Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
418 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | #120 |
| commutator | C22×C4 | C22×C4×C29 |
| Frattini, cosocle | C2×C42 | C23×C29 |
| socle | C22×C29 | #385 |
| prime radical | C22×D4×C29 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C29×#1573 | C2 | semidirect |
| 5. C29×#1302 | C2 | semidirect |
| 85. #8864 | C29 | direct (N × H) |
| 124. #1573 | C2×C29 | semidirect |
| 126. #1302 | C2×C29 | semidirect |
| 199. C29 | #8864 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2973 | 2 |
| C29×#1955 | 2 |
| C29×#1573 | 2 |
| C29×#1460 | 2 |
| C29×#1302 | 2 |
| #8864 | 29 |