Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
404 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#550 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#557 | C2 | semidirect |
| 3. C5×#492 | C2 | semidirect |
| 4. C5×#461 | C2 | semidirect |
| 5. C5×#460 | C2 | semidirect |
| 6. C5×C8×Q8 | C2 | semidirect |
| 7. C4×C5×SD16 | C2 | semidirect |
| 24. #3392 | C5 | direct (N × H) |
| 45. #557 | C2×C5 | semidirect |
| 47. #492 | C2×C5 | semidirect |
| 48. #461 | C2×C5 | semidirect |
| 49. #460 | C2×C5 | semidirect |
| 50. C8×Q8 | C2×C5 | semidirect |
| 51. C4×SD16 | C2×C5 | semidirect |
| 55. C5×Q8 | SD16 | semidirect |
| 68. C5×C8 | #41 | semidirect |
| 123. Q8 | C5×SD16 | semidirect |
| 136. C8 | C5×#41 | semidirect |
| 137. C5 | #3392 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#557 | 2 |
| C5×#550 | 2 |
| C5×#492 | 2 |
| C5×#461 | 2 |
| C5×#460 | 2 |
| C5×C8×Q8 | 2 |
| C4×C5×SD16 | 2 |
| #3392 | 5 |