Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
856 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 856 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#127 | C2×C3 |
| hypercenter | #127 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×D42 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×AGL(1,𝔽7) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#127 | C2 | semidirect |
| 3. #6501⋊C3 | C2 | semidirect |
| 4. #6500⋊C3 | C2 | semidirect |
| 5. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 6. #90408 | C3 | semidirect |
| 7. C23×AGL(1,𝔽7) | C22 | semidirect |
| 8. C2×D4×ΓL(1,𝔽8) | C22 | semidirect |
| 9. C2×D4×ΓL(1,𝔽8) | C22 | semidirect |
| 10. C2×(C22⋊D7)⋊C3 | C22 | semidirect |
| 13. C2×C4×AGL(1,𝔽7) | C22 | semidirect |
| 14. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 15. (C22⋊Dic7)⋊C3 | C22 | semidirect |
| 16. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 17. C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | semidirect |
| 18. C2×D4×D7 | C2×C3 | semidirect |
| 19. C7×#127 | C2×C3 | semidirect |
| 20. #6501 | C2×C3 | semidirect |
| 21. #6500 | C2×C3 | semidirect |
| 22. C4×D28 | C2×C3 | semidirect |
| 34. C4⋊AGL(1,𝔽7) | D4 | semidirect |
| 36. C23×D7 | C22×C3 | semidirect |
| 37. C2×C7×D4 | C22×C3 | semidirect |
| 38. C2×C7×D4 | C22×C3 | semidirect |
| 39. C2×C22⋊D7 | C22×C3 | semidirect |
| 42. C2×C4×D7 | C22×C3 | semidirect |
| 43. C42×C7 | C22×C3 | semidirect |
| 44. C22⋊Dic7 | C22×C3 | semidirect |
| 45. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 46. C4⋊Dic7 | C22×C3 | semidirect |
| 58. C2×AGL(1,𝔽7) | C2×D4 | semidirect |
| 60. C4×ΓL(1,𝔽8) | C2×D4 | semidirect |
| 73. D28 | C3×D4 | semidirect |
| 91. #127 | AGL(1,𝔽7) | semidirect |
| 101. C2×D7 | C2×C3×D4 | semidirect |
| 103. C4×C7 | C2×C3×D4 | semidirect |
| 117. ΓL(1,𝔽8) | D42 | semidirect |
| 122. C2×D4 | C2×AGL(1,𝔽7) | semidirect |
| 123. C2×D4 | C2×AGL(1,𝔽7) | semidirect |
| 128. C42 | C2×AGL(1,𝔽7) | semidirect |
| 155. C7 | C3×D42 | semidirect |
| 169. C4 | D4×AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×#127 | 2 |
| #6501⋊C3 | 2 |
| #6500⋊C3 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #90408 | 3 |
| C3×D42 | 7 |