Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
410 conjugacy classes of subgroups; only the 256 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 256 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C131 | C24 |
| commutator | C22 | C23×C4×C131 |
| Frattini, cosocle | C2×C4 | C24×C131 |
| socle | C22×C131 | C23×C4 |
| prime radical | #532×C131 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #436×C131 | C2 | semidirect |
| 3. C8×D4×C131 | C2 | semidirect |
| 4. #431×C131 | C2 | semidirect |
| 5. #408×C131 | C2 | semidirect |
| 71. C131 | #2979 | direct (N × H) |
| 72. #2979 | C131 | direct (N × H) |
| 74. #436 | C2×C131 | semidirect |
| 75. C8×D4 | C2×C131 | semidirect |
| 76. #431 | C2×C131 | semidirect |
| 77. #408 | C2×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #532×C131 | 2 |
| #436×C131 | 2 |
| C8×D4×C131 | 2 |
| #431×C131 | 2 |
| #408×C131 | 2 |
| #2979 | 131 |