Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 712 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, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C2×C5×#120 | C2 |
| hypercenter | C2×#120 | D5 |
| hypocenter | C5 | #2812 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×#120 | C2 | semidirect |
| 2. C2×#4099 | C2 | semidirect |
| 3. #65536 | C2 | semidirect |
| 4. #65512 | C2 | semidirect |
| 5. #65511 | C2 | semidirect |
| 6. #65248 | C2 | semidirect |
| 7. #65241 | C2 | semidirect |
| 11. C22×C22⋊D5 | C22 | semidirect |
| 18. C2×C22⋊Dic5 | C22 | semidirect |
| 19. C2×#685 | C22 | semidirect |
| 20. C2×#683 | C22 | semidirect |
| 21. #3963 | C22 | semidirect |
| 22. #3963 | C22 | semidirect |
| 26. C24×C5 | D4 | semidirect |
| 49. C2×#120 | D5 | semidirect |
| 124. C24 | C22⋊D5 | semidirect |
| 181. C5 | #2812 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#120 | 2 |
| C2×#4099 | 2 |
| #65536 | 2 |
| #65512 | 2 |
| #65511 | 2 |
| #65248 | 2 |
| #65241 | 2 |
| #2812 | 5 |