Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
1084 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 98 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 |
| 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 |
|---|---|---|
| 1. C2×C29×B(4,𝔽2) | C2 | semidirect |
| 2. C29×#2679 | C2 | semidirect |
| 3. C2×C29×AΣL(1,𝔽16) | C2 | semidirect |
| 4. C29×#1792 | C2 | semidirect |
| 5. C29×#1393 | C2 | semidirect |
| 22. C25×C29 | D4 | semidirect |
| 26. C29×H(5,𝔽2) | D4 | semidirect |
| 45. C2×C4⋊C4×C29 | D4 | semidirect |
| 52. C2×C22⋊C4×C29 | D4 | semidirect |
| 53. C2×C42×C29 | D4 | semidirect |
| 60. C29×#99 | D4 | semidirect |
| 129. #13343 | C29 | direct (N × H) |
| 195. C2×B(4,𝔽2) | C2×C29 | semidirect |
| 196. #2679 | C2×C29 | semidirect |
| 197. C2×AΣL(1,𝔽16) | C2×C29 | semidirect |
| 198. #1792 | C2×C29 | semidirect |
| 199. #1393 | C2×C29 | semidirect |
| 260. C25 | D4×C29 | semidirect |
| 264. H(5,𝔽2) | D4×C29 | semidirect |
| 283. C2×C4⋊C4 | D4×C29 | semidirect |
| 290. C2×C22⋊C4 | D4×C29 | semidirect |
| 291. C2×C42 | D4×C29 | semidirect |
| 298. #99 | D4×C29 | semidirect |
| 300. C29 | #13343 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C29×B(4,𝔽2) | 2 |
| C29×#2679 | 2 |
| C2×C29×AΣL(1,𝔽16) | 2 |
| C29×#1792 | 2 |
| C29×#1393 | 2 |
| #13343 | 29 |