Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
244 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #393 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#533 | C2 | semidirect |
| 5. C5×#344 | C2 | semidirect |
| 6. C5×#339 | C2 | semidirect |
| 7. C5×H(3,ℤ/4) | C2 | semidirect |
| 23. #2031 | C5 | direct (N × H) |
| 29. C2×C5×Q8 | Q8 | semidirect |
| 31. C2×C5×D4 | Q8 | semidirect |
| 55. #533 | C2×C5 | semidirect |
| 59. #344 | C2×C5 | semidirect |
| 60. #339 | C2×C5 | semidirect |
| 61. H(3,ℤ/4) | C2×C5 | semidirect |
| 124. C2×Q8 | C5×Q8 | semidirect |
| 126. C2×D4 | C5×Q8 | semidirect |
| 181. C5 | #2031 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#533 | 2 |
| C5×#421 | 2 |
| C2×C5×#104 | 2 |
| C5×#410 | 2 |
| C5×#344 | 2 |
| C5×#339 | 2 |
| C5×H(3,ℤ/4) | 2 |
| #2031 | 5 |