Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
564 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 94 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 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4×C29 |
| Frattini, cosocle | #127 | C23×C29 |
| prime radical | C2×C29×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C22×C4×C29 ◁ C2×C29×#127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C29×#353 | C2 | semidirect |
| 3. C2×C29×#331 | C2 | semidirect |
| 4. C29×#1402 | C2 | direct (N × H) |
| 5. C2×C29×H(5,𝔽2) | C4 | semidirect |
| 10. C29×#453 | C4 | semidirect |
| 14. C2×C29×#100 | C4 | semidirect |
| 17. C29×#353 | C22 | semidirect |
| 19. C29×#331 | C22 | semidirect |
| 23. C29×H(5,𝔽2) | C2×C4 | semidirect |
| 50. C29×#100 | C2×C4 | semidirect |
| 109. C2×#1402 | C29 | direct (N × H) |
| 163. C2×#353 | C2×C29 | semidirect |
| 164. C2×#331 | C2×C29 | semidirect |
| 165. #1402 | C2×C29 | direct (N × H) |
| 193. C2×H(5,𝔽2) | C4×C29 | semidirect |
| 198. #453 | C4×C29 | semidirect |
| 202. C2×#100 | C4×C29 | semidirect |
| 205. #353 | C22×C29 | semidirect |
| 207. #331 | C22×C29 | semidirect |
| 215. C2×C29 | #1402 | direct (N × H) |
| 220. H(5,𝔽2) | C2×C4×C29 | semidirect |
| 247. #100 | C2×C4×C29 | semidirect |
| 250. C29 | C2×#1402 | direct (N × H) |
| 393. C2 | C29×#1402 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C29×#453 | 2 |
| C2×C29×#353 | 2 |
| C2×C29×#331 | 2 |
| C29×#1402 | 2 |
| C2×#1402 | 29 |