Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
900 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 185 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 | C4×C22⋊C4×C131 | C2 |
| hypercenter | C4×C22⋊C4 | D131 |
| hypocenter | C131 | #2539 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862230 | C2 | semidirect |
| 2. C2×#16862220 | C2 | semidirect |
| 3. C2×#16862216 | C2 | semidirect |
| 4. C22×(C2×D131)⋊C4 | C2 | semidirect |
| 5. #22553249 | C2 | semidirect |
| 6. #22553222 | C2 | semidirect |
| 7. #22553214 | C2 | semidirect |
| 8. C4×C22⋊C4×C131 | C2 | semidirect |
| 11. C22×D524 | C22 | semidirect |
| 17. C2×C4⋊Dic131 | C22 | semidirect |
| 18. C2×C4⋊Dic131 | C22 | semidirect |
| 23. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 24. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 27. C2×C42×C131 | C22 | semidirect |
| 28. #95×C131 | C22 | semidirect |
| 56. C22×C4×C131 | D4 | semidirect |
| 60. C22⋊C4×C131 | D4 | semidirect |
| 129. C131 | #2539 | semidirect |
| 138. C4×C22⋊C4 | D131 | semidirect |
| 157. C2×C42 | C2×D131 | semidirect |
| 158. #95 | C2×D131 | semidirect |
| 179. C22×C4 | D524 | semidirect |
| 190. C22⋊C4 | D524 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862230 | 2 |
| C2×#16862220 | 2 |
| C2×#16862216 | 2 |
| C22×(C2×D131)⋊C4 | 2 |
| #22553249 | 2 |
| #22553222 | 2 |
| #22553214 | 2 |
| C4×C22⋊C4×C131 | 2 |
| #2539 | 131 |