Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D16×D17 |
| commutator, cosocle | C16×C17 | C23 |
| Frattini | C16 | C22×D17 |
| Fitting | C17×SD64 | C2 |
| socle | C2×C17 | C2×D16 |
| hypercenter | SD64 | D17 |
| hypocenter | C17 | #2263 |
| LCS | G ▷ C16×C17 ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763242 | C2 | semidirect |
| 2. D16×D17 | C2 | semidirect |
| 3. #11762646 | C2 | semidirect |
| 4. D544 | C2 | semidirect |
| 5. #11762644 | C2 | semidirect |
| 6. #11762642 | C2 | semidirect |
| 7. C17×SD64 | C2 | semidirect |
| 10. C17×Q32 | C22 | semidirect |
| 15. D272 | C22 | semidirect |
| 17. C17×C32 | C22 | semidirect |
| 18. #10587465 | C22 | semidirect |
| 53. SD64 | D17 | semidirect |
| 64. Q32 | C2×D17 | semidirect |
| 72. C32 | C2×D17 | semidirect |
| 73. C17 | #2263 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763242 | 2 |
| D16×D17 | 2 |
| #11762646 | 2 |
| D544 | 2 |
| #11762644 | 2 |
| #11762642 | 2 |
| C17×SD64 | 2 |
| #2263 | 17 |