Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
142 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C131 |
| prime radical | C2×D8×C131 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C42×C131 ◁ C4×C8×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #500×C131 | C2 | semidirect |
| 3. #358×C131 | C2 | semidirect |
| 5. #346×C131 | C2 | semidirect |
| 56. C131 | #2243 | direct (N × H) |
| 57. #2243 | C131 | direct (N × H) |
| 58. #500 | C2×C131 | semidirect |
| 60. #358 | C2×C131 | semidirect |
| 62. #346 | C2×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #500×C131 | 2 |
| #495×C131 | 2 |
| #358×C131 | 2 |
| #357×C131 | 2 |
| #346×C131 | 2 |
| #2243 | 131 |