Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
1392 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1392 conjugacy classes of subgroups — too many to draw. Only its 222 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 | C2×D4 | C24×C29 |
| LCS | G ▷ C2×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×H(3,𝔽4) | C2 | semidirect |
| 2. C2×C29×B(4,𝔽2) | C2 | semidirect |
| 3. C2×C29×#455 | C2 | semidirect |
| 4. C29×#2199 | C2 | direct (N × H) |
| 5. C2×C29×#354 | C2 | semidirect |
| 6. C2×C29×AΣL(1,𝔽16) | C2 | semidirect |
| 11. C29×H(3,𝔽4) | C22 | semidirect |
| 20. C29×B(4,𝔽2) | C22 | semidirect |
| 22. C29×#455 | C22 | semidirect |
| 27. C29×#354 | C22 | semidirect |
| 28. C29×AΣL(1,𝔽16) | C22 | semidirect |
| 29. C25×C29 | D4 | semidirect |
| 37. C22×Q8×C29 | D4 | semidirect |
| 57. C29×#124 | D4 | semidirect |
| 65. C29×#120 | D4 | semidirect |
| 73. C2×C22⋊C4×C29 | D4 | semidirect |
| 75. C2×C42×C29 | D4 | semidirect |
| 89. C24×C29 | C2×D4 | semidirect |
| 103. C2×Q8×C29 | C2×D4 | semidirect |
| 151. C22⋊C4×C29 | C2×D4 | semidirect |
| 154. C42×C29 | C2×D4 | semidirect |
| 155. C2×#2199 | C29 | direct (N × H) |
| 216. C2×H(3,𝔽4) | C2×C29 | semidirect |
| 217. C2×B(4,𝔽2) | C2×C29 | semidirect |
| 218. C2×#455 | C2×C29 | semidirect |
| 219. #2199 | C2×C29 | direct (N × H) |
| 220. C2×#354 | C2×C29 | semidirect |
| 221. C2×AΣL(1,𝔽16) | C2×C29 | semidirect |
| 255. H(3,𝔽4) | C22×C29 | semidirect |
| 264. B(4,𝔽2) | C22×C29 | semidirect |
| 266. #455 | C22×C29 | semidirect |
| 271. #354 | C22×C29 | semidirect |
| 272. AΣL(1,𝔽16) | C22×C29 | semidirect |
| 279. C2×C29 | #2199 | direct (N × H) |
| 281. C25 | D4×C29 | semidirect |
| 289. C22×Q8 | D4×C29 | semidirect |
| 309. #124 | D4×C29 | semidirect |
| 317. #120 | D4×C29 | semidirect |
| 325. C2×C22⋊C4 | D4×C29 | semidirect |
| 327. C2×C42 | D4×C29 | semidirect |
| 334. C29 | C2×#2199 | direct (N × H) |
| 342. C24 | C2×D4×C29 | semidirect |
| 356. C2×Q8 | C2×D4×C29 | semidirect |
| 404. C22⋊C4 | C2×D4×C29 | semidirect |
| 407. C42 | C2×D4×C29 | semidirect |
| 503. C2 | C29×#2199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C29×H(3,𝔽4) | 2 |
| C2×C29×B(4,𝔽2) | 2 |
| C2×C29×#455 | 2 |
| C29×#2199 | 2 |
| C2×C29×#354 | 2 |
| C2×C29×AΣL(1,𝔽16) | 2 |
| C2×#2199 | 29 |