Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
488 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C2×C4 | C24×C131 |
| prime radical | C2×#127×C131 | C2 |
| LCS | G ▷ C2×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×#125×C131 | C2 | semidirect |
| 3. #490×C131 | C2 | direct (N × H) |
| 4. C2×#102×C131 | C2 | semidirect |
| 5. C2×#97×C131 | C2 | semidirect |
| 12. #125×C131 | C22 | semidirect |
| 16. #102×C131 | C22 | semidirect |
| 17. #97×C131 | C22 | semidirect |
| 60. C2×C131 | #490 | direct (N × H) |
| 63. C131 | C2×#490 | direct (N × H) |
| 64. C2×#490 | C131 | direct (N × H) |
| 66. C2×#125 | C2×C131 | semidirect |
| 67. #490 | C2×C131 | direct (N × H) |
| 68. C2×#102 | C2×C131 | semidirect |
| 69. C2×#97 | C2×C131 | semidirect |
| 76. #125 | C22×C131 | semidirect |
| 80. #102 | C22×C131 | semidirect |
| 81. #97 | C22×C131 | semidirect |
| 124. C2 | #490×C131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C131 | 2 |
| C2×#125×C131 | 2 |
| #490×C131 | 2 |
| C2×#102×C131 | 2 |
| C2×#97×C131 | 2 |
| C2×#490 | 131 |