Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C29 |
| prime radical | C24×C29 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C23×C4×C29 ◁ C23×C8×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C29×Q32 | C2 | semidirect |
| 3. C2×C29×#358 | C2 | semidirect |
| 5. C29×#1380 | C2 | semidirect |
| 80. #13306 | C29 | direct (N × H) |
| 85. C23×C29 | Q32 | semidirect |
| 110. C22×Q32 | C2×C29 | semidirect |
| 112. C2×#358 | C2×C29 | semidirect |
| 114. #1380 | C2×C29 | semidirect |
| 177. C29 | #13306 | direct (N × H) |
| 216. C23 | C29×Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C29×Q32 | 2 |
| C2×C29×#467 | 2 |
| C2×C29×#358 | 2 |
| C2×C29×C22⋊C16 | 2 |
| C29×#1380 | 2 |
| #13306 | 29 |