Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
466 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4076 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#568 | C2 |
| socle | C2×C5 | C2×#121 |
| hypercenter | #568 | D5 |
| hypocenter | C5 | #3068 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #568 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#568 | C2 | semidirect |
| 2. #66156 | C2 | semidirect |
| 3. C2×#4095 | C2 | semidirect |
| 4. C2×#4054 | C2 | semidirect |
| 5. #65431 | C2 | semidirect |
| 6. #65430 | C2 | semidirect |
| 7. #65421 | C2 | semidirect |
| 8. C2×#3955 | C2 | semidirect |
| 9. #65414 | C2 | semidirect |
| 14. C5×#131 | C22 | semidirect |
| 18. #4097 | C22 | semidirect |
| 22. #4095 | C22 | semidirect |
| 31. C2×D40 | C22 | semidirect |
| 32. C2×#677 | C22 | semidirect |
| 33. #3956 | C22 | semidirect |
| 34. #3955 | C22 | semidirect |
| 35. #3955 | C22 | semidirect |
| 36. #3951 | C22 | semidirect |
| 87. #568 | D5 | semidirect |
| 149. #131 | C2×D5 | semidirect |
| 269. C5 | #3068 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#568 | 2 |
| #66156 | 2 |
| C2×#4095 | 2 |
| C2×#4054 | 2 |
| #65431 | 2 |
| #65430 | 2 |
| #65421 | 2 |
| C2×#3955 | 2 |
| #65414 | 2 |
| #3068 | 5 |