Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
2590 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 2590 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C5 | C22×S4 |
| commutator, cosocle | C2×A4 | C24×C5 |
| Frattini | C2 | C23×C5×S4 |
| Fitting | C23×C5×D4 | D3 |
| socle | C24×C5 | C22×D3 |
| hypercenter | C2×C5×D4 | S4 |
| hypocenter | A4 | C22×C5×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C5×S4 | C2 | semidirect |
| 2. C2×C5×D4×A4 | C2 | semidirect |
| 3. C2×C5×GL(2,ℤ/4) | C2 | semidirect |
| 4. C5×D4×S4 | C2 | direct (N × H) |
| 5. C2×C5×C4⋊S4 | C2 | semidirect |
| 6. C2×C4×C5×S4 | C2 | semidirect |
| 8. C23×C5×A4 | C22 | semidirect |
| 11. C22×C5×S4 | C22 | semidirect |
| 12. C22×C5×S4 | C22 | semidirect |
| 14. C5×D4×A4 | C22 | semidirect |
| 15. C2×C4×C5×A4 | C22 | semidirect |
| 16. C5×GL(2,ℤ/4) | C22 | semidirect |
| 17. C2×C5×SL(2,ℤ/4) | C22 | semidirect |
| 18. C5×C4⋊S4 | C22 | semidirect |
| 19. C4×C5×S4 | C22 | semidirect |
| 20. C2×D4×S4 | C5 | direct (N × H) |
| 23. C23×C5×D4 | D3 | semidirect |
| 37. C22×C5×A4 | C23 | semidirect |
| 40. C2×C5×S4 | D4 | direct (N × H) |
| 42. C2×C5×S4 | C23 | semidirect |
| 48. C4×C5×A4 | C23 | semidirect |
| 49. C5×SL(2,ℤ/4) | C23 | semidirect |
| 50. C23×S4 | C2×C5 | semidirect |
| 51. C2×D4×A4 | C2×C5 | semidirect |
| 52. C2×GL(2,ℤ/4) | C2×C5 | semidirect |
| 53. D4×S4 | C2×C5 | direct (N × H) |
| 54. C2×C4⋊S4 | C2×C5 | semidirect |
| 55. C2×C4×S4 | C2×C5 | semidirect |
| 57. C25×C5 | C2×D3 | semidirect |
| 78. C22×C5×D4 | C2×D3 | semidirect |
| 81. C23×C4×C5 | C2×D3 | semidirect |
| 108. C2×C5×A4 | C2×D4 | semidirect |
| 111. C5×S4 | C2×D4 | direct (N × H) |
| 118. C23×A4 | C22×C5 | semidirect |
| 121. C22×S4 | C22×C5 | semidirect |
| 122. C22×S4 | C22×C5 | semidirect |
| 124. D4×A4 | C22×C5 | semidirect |
| 125. C2×C4×A4 | C22×C5 | semidirect |
| 126. GL(2,ℤ/4) | C22×C5 | semidirect |
| 127. C2×SL(2,ℤ/4) | C22×C5 | semidirect |
| 128. C4⋊S4 | C22×C5 | semidirect |
| 129. C4×S4 | C22×C5 | semidirect |
| 144. C24×C5 | C22×D3 | semidirect |
| 182. C2×C5×D4 | S4 | direct (N × H) |
| 196. C22×C4×C5 | C22×D3 | semidirect |
| 208. C23×D4 | C5×D3 | semidirect |
| 227. C5×A4 | C22×D4 | semidirect |
| 233. C22×A4 | C23×C5 | semidirect |
| 236. C2×S4 | C5×D4 | direct (N × H) |
| 238. C2×S4 | C23×C5 | semidirect |
| 244. C4×A4 | C23×C5 | semidirect |
| 245. SL(2,ℤ/4) | C23×C5 | semidirect |
| 279. C23×C5 | D3×D4 | semidirect |
| 280. C23×C5 | C2×S4 | semidirect |
| 298. C5×D4 | C2×S4 | direct (N × H) |
| 316. C2×C4×C5 | C2×S4 | semidirect |
| 318. C25 | C2×C5×D3 | semidirect |
| 339. C22×D4 | C2×C5×D3 | semidirect |
| 342. C23×C4 | C2×C5×D3 | semidirect |
| 373. C2×A4 | C2×C5×D4 | semidirect |
| 376. S4 | C2×C5×D4 | direct (N × H) |
| 411. C22×C5 | C2×D3×D4 | semidirect |
| 413. C22×C5 | C22×S4 | semidirect |
| 419. C4×C5 | C22×S4 | semidirect |
| 434. C24 | C22×C5×D3 | semidirect |
| 472. C2×D4 | C5×S4 | direct (N × H) |
| 486. C22×C4 | C22×C5×D3 | semidirect |
| 505. A4 | C22×C5×D4 | semidirect |
| 515. C2×C5 | D4×S4 | direct (N × H) |
| 550. C23 | C5×D3×D4 | semidirect |
| 551. C23 | C2×C5×S4 | semidirect |
| 569. D4 | C2×C5×S4 | direct (N × H) |
| 587. C2×C4 | C2×C5×S4 | semidirect |
| 593. C5 | C2×D4×S4 | direct (N × H) |
| 622. C22 | C2×C5×D3×D4 | semidirect |
| 624. C22 | C22×C5×S4 | semidirect |
| 630. C4 | C22×C5×S4 | semidirect |
| 639. C2 | C5×D4×S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C5×S4 | 2 |
| C2×C5×D4×A4 | 2 |
| C2×C5×GL(2,ℤ/4) | 2 |
| C5×D4×S4 | 2 |
| C2×C5×C4⋊S4 | 2 |
| C2×C4×C5×S4 | 2 |
| C2×C5×D42 | 3 |
| C2×C5×D3×D4 | 4 |
| C2×D4×S4 | 5 |