Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1956 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×D42 | C2 | semidirect |
| 2. C2×C5×B(4,𝔽2) | C2 | semidirect |
| 3. C2×C5×#453 | C2 | semidirect |
| 4. C5×#2196 | C2 | direct (N × H) |
| 5. C2×C5×#353 | C2 | semidirect |
| 6. C2×C5×AΣL(1,𝔽16) | C2 | semidirect |
| 7. C2×C5×H(5,𝔽2) | C22 | semidirect |
| 12. C5×D42 | C22 | semidirect |
| 20. C5×B(4,𝔽2) | C22 | semidirect |
| 22. C5×#453 | C22 | semidirect |
| 24. C2×C5×#100 | C22 | semidirect |
| 26. C2×C5×#99 | C22 | semidirect |
| 27. C5×#353 | C22 | semidirect |
| 28. C5×AΣL(1,𝔽16) | C22 | semidirect |
| 29. C2×#2196 | C5 | direct (N × H) |
| 31. C25×C5 | D4 | semidirect |
| 35. C5×H(5,𝔽2) | C23 | semidirect |
| 48. C22×C5×D4 | D4 | semidirect |
| 57. C5×#127 | D4 | semidirect |
| 67. C5×#120 | D4 | semidirect |
| 76. C2×C5×C22⋊C4 | D4 | semidirect |
| 77. C2×C42×C5 | D4 | semidirect |
| 80. C5×#100 | C23 | semidirect |
| 83. C5×#99 | C23 | semidirect |
| 84. C2×D42 | C2×C5 | semidirect |
| 85. C2×B(4,𝔽2) | C2×C5 | semidirect |
| 86. C2×#453 | C2×C5 | semidirect |
| 87. #2196 | C2×C5 | direct (N × H) |
| 88. C2×#353 | C2×C5 | semidirect |
| 89. C2×AΣL(1,𝔽16) | C2×C5 | semidirect |
| 102. C24×C5 | C2×D4 | semidirect |
| 142. C2×C5×D4 | C2×D4 | semidirect |
| 165. C5×C22⋊C4 | C2×D4 | semidirect |
| 167. C42×C5 | C2×D4 | semidirect |
| 168. C2×H(5,𝔽2) | C22×C5 | semidirect |
| 173. D42 | C22×C5 | semidirect |
| 181. B(4,𝔽2) | C22×C5 | semidirect |
| 183. #453 | C22×C5 | semidirect |
| 185. C2×#100 | C22×C5 | semidirect |
| 187. C2×#99 | C22×C5 | semidirect |
| 188. #353 | C22×C5 | semidirect |
| 189. AΣL(1,𝔽16) | C22×C5 | semidirect |
| 258. C25 | C5×D4 | semidirect |
| 262. H(5,𝔽2) | C23×C5 | semidirect |
| 275. C22×D4 | C5×D4 | semidirect |
| 284. #127 | C5×D4 | semidirect |
| 294. #120 | C5×D4 | semidirect |
| 303. C2×C22⋊C4 | C5×D4 | semidirect |
| 304. C2×C42 | C5×D4 | semidirect |
| 307. #100 | C23×C5 | semidirect |
| 310. #99 | C23×C5 | semidirect |
| 356. C24 | C2×C5×D4 | semidirect |
| 396. C2×D4 | C2×C5×D4 | semidirect |
| 419. C22⋊C4 | C2×C5×D4 | semidirect |
| 421. C42 | C2×C5×D4 | semidirect |
| 429. C2×C5 | #2196 | direct (N × H) |
| 498. C5 | C2×#2196 | direct (N × H) |
| 539. C2 | C5×#2196 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×D42 | 2 |
| C2×C5×B(4,𝔽2) | 2 |
| C2×C5×#453 | 2 |
| C5×#2196 | 2 |
| C2×C5×#353 | 2 |
| C2×C5×AΣL(1,𝔽16) | 2 |
| C2×#2196 | 5 |