Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
412 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 129 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 | #399×C131 | C2 |
| prime radical | C24×C131 | C23 |
| hypercenter | #399 | D131 |
| hypocenter | C131 | #2517 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #22553198 | C2 | semidirect |
| 4. #22553195 | C2 | semidirect |
| 5. #22553193 | C2 | semidirect |
| 7. #22553177 | C2 | semidirect |
| 106. C131 | #2517 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22553237 | 2 |
| #22553198 | 2 |
| #22553195 | 2 |
| #22553193 | 2 |
| #22553184 | 2 |
| #22553177 | 2 |
| C22⋊C4×Dic131 | 2 |
| #399×C131 | 2 |
| #2517 | 131 |