393 | C2 | semidirect |
| 7. C22×Q8×C17 | C22 | semidirect |
| 8. C22×Dic34 | C22 | semidirect |
| 13. C17×#95 | C22 | semidirect |
| 14. #10587502 | C22 | semidirect |
| 15. #10587502 | C22 | semidirect |
| 19. C23×D17 | Q8 | semidirect |
| 67. #393 | D17 | semidirect |
| 76. C22×Q8 | C2×D17 | semidirect |
| 83. #95 | C2×D17 | semidirect |
| 84. C17 | #2840 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11762722 | 2 |
| #11762721 | 2 |
| #11762720 | 2 |
| #11762664 | 2 |
| C17×#393 | 2 |
| #2840 | 17 |