Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C53 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ #41×C53 ◁ #131×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#576 | C2 | semidirect |
| 2. C53×#509 | C2 | semidirect |
| 3. C53×#508 | C2 | semidirect |
| 4. C2×D16×C53 | C2 | semidirect |
| 5. C53×#504 | C2 | semidirect |
| 8. #135×C53 | C22 | semidirect |
| 9. C2×D8×C53 | C22 | semidirect |
| 11. Q32×C53 | C22 | semidirect |
| 12. SD32×C53 | C22 | semidirect |
| 13. D16×C53 | C22 | semidirect |
| 14. D16×C53 | C22 | semidirect |
| 15. M32×C53 | C22 | semidirect |
| 16. C2×C16×C53 | C22 | semidirect |
| 47. #3415 | C53 | direct (N × H) |
| 51. #576 | C2×C53 | semidirect |
| 52. #509 | C2×C53 | semidirect |
| 53. #508 | C2×C53 | semidirect |
| 54. C2×D16 | C2×C53 | semidirect |
| 55. #504 | C2×C53 | semidirect |
| 56. C53 | #3415 | direct (N × H) |
| 59. #135 | C22×C53 | semidirect |
| 60. C2×D8 | C22×C53 | semidirect |
| 62. Q32 | C22×C53 | semidirect |
| 63. SD32 | C22×C53 | semidirect |
| 64. D16 | C22×C53 | semidirect |
| 65. D16 | C22×C53 | semidirect |
| 66. M32 | C22×C53 | semidirect |
| 67. C2×C16 | C22×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#576 | 2 |
| C53×#509 | 2 |
| C53×#508 | 2 |
| C2×D16×C53 | 2 |
| C53×#504 | 2 |
| #3415 | 53 |