Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1720 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C23×S4 |
| commutator, cosocle | C2×A4 | C24×C5 |
| Frattini | C2 | C23×C5×S4 |
| Fitting | C22×C5×#41 | D3 |
| socle | C23×C5 | C23×D3 |
| hypercenter | C5×#41 | S4 |
| hypocenter | A4 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C5 ◁ C5×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×A4×#41 | C2 | semidirect |
| 2. C5×#5919 | C2 | semidirect |
| 3. C5×D4×S4 | C2 | semidirect |
| 4. C5×#5912 | C2 | semidirect |
| 5. C2×C5×C4⋊S4 | C2 | semidirect |
| 7. C22×C5×S4 | C22 | semidirect |
| 9. C5×Q8×A4 | C22 | semidirect |
| 10. C5×D4×A4 | C22 | semidirect |
| 11. C2×C4×C5×A4 | C22 | semidirect |
| 12. C5×GL(2,ℤ/4) | C22 | semidirect |
| 13. C5×C4⋊S4 | C22 | semidirect |
| 14. C5×C4⋊S4 | C22 | semidirect |
| 15. C4×C5×S4 | C22 | semidirect |
| 16. C5×A4⋊Q8 | C22 | semidirect |
| 17. #87489 | C5 | direct (N × H) |
| 19. C22×C5×#41 | D3 | semidirect |
| 40. A4×#41 | C2×C5 | semidirect |
| 41. #5919 | C2×C5 | semidirect |
| 42. D4×S4 | C2×C5 | semidirect |
| 43. #5912 | C2×C5 | semidirect |
| 44. C2×C4⋊S4 | C2×C5 | semidirect |
| 55. C22×C5×Q8 | C2×D3 | semidirect |
| 59. C22×C5×D4 | C2×D3 | semidirect |
| 60. C23×C4×C5 | C2×D3 | semidirect |
| 96. C22×S4 | C22×C5 | semidirect |
| 98. Q8×A4 | C22×C5 | semidirect |
| 99. D4×A4 | C22×C5 | semidirect |
| 100. C2×C4×A4 | C22×C5 | semidirect |
| 101. GL(2,ℤ/4) | C22×C5 | semidirect |
| 102. C4⋊S4 | C22×C5 | semidirect |
| 103. C4⋊S4 | C22×C5 | semidirect |
| 104. C4×S4 | C22×C5 | semidirect |
| 105. A4⋊Q8 | C22×C5 | semidirect |
| 120. C5×#41 | S4 | semidirect |
| 163. C22×#41 | C5×D3 | semidirect |
| 176. C5×A4 | H(5,𝔽2) | semidirect |
| 205. C5×Q8 | C2×S4 | semidirect |
| 219. C5×D4 | C2×S4 | semidirect |
| 234. C2×C4×C5 | C2×S4 | semidirect |
| 245. C22×Q8 | C2×C5×D3 | semidirect |
| 249. C22×D4 | C2×C5×D3 | semidirect |
| 250. C23×C4 | C2×C5×D3 | semidirect |
| 300. C22×C5 | #1009 | semidirect |
| 323. #41 | C5×S4 | semidirect |
| 369. A4 | C5×H(5,𝔽2) | semidirect |
| 393. Q8 | C2×C5×S4 | semidirect |
| 407. D4 | C2×C5×S4 | semidirect |
| 422. C2×C4 | C2×C5×S4 | semidirect |
| 426. C5 | #87489 | direct (N × H) |
| 439. C22 | C5×#1009 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×A4×#41 | 2 |
| C5×#5919 | 2 |
| C5×D4×S4 | 2 |
| C5×#5912 | 2 |
| C2×C5×C4⋊S4 | 2 |
| C5×#3488 | 3 |
| C5×#1009 | 4 |
| #87489 | 5 |