Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1008 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 211 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×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C52×#123 | C2 |
| hypercenter | #123 | C5⋊D5 |
| hypocenter | C52 | #560 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×#123 | C2 | semidirect |
| 2. #11717260 | C2 | semidirect |
| 3. #11717257 | C2 | semidirect |
| 4. #11717250 | C2 | semidirect |
| 5. #11717218 | C2 | semidirect |
| 6. #11717215 | C2 | semidirect |
| 7. #11717204 | C2 | semidirect |
| 8. #11717203 | C2 | semidirect |
| 9. #11717202 | C2 | semidirect |
| 12. C22×C4×C52 | C22 | semidirect |
| 15. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 18. C2×C4⋊(C5⋊D5) | C22 | semidirect |
| 19. C2×C4×C5⋊D5 | C22 | semidirect |
| 20. C52×C4⋊C4 | C22 | semidirect |
| 22. C52×C22⋊C4 | C22 | semidirect |
| 27. #87808 | C22 | semidirect |
| 28. C4⋊#1039 | C22 | semidirect |
| 29. #87806 | C22 | semidirect |
| 30. C4×#1039 | C22 | semidirect |
| 52. C5×#123 | D5 | semidirect |
| 80. C22×C4×C5 | C2×D5 | semidirect |
| 88. C5×C4⋊C4 | C2×D5 | semidirect |
| 90. C5×C22⋊C4 | C2×D5 | semidirect |
| 129. #123 | C5⋊D5 | semidirect |
| 135. C52 | #560 | semidirect |
| 162. C22×C4 | C2×C5⋊D5 | semidirect |
| 165. C4⋊C4 | C2×C5⋊D5 | semidirect |
| 172. C22⋊C4 | C2×C5⋊D5 | semidirect |
| 200. C5 | #66000 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×#123 | 2 |
| #11717260 | 2 |
| #11717257 | 2 |
| #11717250 | 2 |
| #11717218 | 2 |
| #11717215 | 2 |
| #11717204 | 2 |
| #11717203 | 2 |
| #11717202 | 2 |
| #66000 | 5 |