Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
502 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 502 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 | C23×C29 | D16 |
| commutator, Frattini, cosocle | C16 | C24×C29 |
| prime radical | C22×C29×D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C22×C4×C29 ◁ C22×C8×C29 ◁ C22×C16×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×SD64 | C2 | direct (N × H) |
| 4. C22×C29×C32 | C2 | semidirect |
| 8. C2×C29×Q32 | C22 | semidirect |
| 12. C29×SD64 | C22 | direct (N × H) |
| 13. C2×C29×C32 | C22 | semidirect |
| 22. C29×Q32 | C23 | semidirect |
| 27. C29×C32 | C23 | semidirect |
| 43. C22×SD64 | C29 | direct (N × H) |
| 56. C22×Q32 | C2×C29 | semidirect |
| 58. C2×SD64 | C2×C29 | direct (N × H) |
| 59. C22×C32 | C2×C29 | semidirect |
| 62. C22×C29 | SD64 | direct (N × H) |
| 70. C2×Q32 | C22×C29 | semidirect |
| 74. SD64 | C22×C29 | direct (N × H) |
| 75. C2×C32 | C22×C29 | semidirect |
| 77. C2×C29 | C2×SD64 | direct (N × H) |
| 87. Q32 | C23×C29 | semidirect |
| 92. C32 | C23×C29 | semidirect |
| 93. C29 | C22×SD64 | direct (N × H) |
| 123. C22 | C29×SD64 | direct (N × H) |
| 129. C2 | C2×C29×SD64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C29×Q32 | 2 |
| C22×C29×D16 | 2 |
| C2×C29×SD64 | 2 |
| C22×C29×C32 | 2 |
| C22×SD64 | 29 |