Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
516 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 165 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×D131 |
| commutator, socle, cosocle | C23×C131 | C24 |
| Fitting | C43×C131 | C2 |
| prime radical | C22×D524 | C22 |
| hypercenter | C43 | D131 |
| hypocenter | C131 | #2860 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862240 | C2 | semidirect |
| 3. #22553210 | C2 | semidirect |
| 4. #22553203 | C2 | semidirect |
| 5. C43×C131 | C2 | semidirect |
| 28. C42×C131 | D4 | semidirect |
| 52. C4×C131 | #126 | semidirect |
| 58. C131 | #2860 | semidirect |
| 64. C43 | D131 | semidirect |
| 87. C42 | D524 | semidirect |
| 112. C4 | #16862240 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862240 | 2 |
| #22553214 | 2 |
| #22553210 | 2 |
| #22553203 | 2 |
| C43×C131 | 2 |
| #2860 | 131 |