Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
7014 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.
This group has 7014 conjugacy classes of subgroups — too many to draw. Only its 1134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #120 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C25×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×H(5,𝔽2) | C2 | semidirect |
| 2. C22×C7×#120 | C2 | semidirect |
| 3. C2×C7×B(4,𝔽2) | C2 | direct (N × H) |
| 4. C22×C7×#99 | C2 | semidirect |
| 5. C26×C7 | C22 | semidirect |
| 7. C2×C7×H(5,𝔽2) | C22 | semidirect |
| 13. C2×C7×#120 | C22 | semidirect |
| 15. C22×C7×C22⋊C4 | C22 | semidirect |
| 16. C7×B(4,𝔽2) | C22 | direct (N × H) |
| 17. C2×C7×#99 | C22 | semidirect |
| 18. C22×B(4,𝔽2) | C7 | direct (N × H) |
| 23. C25×C7 | D4 | semidirect |
| 25. C25×C7 | C23 | semidirect |
| 28. C7×H(5,𝔽2) | C23 | semidirect |
| 39. C22×C7×D4 | D4 | semidirect |
| 43. C23×C4×C7 | D4 | semidirect |
| 45. C7×#120 | C23 | semidirect |
| 48. C2×C7×C22⋊C4 | C23 | semidirect |
| 49. C7×#99 | C23 | semidirect |
| 50. C22×H(5,𝔽2) | C2×C7 | semidirect |
| 51. C22×#120 | C2×C7 | semidirect |
| 52. C2×B(4,𝔽2) | C2×C7 | direct (N × H) |
| 53. C22×#99 | C2×C7 | semidirect |
| 71. C24×C7 | C2×D4 | semidirect |
| 73. C24×C7 | C24 | semidirect |
| 86. C2×C7×D4 | C2×D4 | semidirect |
| 93. C22×C4×C7 | C2×D4 | semidirect |
| 96. C7×C22⋊C4 | C24 | semidirect |
| 97. C26 | C22×C7 | semidirect |
| 99. C2×H(5,𝔽2) | C22×C7 | semidirect |
| 105. C2×#120 | C22×C7 | semidirect |
| 107. C22×C22⋊C4 | C22×C7 | semidirect |
| 108. B(4,𝔽2) | C22×C7 | direct (N × H) |
| 109. C2×#99 | C22×C7 | semidirect |
| 135. C23×C7 | C22×D4 | semidirect |
| 147. C2×C4×C7 | C22×D4 | semidirect |
| 152. C25 | C7×D4 | semidirect |
| 154. C25 | C23×C7 | semidirect |
| 157. H(5,𝔽2) | C23×C7 | semidirect |
| 168. C22×D4 | C7×D4 | semidirect |
| 172. C23×C4 | C7×D4 | semidirect |
| 174. #120 | C23×C7 | semidirect |
| 177. C2×C22⋊C4 | C23×C7 | semidirect |
| 178. #99 | C23×C7 | semidirect |
| 194. C22×C7 | B(4,𝔽2) | direct (N × H) |
| 217. C24 | C2×C7×D4 | semidirect |
| 219. C24 | C24×C7 | semidirect |
| 232. C2×D4 | C2×C7×D4 | semidirect |
| 239. C22×C4 | C2×C7×D4 | semidirect |
| 242. C22⋊C4 | C24×C7 | semidirect |
| 247. C2×C7 | C2×B(4,𝔽2) | direct (N × H) |
| 274. C23 | C22×C7×D4 | semidirect |
| 286. C2×C4 | C22×C7×D4 | semidirect |
| 287. C7 | C22×B(4,𝔽2) | direct (N × H) |
| 303. C22 | C7×B(4,𝔽2) | direct (N × H) |
| 313. C2 | C2×C7×B(4,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×H(5,𝔽2) | 2 |
| C22×C7×#120 | 2 |
| C2×C7×B(4,𝔽2) | 2 |
| C22×C7×#99 | 2 |
| C22×B(4,𝔽2) | 7 |