Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
676 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 173 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×#127×C131 | C2 |
| hypercenter | C2×#127 | D131 |
| hypocenter | C131 | #2548 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862233 | C2 | semidirect |
| 2. C2×C4×Dic262 | C2 | semidirect |
| 3. #22553226 | C2 | semidirect |
| 4. #22553215 | C2 | semidirect |
| 5. #22553215 | C2 | semidirect |
| 6. C4×C4⋊Dic131 | C2 | semidirect |
| 26. C2×Dic262 | D4 | semidirect |
| 86. C4×C131 | #124 | semidirect |
| 96. C131 | #2548 | semidirect |
| 186. C4 | #16862233 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862233 | 2 |
| C2×C4×Dic262 | 2 |
| #22553226 | 2 |
| #22553215 | 2 |
| C4×C4⋊Dic131 | 2 |
| C2×#127×C131 | 2 |
| #2548 | 131 |