Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 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×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#124 | C2 |
| hypercenter | #124 | D5 |
| hypocenter | C5 | H(3,𝔽4) |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#124 | C2 | semidirect |
| 2. #4093 | C2 | semidirect |
| 3. #4083 | C2 | semidirect |
| 4. #4031 | C2 | semidirect |
| 5. #4028 | C2 | semidirect |
| 6. #4021 | C2 | semidirect |
| 7. C23×D5 | C22 | semidirect |
| 8. C2×C5×Q8 | C22 | semidirect |
| 13. C2×Dic10 | C22 | semidirect |
| 14. C5×C22⋊C4 | C22 | semidirect |
| 15. C42×C5 | C22 | semidirect |
| 16. C22⋊Dic5 | C22 | semidirect |
| 17. #685 | C22 | semidirect |
| 18. #685 | C22 | semidirect |
| 19. #685 | C22 | semidirect |
| 20. C4×Dic5 | C22 | semidirect |
| 39. #124 | D5 | semidirect |
| 55. C2×Q8 | C2×D5 | semidirect |
| 64. C22⋊C4 | C2×D5 | semidirect |
| 66. C42 | C2×D5 | semidirect |
| 85. C5 | H(3,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#124 | 2 |
| #4093 | 2 |
| #4083 | 2 |
| #4031 | 2 |
| #4028 | 2 |
| #4021 | 2 |
| H(3,𝔽4) | 5 |