Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#518 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #518 | D5 |
| hypocenter | C5 | #3041 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #518 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#518 | C2 | semidirect |
| 2. C2×#4078 | C2 | semidirect |
| 3. C22×#686 | C2 | semidirect |
| 4. #65859 | C2 | semidirect |
| 5. #65305 | C2 | semidirect |
| 6. #65304 | C2 | semidirect |
| 7. #65291 | C2 | semidirect |
| 8. #65290 | C2 | semidirect |
| 9. #65288 | C2 | semidirect |
| 14. C4×C5×D4 | C22 | semidirect |
| 18. C2×#687 | C22 | semidirect |
| 22. C2×#686 | C22 | semidirect |
| 27. #4021 | C22 | semidirect |
| 28. #4020 | C22 | semidirect |
| 29. #3942 | C22 | semidirect |
| 30. #3941 | C22 | semidirect |
| 31. #3941 | C22 | semidirect |
| 32. C4⋊#168 | C22 | semidirect |
| 38. C2×C5×D4 | D4 | semidirect |
| 71. #518 | D5 | semidirect |
| 123. C4×D4 | C2×D5 | semidirect |
| 162. C2×D4 | D20 | semidirect |
| 219. C5 | #3041 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#518 | 2 |
| C2×#4078 | 2 |
| C22×#686 | 2 |
| #65859 | 2 |
| #65305 | 2 |
| #65304 | 2 |
| #65291 | 2 |
| #65290 | 2 |
| #65288 | 2 |
| #3041 | 5 |