Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 386 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×#436 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #436 | D5 |
| hypocenter | C5 | #3354 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #436 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #65897 | C2 | semidirect |
| 2. #65889 | C2 | semidirect |
| 3. C5×#436 | C2 | semidirect |
| 4. #65424 | C2 | semidirect |
| 5. #65410 | C2 | semidirect |
| 6. #65135 | C2 | semidirect |
| 7. #65131 | C2 | semidirect |
| 8. #65026 | C2 | semidirect |
| 9. #65024 | C2 | semidirect |
| 10. C4×D40 | C2 | semidirect |
| 11. #64969 | C2 | semidirect |
| 84. #436 | D5 | semidirect |
| 113. C5×C8 | #41 | semidirect |
| 228. C8 | #711 | 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 |
| #65889 | 2 |
| C5×#436 | 2 |
| #65424 | 2 |
| #65410 | 2 |
| #65135 | 2 |
| #65131 | 2 |
| #65026 | 2 |
| #65024 | 2 |
| C4×D40 | 2 |
| #64969 | 2 |
| #3354 | 5 |