Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
352 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 74 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×C4⋊C4 | C23×C29 |
| prime radical | C23×C29 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C2×C4⋊C4×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 99. C42×C29 | Q16 | semidirect |
| 100. #12258 | C29 | direct (N × H) |
| 220. C29 | #12258 | direct (N × H) |
| 258. C42 | Q16×C29 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2612 | 2 |
| C29×#2061 | 2 |
| C29×#1617 | 2 |
| C29×#1605 | 2 |
| C29×#1354 | 2 |
| C29×#1353 | 2 |
| C29×#1324 | 2 |
| #12258 | 29 |