Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
496 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×C4⋊Q8 | C4 |
| prime radical, cosocle | C22×D5 | C24 |
| hypercenter | C4⋊Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #2369 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. C5×C4⋊Q8 | C4 | semidirect |
| 13. #4090 | C4 | semidirect |
| 15. #4034 | C4 | semidirect |
| 17. #4015 | C4 | semidirect |
| 33. C4⋊AGL(1,𝔽5) | Q8 | semidirect |
| 73. C4⋊Q8 | AGL(1,𝔽5) | semidirect |
| 144. C5 | #2369 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×C4⋊Q8 | 2 |
| (C2×AGL(1,𝔽5))⋊Q8 | 2 |
| #65712 | 2 |
| C4⋊C4×AGL(1,𝔽5) | 2 |
| C4×C4⋊AGL(1,𝔽5) | 2 |
| #2369 | 5 |