Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
776 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23×C29 |
| prime radical | C29×#2679 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C22×D4×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C29×AΣL(1,𝔽16) | C2 | semidirect |
| 3. C29×#1889 | C2 | semidirect |
| 4. C29×#1856 | C2 | semidirect |
| 5. C29×#1393 | C2 | semidirect |
| 49. C29×#100 | D4 | semidirect |
| 104. #13419 | C29 | direct (N × H) |
| 156. C2×AΣL(1,𝔽16) | C2×C29 | semidirect |
| 157. #1889 | C2×C29 | semidirect |
| 158. #1856 | C2×C29 | semidirect |
| 159. #1393 | C2×C29 | semidirect |
| 242. #100 | D4×C29 | semidirect |
| 246. C29 | #13419 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2679 | 2 |
| C2×C29×AΣL(1,𝔽16) | 2 |
| C29×#1889 | 2 |
| C29×#1856 | 2 |
| C29×#1393 | 2 |
| #13419 | 29 |