Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
396 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 120 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×C22⋊C4 |
| commutator | C23 | C22×C4×C131 |
| Frattini, cosocle | C22×C4 | C23×C131 |
| prime radical | C2×#127×C131 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×C42×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #423×C131 | C2 | semidirect |
| 3. C2×#100×C131 | C2 | semidirect |
| 4. C4×M16×C131 | C2 | semidirect |
| 29. M16×C131 | D4 | semidirect |
| 60. C4×C131 | #100 | semidirect |
| 68. C131 | #1965 | direct (N × H) |
| 69. #1965 | C131 | direct (N × H) |
| 71. #423 | C2×C131 | semidirect |
| 72. C2×#100 | C2×C131 | semidirect |
| 73. C4×M16 | C2×C131 | semidirect |
| 98. M16 | D4×C131 | semidirect |
| 129. C4 | #100×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C131 | 2 |
| #423×C131 | 2 |
| C2×#100×C131 | 2 |
| C4×M16×C131 | 2 |
| #1965 | 131 |