Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D5 |
| hypocenter | C5 | #3529 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×H(3,𝔽4) | C2 | semidirect |
| 2. #66018 | C2 | semidirect |
| 3. #66014 | C2 | semidirect |
| 4. #66006 | C2 | semidirect |
| 5. #66002 | C2 | semidirect |
| 6. #65929 | C2 | semidirect |
| 7. #65920 | C2 | semidirect |
| 8. C23×Dic5 | C22 | semidirect |
| 9. C2×#713 | C22 | semidirect |
| 14. #4099 | C22 | semidirect |
| 15. #4092 | C22 | semidirect |
| 16. Q8×Dic5 | C22 | semidirect |
| 17. #4090 | C22 | semidirect |
| 23. #4032 | C22 | semidirect |
| 24. #4030 | C22 | semidirect |
| 25. #4029 | C22 | semidirect |
| 26. #4027 | C22 | semidirect |
| 27. #4025 | C22 | semidirect |
| 28. #4024 | C22 | semidirect |
| 29. #4023 | C22 | semidirect |
| 30. #4021 | C22 | semidirect |
| 31. C42×D5 | C22 | semidirect |
| 32. #4016 | C22 | semidirect |
| 64. H(3,𝔽4) | D5 | semidirect |
| 179. C5 | #3529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×H(3,𝔽4) | 2 |
| #66018 | 2 |
| #66014 | 2 |
| #66006 | 2 |
| #66002 | 2 |
| #65929 | 2 |
| #65920 | 2 |
| #3529 | 5 |