Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C167 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C167 |
| socle | C2×C167 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C167 ◁ C22×C4×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×C167 | C2 | semidirect |
| 2. #137×C167 | C2 | semidirect |
| 3. AGL(1,ℤ/8)×C167 | C2 | semidirect |
| 4. #135×C167 | C2 | semidirect |
| 5. C2×M16×C167 | C2 | semidirect |
| 9. #41×C167 | C22 | semidirect |
| 10. C2×Q8×C167 | C22 | semidirect |
| 12. C2×D4×C167 | C22 | semidirect |
| 15. Q16×C167 | C22 | semidirect |
| 16. SD16×C167 | C22 | semidirect |
| 17. D8×C167 | C22 | semidirect |
| 18. M16×C167 | C22 | semidirect |
| 19. C2×C8×C167 | C22 | semidirect |
| 49. C167 | #575 | direct (N × H) |
| 50. #575 | C167 | direct (N × H) |
| 51. C2×#41 | C2×C167 | semidirect |
| 52. #137 | C2×C167 | semidirect |
| 53. AGL(1,ℤ/8) | C2×C167 | semidirect |
| 54. #135 | C2×C167 | semidirect |
| 55. C2×M16 | C2×C167 | semidirect |
| 59. #41 | C22×C167 | semidirect |
| 60. C2×Q8 | C22×C167 | semidirect |
| 62. C2×D4 | C22×C167 | semidirect |
| 65. Q16 | C22×C167 | semidirect |
| 66. SD16 | C22×C167 | semidirect |
| 67. D8 | C22×C167 | semidirect |
| 68. M16 | C22×C167 | semidirect |
| 69. C2×C8 | C22×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×C167 | 2 |
| #137×C167 | 2 |
| AGL(1,ℤ/8)×C167 | 2 |
| #135×C167 | 2 |
| C2×M16×C167 | 2 |
| #575 | 167 |