Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
1786 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 366 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C29 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C24×C4×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C8×C29 | C2 | semidirect |
| 2. C2×C29×#466 | C2 | semidirect |
| 3. C29×#2847 | C2 | semidirect |
| 4. C29×#1893 | C2 | semidirect |
| 5. C29×#1808 | C2 | semidirect |
| 7. C22×D8×C29 | C22 | semidirect |
| 14. C2×C29×#107 | C22 | semidirect |
| 15. C2×C29×#102 | C22 | semidirect |
| 18. C29×#340 | C22 | semidirect |
| 31. C22×C8×C29 | D4 | semidirect |
| 46. C24×C29 | D8 | semidirect |
| 74. #32930 | C29 | direct (N × H) |
| 102. C8×C29 | #120 | semidirect |
| 103. C24×C8 | C2×C29 | semidirect |
| 104. C2×#466 | C2×C29 | semidirect |
| 105. #2847 | C2×C29 | semidirect |
| 106. #1893 | C2×C29 | semidirect |
| 107. #1808 | C2×C29 | semidirect |
| 116. C22×C29 | #466 | semidirect |
| 125. C22×D8 | C22×C29 | semidirect |
| 132. C2×#107 | C22×C29 | semidirect |
| 133. C2×#102 | C22×C29 | semidirect |
| 136. #340 | C22×C29 | semidirect |
| 154. C22×C8 | D4×C29 | semidirect |
| 166. C29 | #32930 | direct (N × H) |
| 170. C24 | D8×C29 | semidirect |
| 225. C8 | C29×#120 | semidirect |
| 234. C22 | C29×#466 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8×C29 | 2 |
| C2×C29×#466 | 2 |
| C29×#2847 | 2 |
| C29×#1893 | 2 |
| C29×#1808 | 2 |
| #32930 | 29 |