Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
438 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 438 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 | C22×D20 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#431 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #431 | D5 |
| hypocenter | C5 | #3207 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #431 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #65811 | C2 | semidirect |
| 2. #65811 | C2 | semidirect |
| 3. C5×#431 | C2 | semidirect |
| 4. #65133 | C2 | semidirect |
| 5. #65023 | C2 | semidirect |
| 6. #64984 | C2 | semidirect |
| 7. #64983 | C2 | semidirect |
| 8. #64982 | C2 | semidirect |
| 9. #64970 | C2 | semidirect |
| 77. #431 | D5 | semidirect |
| 216. C5 | #3207 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #65811 | 2 |
| C5×#431 | 2 |
| #65133 | 2 |
| #65023 | 2 |
| #64984 | 2 |
| #64983 | 2 |
| #64982 | 2 |
| #64970 | 2 |
| #3207 | 5 |