Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
708 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 181 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, prime radical | C2×#121×C131 | C2 |
| hypercenter | C2×#121 | D131 |
| hypocenter | C131 | #2501 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862227 | C2 | semidirect |
| 2. C22×C4⋊Dic131 | C2 | semidirect |
| 3. #22553248 | C2 | semidirect |
| 4. #22553215 | C2 | semidirect |
| 5. #22553181 | C2 | semidirect |
| 6. C4×C22⋊Dic131 | C2 | semidirect |
| 32. C22⋊Dic131 | D4 | semidirect |
| 85. C4×C131 | #123 | semidirect |
| 94. C131 | #2501 | semidirect |
| 182. C4 | #16862227 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862227 | 2 |
| C22×C4⋊Dic131 | 2 |
| #22553248 | 2 |
| #22553215 | 2 |
| #22553181 | 2 |
| C4×C22⋊Dic131 | 2 |
| C2×#121×C131 | 2 |
| #2501 | 131 |