Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
60570 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.
This group has 60570 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D7 |
| commutator, socle, cosocle | C2×C7 | C27 |
| Fitting | C7×H(7,𝔽2) | C2 |
| hypercenter | H(7,𝔽2) | D7 |
| hypocenter | C7 | C2×H(7,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ H(7,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×H(7,𝔽2) | C2 | semidirect |
| 2. #11744566 | C2 | semidirect |
| 3. D7×#585 | C2 | semidirect |
| 4. #11744560 | C2 | semidirect |
| 5. C2×D7×H(5,𝔽2) | C2 | semidirect |
| 6. C7×#585 | C22 | semidirect |
| 7. C2×C7×H(5,𝔽2) | C22 | semidirect |
| 8. #90620 | C22 | semidirect |
| 9. D7×2−1+4 | C22 | semidirect |
| 10. #90618 | C22 | semidirect |
| 11. D7×H(5,𝔽2) | C22 | semidirect |
| 12. D7×H(5,𝔽2) | C22 | semidirect |
| 13. #90616 | C22 | semidirect |
| 14. C2×#6553 | C22 | semidirect |
| 15. C2×D7×#41 | C22 | semidirect |
| 16. C2×#6548 | C22 | semidirect |
| 17. C22×D4×D7 | C22 | semidirect |
| 19. C24×D7 | C23 | semidirect |
| 20. C7×2−1+4 | C23 | semidirect |
| 21. C7×H(5,𝔽2) | C23 | semidirect |
| 22. C2×C7×#41 | C23 | semidirect |
| 23. C22×C7×D4 | C23 | semidirect |
| 24. C22×C22⋊D7 | C23 | semidirect |
| 25. #6554 | C23 | semidirect |
| 26. #6553 | C23 | semidirect |
| 27. D7×#41 | C23 | semidirect |
| 28. D7×#41 | C23 | semidirect |
| 29. #6551 | C23 | semidirect |
| 30. C2×#1161 | C23 | semidirect |
| 31. C2×Q8×D7 | C23 | semidirect |
| 32. #6548 | C23 | semidirect |
| 33. C2×#1159 | C23 | semidirect |
| 34. C2×D4×D7 | C23 | semidirect |
| 35. C2×D4×D7 | C23 | semidirect |
| 36. C2×#1157 | C23 | semidirect |
| 37. C22×D28 | C23 | semidirect |
| 38. C22×C4×D7 | C23 | semidirect |
| 41. H(7,𝔽2) | D7 | direct (N × H) |
| 44. C24×C7 | C24 | semidirect |
| 47. C23×D7 | C24 | semidirect |
| 48. C7×#41 | C24 | semidirect |
| 49. C2×C7×Q8 | C24 | semidirect |
| 50. C2×C7×D4 | C24 | semidirect |
| 51. C22×C4×C7 | C24 | semidirect |
| 52. C2×C22⋊D7 | C24 | semidirect |
| 53. C22×Dic7 | C24 | semidirect |
| 54. #1161 | C24 | semidirect |
| 56. Q8×D7 | C24 | semidirect |
| 57. #1159 | C24 | semidirect |
| 59. D4×D7 | C24 | semidirect |
| 60. #1157 | C24 | semidirect |
| 61. C2×D28 | C24 | semidirect |
| 63. C2×C4×D7 | C24 | semidirect |
| 64. C2×Dic14 | C24 | semidirect |
| 68. #585 | C2×D7 | semidirect |
| 74. C2×H(5,𝔽2) | C2×D7 | semidirect |
| 95. 2−1+4 | C22×D7 | semidirect |
| 98. H(5,𝔽2) | C22×D7 | semidirect |
| 104. C2×#41 | C22×D7 | semidirect |
| 110. C22×D4 | C22×D7 | semidirect |
| 123. C24 | C23×D7 | semidirect |
| 127. #41 | C23×D7 | semidirect |
| 131. C2×Q8 | C23×D7 | semidirect |
| 136. C2×D4 | C23×D7 | semidirect |
| 140. C22×C4 | C23×D7 | semidirect |
| 144. D7 | H(7,𝔽2) | direct (N × H) |
| 160. C7 | C2×H(7,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(7,𝔽2) | 2 |
| #11744566 | 2 |
| D7×#585 | 2 |
| #11744560 | 2 |
| C2×D7×H(5,𝔽2) | 2 |
| C2×H(7,𝔽2) | 7 |