Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
788 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 181 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 | C5×#384 | C2 |
| hypercenter | #384 | D5 |
| hypocenter | C5 | #2537 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4041 | C2 | semidirect |
| 2. C2×C4×D20 | C2 | semidirect |
| 3. C5×#384 | C2 | semidirect |
| 4. #65280 | C2 | semidirect |
| 5. #65276 | C2 | semidirect |
| 28. C2×D20 | D4 | semidirect |
| 34. C42×C5 | D4 | semidirect |
| 41. #384 | D5 | semidirect |
| 91. C4×C5 | #121 | semidirect |
| 111. C42 | D20 | semidirect |
| 140. C5 | #2537 | semidirect |
| 156. C4 | #4041 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4041 | 2 |
| C2×C4×D20 | 2 |
| C5×#384 | 2 |
| #65280 | 2 |
| #65276 | 2 |
| #2537 | 5 |