Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4086 |
| commutator, cosocle | C42×C5 | C23 |
| Frattini | C42 | C22×D5 |
| Fitting | C5×#493 | C2 |
| socle | C22×C5 | #127 |
| hypercenter | #493 | D5 |
| hypocenter | C5 | #1721 |
| LCS | G ▷ C42×C5 ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#493 | C2 | semidirect |
| 2. #65365 | C2 | semidirect |
| 3. #65362 | C2 | semidirect |
| 4. #65362 | C2 | semidirect |
| 5. #65359 | C2 | semidirect |
| 6. #64983 | C2 | semidirect |
| 7. #64674 | C2 | semidirect |
| 44. #493 | D5 | semidirect |
| 68. #168 | D8 | semidirect |
| 142. C5 | #1721 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#493 | 2 |
| #65365 | 2 |
| #65362 | 2 |
| #65359 | 2 |
| #64983 | 2 |
| #64674 | 2 |
| #1721 | 5 |