Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
136 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#237 | C2 | semidirect |
| 2. D3×C7×D4 | C2 | semidirect |
| 3. C3×C7×D8 | C2 | semidirect |
| 4. C7×#214 | C2 | semidirect |
| 5. C7×#213 | C2 | semidirect |
| 6. C7×#204 | C2 | semidirect |
| 7. C7×ΓL(1,𝔽25) | C2 | semidirect |
| 11. C3×C7×D4 | C22 | semidirect |
| 17. C7×Dic6 | C22 | semidirect |
| 18. C3×C7×C8 | C22 | semidirect |
| 19. C7×C3⋊C8 | C22 | semidirect |
| 25. C7×D8 | D3 | semidirect |
| 27. #911 | C7 | direct (N × H) |
| 42. C7×D4 | C2×D3 | semidirect |
| 46. C7×C8 | C2×D3 | semidirect |
| 47. #237 | C2×C7 | semidirect |
| 48. D3×D4 | C2×C7 | semidirect |
| 49. C3×D8 | C2×C7 | semidirect |
| 50. #214 | C2×C7 | semidirect |
| 51. #213 | C2×C7 | semidirect |
| 52. #204 | C2×C7 | semidirect |
| 53. ΓL(1,𝔽25) | C2×C7 | semidirect |
| 70. C3×D4 | C22×C7 | semidirect |
| 76. Dic6 | C22×C7 | semidirect |
| 77. C3×C8 | C22×C7 | semidirect |
| 78. C3⋊C8 | C22×C7 | semidirect |
| 79. C3×C7 | AGL(1,ℤ/8) | semidirect |
| 85. D8 | D3×C7 | semidirect |
| 106. D4 | C2×D3×C7 | semidirect |
| 110. C8 | C2×D3×C7 | semidirect |
| 111. C7 | #911 | direct (N × H) |
| 125. C3 | C7×AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#237 | 2 |
| D3×C7×D4 | 2 |
| C3×C7×D8 | 2 |
| C7×#214 | 2 |
| C7×#213 | 2 |
| C7×#204 | 2 |
| C7×ΓL(1,𝔽25) | 2 |
| C7×AGL(1,ℤ/8) | 3 |
| #911 | 7 |