Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
524 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 133 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×#418 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #418 | D5 |
| hypocenter | C5 | #3058 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #418 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #66139 | C2 | semidirect |
| 2. C22×#686 | C2 | semidirect |
| 3. C2×#4053 | C2 | semidirect |
| 4. #65859 | C2 | semidirect |
| 5. C5×#418 | C2 | semidirect |
| 6. #65420 | C2 | semidirect |
| 7. #65287 | C2 | semidirect |
| 8. #65071 | C2 | semidirect |
| 9. #65064 | C2 | semidirect |
| 17. #4084 | C22 | semidirect |
| 25. C2×#676 | C22 | semidirect |
| 26. C2×C8×D5 | C22 | semidirect |
| 27. #4041 | C22 | semidirect |
| 31. C5×#102 | C22 | semidirect |
| 32. #3953 | C22 | semidirect |
| 33. #3940 | C22 | semidirect |
| 34. #3939 | C22 | semidirect |
| 67. C2×#168 | D4 | semidirect |
| 80. #418 | D5 | semidirect |
| 136. #102 | C2×D5 | semidirect |
| 227. C5 | #3058 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66139 | 2 |
| C22×#686 | 2 |
| C2×#4053 | 2 |
| #65859 | 2 |
| C5×#418 | 2 |
| #65420 | 2 |
| #65287 | 2 |
| #65071 | 2 |
| #65064 | 2 |
| #3058 | 5 |