Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
840 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 189 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 | #380×C131 | C2 |
| prime radical | C22×C22⋊D131 | C22 |
| hypercenter | #380 | D131 |
| hypocenter | C131 | #2409 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic262 | C2 | semidirect |
| 2. C2×#16862226 | C2 | semidirect |
| 4. #22553233 | C2 | semidirect |
| 5. #22553225 | C2 | semidirect |
| 6. #22553217 | C2 | semidirect |
| 8. #22553187 | C2 | semidirect |
| 9. #380×C131 | C2 | semidirect |
| 37. C2×Dic262 | D4 | semidirect |
| 40. C2×C22⋊D131 | Q8 | semidirect |
| 67. C22×D131 | C2×Q8 | semidirect |
| 81. C2×Dic131 | C2×Q8 | semidirect |
| 101. C22×C131 | C22⋊Q8 | semidirect |
| 126. C131 | #2409 | semidirect |
| 136. #380 | D131 | semidirect |
| 233. C22 | #16862226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic262 | 2 |
| C2×#16862226 | 2 |
| C2×C4×C22⋊D131 | 2 |
| #22553233 | 2 |
| #22553225 | 2 |
| #22553217 | 2 |
| #22553189 | 2 |
| #22553187 | 2 |
| #380×C131 | 2 |
| #2409 | 131 |