Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
984 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 984 conjugacy classes of subgroups — too many to draw. Only its 221 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×#126 | C2 |
| prime radical | C23×C5⋊D5 | C22 |
| hypercenter | #126 | C5⋊D5 |
| hypocenter | C52 | C2×#126 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×#126 | C2 | semidirect |
| 2. #11717218 | C2 | semidirect |
| 3. C4⋊C4×C5⋊D5 | C2 | semidirect |
| 5. #11717199 | C2 | semidirect |
| 6. #11717196 | C2 | semidirect |
| 7. C42×C5⋊D5 | C2 | semidirect |
| 11. C52×C4⋊C4 | C22 | semidirect |
| 13. C42×C52 | C22 | semidirect |
| 17. C4⋊#1039 | C22 | semidirect |
| 18. #87806 | C22 | semidirect |
| 19. #87806 | C22 | semidirect |
| 20. C4×#1039 | C22 | semidirect |
| 32. C5×#126 | D5 | semidirect |
| 51. C5×C4⋊C4 | C2×D5 | semidirect |
| 53. C42×C5 | C2×D5 | semidirect |
| 65. C5⋊D5 | #126 | direct (N × H) |
| 79. #126 | C5⋊D5 | direct (N × H) |
| 83. C52 | C2×#126 | semidirect |
| 99. C4⋊C4 | C2×C5⋊D5 | semidirect |
| 105. C42 | C2×C5⋊D5 | semidirect |
| 120. C5 | D5×#126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×#126 | 2 |
| #11717218 | 2 |
| C4⋊C4×C5⋊D5 | 2 |
| C22⋊C4×C5⋊D5 | 2 |
| #11717199 | 2 |
| #11717196 | 2 |
| C42×C5⋊D5 | 2 |
| D5×#126 | 5 |