Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
544 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 70 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×C22⋊C4 | C23×C29 |
| prime radical | C29×#2021 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C2×C22⋊C4×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×#2735 | C2 | semidirect |
| 3. C29×#1613 | C2 | semidirect |
| 4. C29×#1607 | C2 | semidirect |
| 5. C29×#1352 | C2 | semidirect |
| 6. C29×#1344 | C2 | semidirect |
| 7. C29×#1316 | C2 | semidirect |
| 38. C2×C4⋊C4×C29 | D4 | semidirect |
| 39. C2×C4⋊C4×C29 | D4 | semidirect |
| 63. C29×#95 | D4 | semidirect |
| 64. C29×#95 | D4 | semidirect |
| 119. #12192 | C29 | direct (N × H) |
| 182. #2735 | C2×C29 | semidirect |
| 184. #1613 | C2×C29 | semidirect |
| 185. #1607 | C2×C29 | semidirect |
| 186. #1352 | C2×C29 | semidirect |
| 187. #1344 | C2×C29 | semidirect |
| 188. #1316 | C2×C29 | semidirect |
| 265. C2×C4⋊C4 | D4×C29 | semidirect |
| 266. C2×C4⋊C4 | D4×C29 | semidirect |
| 290. #95 | D4×C29 | semidirect |
| 291. #95 | D4×C29 | semidirect |
| 292. C29 | #12192 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2735 | 2 |
| C29×#2021 | 2 |
| C29×#1613 | 2 |
| C29×#1607 | 2 |
| C29×#1352 | 2 |
| C29×#1344 | 2 |
| C29×#1316 | 2 |
| #12192 | 29 |