Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
348 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 216 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C131 |
| prime radical | C23×C131 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ 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 |
|---|---|---|
| 3. #501×C131 | C2 | semidirect |
| 4. #499×C131 | C2 | semidirect |
| 27. M16×C131 | Q8 | semidirect |
| 65. C131 | #3166 | direct (N × H) |
| 66. #3166 | C131 | direct (N × H) |
| 69. #501 | C2×C131 | semidirect |
| 70. #499 | C2×C131 | semidirect |
| 93. M16 | Q8×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533×C131 | 2 |
| C2×#125×C131 | 2 |
| #501×C131 | 2 |
| #499×C131 | 2 |
| #428×C131 | 2 |
| C4×M16×C131 | 2 |
| #3166 | 131 |