Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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×#126 | C2 |
| hypercenter | #126 | D5 |
| hypocenter | C5 | #560 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#126 | C2 | semidirect |
| 2. #4041 | C2 | semidirect |
| 3. #4040 | C2 | semidirect |
| 4. #4030 | C2 | semidirect |
| 5. #4028 | C2 | semidirect |
| 6. #4022 | C2 | semidirect |
| 7. #4020 | C2 | semidirect |
| 12. C2×D20 | C22 | semidirect |
| 13. C2×C4×D5 | C22 | semidirect |
| 14. C5×C4⋊C4 | C22 | semidirect |
| 16. C42×C5 | C22 | semidirect |
| 19. #685 | C22 | semidirect |
| 20. #683 | C22 | semidirect |
| 37. #126 | D5 | semidirect |
| 60. C4⋊C4 | C2×D5 | semidirect |
| 65. C42 | C2×D5 | semidirect |
| 84. C5 | #560 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#126 | 2 |
| #4041 | 2 |
| #4040 | 2 |
| #4030 | 2 |
| #4028 | 2 |
| #4022 | 2 |
| #4020 | 2 |
| #560 | 5 |