Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
910 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C29 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C2×C42×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×#3568 | C2 | semidirect |
| 2. C29×#3217 | C2 | semidirect |
| 3. C29×#1657 | C2 | semidirect |
| 4. C29×#1655 | C2 | semidirect |
| 5. C29×#1535 | C2 | semidirect |
| 6. C29×#563 | C22 | semidirect |
| 7. C29×H(3,𝔽4) | C22 | semidirect |
| 12. C2×C29×#127 | C22 | semidirect |
| 15. C29×#490 | C22 | semidirect |
| 20. C29×#432 | C22 | semidirect |
| 21. C29×#330 | C22 | semidirect |
| 22. C29×#329 | C22 | semidirect |
| 102. #19094 | C29 | direct (N × H) |
| 125. #3568 | C2×C29 | semidirect |
| 126. #3217 | C2×C29 | semidirect |
| 127. #1657 | C2×C29 | semidirect |
| 128. #1655 | C2×C29 | semidirect |
| 129. #1535 | C2×C29 | semidirect |
| 143. #563 | C22×C29 | semidirect |
| 144. H(3,𝔽4) | C22×C29 | semidirect |
| 149. C2×#127 | C22×C29 | semidirect |
| 152. #490 | C22×C29 | semidirect |
| 157. #432 | C22×C29 | semidirect |
| 158. #330 | C22×C29 | semidirect |
| 159. #329 | C22×C29 | semidirect |
| 205. C29 | #19094 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#3568 | 2 |
| C29×#3217 | 2 |
| C29×#1657 | 2 |
| C29×#1655 | 2 |
| C29×#1535 | 2 |
| #19094 | 29 |