Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C22×C5 | C23×C4 |
| Fitting | C5×#390 | C2 |
| socle, cosocle | C23×C5 | C24 |
| prime radical | C2×#4086 | C2 |
| hypercenter | #390 | D5 |
| hypocenter | C5 | C4×#127 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×D4×Dic5 | C2 | semidirect |
| 3. C2×#4039 | C2 | semidirect |
| 4. C2×#4027 | C2 | semidirect |
| 5. C5×#390 | C2 | semidirect |
| 6. #65265 | C2 | semidirect |
| 7. #65246 | C2 | semidirect |
| 8. #65228 | C2 | semidirect |
| 9. C42×Dic5 | C2 | semidirect |
| 19. #4086 | C4 | semidirect |
| 36. C2×C5×D4 | C2×C4 | semidirect |
| 42. C2×C22⋊D5 | C2×C4 | semidirect |
| 51. C2×D20 | C2×C4 | semidirect |
| 63. C4×Dic5 | C2×C4 | semidirect |
| 64. C4×Dic5 | D4 | semidirect |
| 65. C4×Dic5 | D4 | semidirect |
| 72. #390 | D5 | semidirect |
| 146. C4×C5 | C4×D4 | semidirect |
| 149. Dic5 | C4×D4 | semidirect |
| 150. Dic5 | #127 | semidirect |
| 161. C2×D4 | C4×D5 | semidirect |
| 225. C5 | C4×#127 | semidirect |
| 247. C4 | #4027 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4086 | 2 |
| C2×D4×Dic5 | 2 |
| C2×#4039 | 2 |
| C2×#4027 | 2 |
| C5×#390 | 2 |
| #65265 | 2 |
| #65246 | 2 |
| #65228 | 2 |
| C42×Dic5 | 2 |
| C4×#127 | 5 |