Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
532 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#493 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #493 | D5 |
| hypocenter | C5 | #3205 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #66050 | C2 | semidirect |
| 2. D5×#127 | C2 | semidirect |
| 3. C5×#493 | C2 | semidirect |
| 4. #65445 | C2 | semidirect |
| 5. #65439 | C2 | semidirect |
| 6. #65362 | C2 | semidirect |
| 7. #65359 | C2 | semidirect |
| 8. #64983 | C2 | semidirect |
| 9. #64974 | C2 | semidirect |
| 14. C2×C5×D8 | C22 | semidirect |
| 16. C5×#127 | C22 | semidirect |
| 24. D4×Dic5 | C22 | semidirect |
| 25. C2×#686 | C22 | semidirect |
| 28. #4015 | C22 | semidirect |
| 29. C4×C5×C8 | C22 | semidirect |
| 30. #3964 | C22 | semidirect |
| 31. #3953 | C22 | semidirect |
| 33. #3946 | C22 | semidirect |
| 34. C4×#168 | C22 | semidirect |
| 74. #493 | D5 | semidirect |
| 110. C4×D5 | D8 | semidirect |
| 119. C2×D8 | C2×D5 | semidirect |
| 126. #127 | C2×D5 | semidirect |
| 138. C4×C8 | C2×D5 | semidirect |
| 236. C5 | #3205 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66050 | 2 |
| D5×#127 | 2 |
| C5×#493 | 2 |
| #65445 | 2 |
| #65439 | 2 |
| #65362 | 2 |
| #65359 | 2 |
| #64983 | 2 |
| #64974 | 2 |
| #3205 | 5 |