Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
840 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 126 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 | C2×#99 |
| commutator | C24 | C22×C4×C29 |
| Frattini, cosocle | C22×D4 | C23×C29 |
| prime radical | C29×#2408 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ C2×C4×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×#1799 | C2 | semidirect |
| 4. C4×C29×#100 | C2 | semidirect |
| 5. C23×D4×C29 | C4 | semidirect |
| 16. C29×#380 | C4 | semidirect |
| 17. C29×#379 | C4 | semidirect |
| 49. C29×#100 | D4 | semidirect |
| 109. #13218 | C29 | direct (N × H) |
| 166. C2×AΣL(1,𝔽16) | C2×C29 | semidirect |
| 167. #1799 | C2×C29 | semidirect |
| 168. C4×#100 | C2×C29 | semidirect |
| 202. C4×C29 | AΣL(1,𝔽16) | semidirect |
| 203. C23×D4 | C4×C29 | semidirect |
| 214. #380 | C4×C29 | semidirect |
| 215. #379 | C4×C29 | semidirect |
| 256. #100 | D4×C29 | semidirect |
| 262. C29 | #13218 | direct (N × H) |
| 405. C4 | C29×AΣL(1,𝔽16) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2408 | 2 |
| C2×C29×AΣL(1,𝔽16) | 2 |
| C29×#1799 | 2 |
| C4×C29×#100 | 2 |
| #13218 | 29 |