Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
676 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 173 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, prime radical | C2×C5×#127 | C2 |
| hypercenter | C2×#127 | D5 |
| hypocenter | C5 | #2548 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4082 | C2 | semidirect |
| 3. C2×C4×Dic10 | C2 | semidirect |
| 4. #65511 | C2 | semidirect |
| 5. #65510 | C2 | semidirect |
| 6. #65510 | C2 | semidirect |
| 7. C4×C4⋊Dic5 | C2 | semidirect |
| 41. C2×Dic10 | D4 | semidirect |
| 110. C4×C5 | #124 | semidirect |
| 170. C5 | #2548 | semidirect |
| 185. C4 | #4082 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#127 | 2 |
| C2×#4082 | 2 |
| C2×C4×Dic10 | 2 |
| #65511 | 2 |
| #65510 | 2 |
| C4×C4⋊Dic5 | 2 |
| #2548 | 5 |