Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
688 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 202 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C29 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C4×C29 ◁ C23×C8×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C29×#466 | C2 | semidirect |
| 2. C29×#2233 | C2 | semidirect |
| 3. C29×#2218 | C2 | semidirect |
| 4. C29×#2184 | C2 | semidirect |
| 5. C29×#2112 | C2 | semidirect |
| 6. C22×D8×C29 | C22 | semidirect |
| 9. C2×C29×D16 | C22 | semidirect |
| 10. C2×C29×M32 | C22 | semidirect |
| 14. C29×#466 | C22 | semidirect |
| 15. C2×C29×#107 | C22 | semidirect |
| 17. C29×#367 | C22 | semidirect |
| 18. C29×#357 | C22 | semidirect |
| 19. C29×C22⋊C16 | C22 | semidirect |
| 74. #33758 | C29 | direct (N × H) |
| 102. C2×#466 | C2×C29 | semidirect |
| 103. #2233 | C2×C29 | semidirect |
| 104. #2218 | C2×C29 | semidirect |
| 105. #2184 | C2×C29 | semidirect |
| 106. #2112 | C2×C29 | semidirect |
| 123. C22×D8 | C22×C29 | semidirect |
| 126. C2×D16 | C22×C29 | semidirect |
| 127. C2×M32 | C22×C29 | semidirect |
| 131. #466 | C22×C29 | semidirect |
| 132. C2×#107 | C22×C29 | semidirect |
| 134. #367 | C22×C29 | semidirect |
| 135. #357 | C22×C29 | semidirect |
| 136. C22⋊C16 | C22×C29 | semidirect |
| 166. C29 | #33758 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C29×#466 | 2 |
| C29×#2233 | 2 |
| C29×#2218 | 2 |
| C29×#2184 | 2 |
| C29×#2112 | 2 |
| #33758 | 29 |