Finite Group DatabasebetaFinite Group DatabasebetaOrder 3200.
7362 conjugacy classes of subgroups; only the 1854 normal subgroups are shown.
This group has 7362 conjugacy classes of subgroups — too many to draw. Only its 1854 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | #1044 |
| commutator, hypocenter | C52 | C23×C42 |
| Frattini | C2 | C24×#1044 |
| Fitting | C23×C4×C52 | C4 |
| socle | C24×C52 | C2×C4 |
| prime radical | C24×C5⋊D5 | C22 |
| cosocle | C2×C5⋊D5 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C4×#1044 | C2 | direct (N × H) |
| 5. C23×#1044 | C4 | direct (N × H) |
| 7. C23×C4×C52 | C4 | semidirect |
| 8. C23×#1039 | C4 | semidirect |
| 9. C22×C4×C5⋊D5 | C4 | semidirect |
| 11. C2×C4×#1044 | C22 | direct (N × H) |
| 17. C22×#1044 | C2×C4 | direct (N × H) |
| 19. C22×C4×C52 | C2×C4 | semidirect |
| 20. C22×#1039 | C2×C4 | semidirect |
| 21. C2×C4×C5⋊D5 | C2×C4 | semidirect |
| 23. C4×#1044 | C23 | direct (N × H) |
| 28. C23×C52 | C42 | semidirect |
| 30. C22×C5⋊D5 | C42 | semidirect |
| 34. C2×#1044 | C22×C4 | direct (N × H) |
| 36. C2×C4×C52 | C22×C4 | semidirect |
| 37. C2×#1039 | C22×C4 | semidirect |
| 38. C4×C5⋊D5 | C22×C4 | semidirect |
| 45. C23×C4×C5 | AGL(1,𝔽5) | semidirect |
| 54. C22×C52 | C2×C42 | semidirect |
| 56. C2×C5⋊D5 | C2×C42 | semidirect |
| 60. #1044 | C23×C4 | direct (N × H) |
| 61. C4×C52 | C23×C4 | semidirect |
| 62. #1039 | C23×C4 | semidirect |
| 74. C22×C4×C5 | C2×AGL(1,𝔽5) | semidirect |
| 85. C2×C52 | C22×C42 | semidirect |
| 87. C5⋊D5 | C22×C42 | semidirect |
| 91. C23×C5 | C4×AGL(1,𝔽5) | semidirect |
| 104. C2×C4×C5 | C22×AGL(1,𝔽5) | semidirect |
| 117. C23×C4 | #1044 | direct (N × H) |
| 119. C52 | C23×C42 | semidirect |
| 122. C22×C5 | C2×C4×AGL(1,𝔽5) | semidirect |
| 134. C4×C5 | C23×AGL(1,𝔽5) | semidirect |
| 146. C22×C4 | C2×#1044 | direct (N × H) |
| 150. C2×C5 | C22×C4×AGL(1,𝔽5) | semidirect |
| 160. C23 | C4×#1044 | direct (N × H) |
| 167. C2×C4 | C22×#1044 | direct (N × H) |
| 169. C5 | C23×C4×AGL(1,𝔽5) | semidirect |
| 174. C22 | C2×C4×#1044 | direct (N × H) |
| 178. C4 | C23×#1044 | direct (N × H) |
| 181. C2 | C22×C4×#1044 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#1044 | 2 |
| C23×C4×C5⋊D5 | 2 |
| C22×C4×#1044 | 2 |
| C23×C4×AGL(1,𝔽5) | 5 |