Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | #538 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C22⋊Q8 | C2 | semidirect |
| 2. #4093 | C2 | semidirect |
| 3. #4076 | C2 | semidirect |
| 4. C2×#685 | C2 | semidirect |
| 5. #4044 | C2 | semidirect |
| 6. #4041 | C2 | semidirect |
| 7. #4040 | C2 | semidirect |
| 8. #4028 | C2 | semidirect |
| 9. #4027 | C2 | semidirect |
| 11. C2×C5×Q8 | C22 | semidirect |
| 17. C2×D20 | C22 | semidirect |
| 18. C2×C4×D5 | C22 | semidirect |
| 19. C5×C4⋊C4 | C22 | semidirect |
| 20. C5×C4⋊C4 | C22 | semidirect |
| 26. #685 | C22 | semidirect |
| 27. #685 | C22 | semidirect |
| 28. C4⋊Dic5 | C22 | semidirect |
| 29. #683 | C22 | semidirect |
| 30. C4×Dic5 | C22 | semidirect |
| 55. C22⋊Q8 | D5 | semidirect |
| 79. C2×Q8 | C2×D5 | semidirect |
| 89. C4⋊C4 | C2×D5 | semidirect |
| 90. C4⋊C4 | C2×D5 | semidirect |
| 125. C5 | #538 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C22⋊Q8 | 2 |
| #4093 | 2 |
| #4076 | 2 |
| C2×#685 | 2 |
| #4044 | 2 |
| #4041 | 2 |
| #4040 | 2 |
| #4028 | 2 |
| #4027 | 2 |
| #538 | 5 |