Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
360 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C131 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C131 |
| prime radical | C22×D8×C131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ C2×C42×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4⋊C8×C131 | C2 | semidirect |
| 3. C2×#102×C131 | C2 | semidirect |
| 4. #390×C131 | C2 | semidirect |
| 5. #340×C131 | C2 | semidirect |
| 58. C2×C4×C131 | D8 | semidirect |
| 89. C131 | #2054 | direct (N × H) |
| 90. #2054 | C131 | direct (N × H) |
| 92. C2×C4⋊C8 | C2×C131 | semidirect |
| 93. C2×#102 | C2×C131 | semidirect |
| 94. #390 | C2×C131 | semidirect |
| 95. #340 | C2×C131 | semidirect |
| 148. C2×C4 | D8×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D8×C131 | 2 |
| C2×C4⋊C8×C131 | 2 |
| C2×#102×C131 | 2 |
| #390×C131 | 2 |
| #340×C131 | 2 |
| #2054 | 131 |