Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1392 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1392 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×H(3,𝔽4) | C2 | semidirect |
| 2. C2×C5×B(4,𝔽2) | C2 | semidirect |
| 3. C2×C5×#455 | C2 | semidirect |
| 4. C5×#2199 | C2 | direct (N × H) |
| 5. C2×C5×#354 | C2 | semidirect |
| 6. C2×C5×AΣL(1,𝔽16) | C2 | semidirect |
| 11. C5×H(3,𝔽4) | C22 | semidirect |
| 20. C5×B(4,𝔽2) | C22 | semidirect |
| 22. C5×#455 | C22 | semidirect |
| 27. C5×#354 | C22 | semidirect |
| 28. C5×AΣL(1,𝔽16) | C22 | semidirect |
| 29. C2×#2199 | C5 | direct (N × H) |
| 30. C25×C5 | D4 | semidirect |
| 38. C22×C5×Q8 | D4 | semidirect |
| 58. C5×#124 | D4 | semidirect |
| 66. C5×#120 | D4 | semidirect |
| 74. C2×C5×C22⋊C4 | D4 | semidirect |
| 76. C2×C42×C5 | D4 | semidirect |
| 83. C2×H(3,𝔽4) | C2×C5 | semidirect |
| 84. C2×B(4,𝔽2) | C2×C5 | semidirect |
| 85. C2×#455 | C2×C5 | semidirect |
| 86. #2199 | C2×C5 | direct (N × H) |
| 87. C2×#354 | C2×C5 | semidirect |
| 88. C2×AΣL(1,𝔽16) | C2×C5 | semidirect |
| 96. C24×C5 | C2×D4 | semidirect |
| 110. C2×C5×Q8 | C2×D4 | semidirect |
| 158. C5×C22⋊C4 | C2×D4 | semidirect |
| 161. C42×C5 | C2×D4 | semidirect |
| 166. H(3,𝔽4) | C22×C5 | semidirect |
| 175. B(4,𝔽2) | C22×C5 | semidirect |
| 177. #455 | C22×C5 | semidirect |
| 182. #354 | C22×C5 | semidirect |
| 183. AΣL(1,𝔽16) | C22×C5 | semidirect |
| 244. C25 | C5×D4 | semidirect |
| 252. C22×Q8 | C5×D4 | semidirect |
| 272. #124 | C5×D4 | semidirect |
| 280. #120 | C5×D4 | semidirect |
| 288. C2×C22⋊C4 | C5×D4 | semidirect |
| 290. C2×C42 | C5×D4 | semidirect |
| 333. C24 | C2×C5×D4 | semidirect |
| 347. C2×Q8 | C2×C5×D4 | semidirect |
| 395. C22⋊C4 | C2×C5×D4 | semidirect |
| 398. C42 | C2×C5×D4 | semidirect |
| 405. C2×C5 | #2199 | direct (N × H) |
| 467. C5 | C2×#2199 | direct (N × H) |
| 503. C2 | C5×#2199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×H(3,𝔽4) | 2 |
| C2×C5×B(4,𝔽2) | 2 |
| C2×C5×#455 | 2 |
| C5×#2199 | 2 |
| C2×C5×#354 | 2 |
| C2×C5×AΣL(1,𝔽16) | 2 |
| C2×#2199 | 5 |