Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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 | C4×C5×D8 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | C4×D8 | D5 |
| hypocenter | C5 | #3354 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #65897 | C2 | semidirect |
| 2. #65885 | C2 | semidirect |
| 3. C4×C5×D8 | C2 | semidirect |
| 4. #65448 | C2 | semidirect |
| 5. #65297 | C2 | semidirect |
| 6. #65296 | C2 | semidirect |
| 7. #65170 | C2 | semidirect |
| 8. #65068 | C2 | semidirect |
| 9. #65066 | C2 | semidirect |
| 10. #64977 | C2 | semidirect |
| 11. #64969 | C2 | semidirect |
| 83. C4×D8 | D5 | semidirect |
| 112. C5×C8 | #41 | semidirect |
| 228. C8 | #709 | semidirect |
| 229. C5 | #3354 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #65897 | 2 |
| #65885 | 2 |
| C4×C5×D8 | 2 |
| #65448 | 2 |
| #65297 | 2 |
| #65296 | 2 |
| #65170 | 2 |
| #65068 | 2 |
| #65066 | 2 |
| #64977 | 2 |
| #64969 | 2 |
| #3354 | 5 |