Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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 | C2×D4 |
| commutator | C2×C4 | C22×C4×C131 |
| Frattini, cosocle | C42 | C23×C131 |
| socle | C22×C131 | C2×C22⋊C4 |
| prime radical | #532×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ C2×C42×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #431×C131 | C2 | semidirect |
| 3. C2×C4⋊C8×C131 | C2 | semidirect |
| 4. D4⋊C8×C131 | C2 | semidirect |
| 84. C131 | #1480 | direct (N × H) |
| 85. #1480 | C131 | direct (N × H) |
| 87. #431 | C2×C131 | semidirect |
| 88. C2×C4⋊C8 | C2×C131 | semidirect |
| 89. D4⋊C8 | 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 |
| #431×C131 | 2 |
| C2×C4⋊C8×C131 | 2 |
| D4⋊C8×C131 | 2 |
| #1480 | 131 |