Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
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×C17 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C17 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ #41×C17 ◁ C17×#131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C17×#576 | C2 | semidirect |
| 2. C17×#509 | C2 | semidirect |
| 3. C17×#508 | C2 | semidirect |
| 4. C2×C17×D16 | C2 | semidirect |
| 5. C17×#504 | C2 | semidirect |
| 8. C17×#135 | C22 | semidirect |
| 9. C2×D8×C17 | C22 | semidirect |
| 11. C17×Q32 | C22 | semidirect |
| 12. C17×SD32 | C22 | semidirect |
| 13. C17×D16 | C22 | semidirect |
| 14. C17×D16 | C22 | semidirect |
| 15. C17×M32 | C22 | semidirect |
| 16. C2×C16×C17 | C22 | semidirect |
| 41. #3415 | C17 | direct (N × H) |
| 48. #576 | C2×C17 | semidirect |
| 49. #509 | C2×C17 | semidirect |
| 50. #508 | C2×C17 | semidirect |
| 51. C2×D16 | C2×C17 | semidirect |
| 52. #504 | C2×C17 | semidirect |
| 58. #135 | C22×C17 | semidirect |
| 59. C2×D8 | C22×C17 | semidirect |
| 61. Q32 | C22×C17 | semidirect |
| 62. SD32 | C22×C17 | semidirect |
| 63. D16 | C22×C17 | semidirect |
| 64. D16 | C22×C17 | semidirect |
| 65. M32 | C22×C17 | semidirect |
| 66. C2×C16 | C22×C17 | semidirect |
| 67. C17 | #3415 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C17×#576 | 2 |
| C17×#509 | 2 |
| C17×#508 | 2 |
| C2×C17×D16 | 2 |
| C17×#504 | 2 |
| #3415 | 17 |