Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | #576 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #4149 | C2 | semidirect |
| 2. C5×#135 | C2 | semidirect |
| 3. #4095 | C2 | semidirect |
| 4. #4065 | C2 | semidirect |
| 5. #4060 | C2 | semidirect |
| 6. D5×D8 | C2 | semidirect |
| 7. C2×D40 | C2 | semidirect |
| 8. #4047 | C2 | semidirect |
| 9. C5×#41 | C22 | semidirect |
| 10. #713 | C22 | semidirect |
| 12. D4×D5 | C22 | semidirect |
| 16. C2×D20 | C22 | semidirect |
| 17. C5×Q16 | C22 | semidirect |
| 18. C5×SD16 | C22 | semidirect |
| 19. C5×D8 | C22 | semidirect |
| 20. C2×C5×C8 | C22 | semidirect |
| 21. #688 | C22 | semidirect |
| 22. #686 | C22 | semidirect |
| 23. #681 | C22 | semidirect |
| 24. D40 | C22 | semidirect |
| 25. D40 | C22 | semidirect |
| 26. #676 | C22 | semidirect |
| 27. C8×D5 | C22 | semidirect |
| 52. #135 | D5 | semidirect |
| 69. #41 | C2×D5 | semidirect |
| 74. Q16 | C2×D5 | semidirect |
| 76. SD16 | C2×D5 | semidirect |
| 80. D8 | C2×D5 | semidirect |
| 84. C2×C8 | C2×D5 | semidirect |
| 109. C5 | #576 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #4149 | 2 |
| C5×#135 | 2 |
| #4095 | 2 |
| #4065 | 2 |
| #4060 | 2 |
| D5×D8 | 2 |
| C2×D40 | 2 |
| #4047 | 2 |
| #576 | 5 |