Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C29 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×Q8×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×2−1+6 | C2 | semidirect |
| 2. C29×#3487 | C2 | semidirect |
| 3. C29×#3027 | C2 | semidirect |
| 4. C29×#3025 | C2 | semidirect |
| 5. C29×#2884 | C2 | semidirect |
| 10. C22×#41×C29 | C22 | semidirect |
| 13. C29×#535 | C22 | semidirect |
| 14. C29×#525 | C22 | semidirect |
| 15. C29×#519 | C22 | semidirect |
| 16. C29×#458 | C22 | semidirect |
| 17. C29×B(4,𝔽2) | C22 | semidirect |
| 18. C29×#410 | C22 | semidirect |
| 28. C2×#41×C29 | D4 | semidirect |
| 31. C22×Q8×C29 | D4 | semidirect |
| 74. #60402 | C29 | direct (N × H) |
| 95. 2−1+6 | C2×C29 | semidirect |
| 96. #3487 | C2×C29 | semidirect |
| 97. #3027 | C2×C29 | semidirect |
| 98. #3025 | C2×C29 | semidirect |
| 99. #2884 | C2×C29 | semidirect |
| 115. C22×#41 | C22×C29 | semidirect |
| 118. #535 | C22×C29 | semidirect |
| 119. #525 | C22×C29 | semidirect |
| 120. #519 | C22×C29 | semidirect |
| 121. #458 | C22×C29 | semidirect |
| 122. B(4,𝔽2) | C22×C29 | semidirect |
| 123. #410 | C22×C29 | semidirect |
| 137. C2×#41 | D4×C29 | semidirect |
| 140. C22×Q8 | D4×C29 | semidirect |
| 155. C29 | #60402 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×2−1+6 | 2 |
| C29×#3487 | 2 |
| C29×#3027 | 2 |
| C29×#3025 | 2 |
| C29×#2884 | 2 |
| #60402 | 29 |