Finite Group DatabasebetaFinite Group DatabasebetaOrder 5248.
844 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 844 conjugacy classes of subgroups — too many to draw. Only its 216 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×C41 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C22×C4×C41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(5,𝔽2)×C41 | C2 | semidirect |
| 2. C2×AGL(1,ℤ/8)×C41 | C2 | semidirect |
| 3. C2×#127×C41 | C2 | semidirect |
| 4. C41×#453 | C2 | direct (N × H) |
| 5. C2×#104×C41 | C2 | semidirect |
| 6. C2×#100×C41 | C2 | semidirect |
| 9. H(5,𝔽2)×C41 | C22 | semidirect |
| 16. C22×D4×C41 | C22 | semidirect |
| 18. AGL(1,ℤ/8)×C41 | C22 | semidirect |
| 21. C2×M16×C41 | C22 | semidirect |
| 23. #127×C41 | C22 | semidirect |
| 24. C2×C42×C41 | C22 | semidirect |
| 25. #104×C41 | C22 | semidirect |
| 26. #100×C41 | C22 | semidirect |
| 34. #41×C41 | D4 | semidirect |
| 36. C2×Q8×C41 | D4 | semidirect |
| 51. C2×D4×C41 | D4 | semidirect |
| 53. C2×D4×C41 | C23 | semidirect |
| 61. M16×C41 | C23 | semidirect |
| 64. C42×C41 | C23 | semidirect |
| 78. Q8×C41 | C2×D4 | semidirect |
| 88. D4×C41 | C2×D4 | semidirect |
| 119. C2×#453 | C41 | direct (N × H) |
| 125. C2×C41 | #453 | direct (N × H) |
| 127. C2×H(5,𝔽2) | C2×C41 | semidirect |
| 128. C2×AGL(1,ℤ/8) | C2×C41 | semidirect |
| 129. C2×#127 | C2×C41 | semidirect |
| 130. #453 | C2×C41 | direct (N × H) |
| 131. C2×#104 | C2×C41 | semidirect |
| 132. C2×#100 | C2×C41 | semidirect |
| 133. C41 | C2×#453 | direct (N × H) |
| 136. H(5,𝔽2) | C22×C41 | semidirect |
| 143. C22×D4 | C22×C41 | semidirect |
| 145. AGL(1,ℤ/8) | C22×C41 | semidirect |
| 148. C2×M16 | C22×C41 | semidirect |
| 150. #127 | C22×C41 | semidirect |
| 151. C2×C42 | C22×C41 | semidirect |
| 152. #104 | C22×C41 | semidirect |
| 153. #100 | C22×C41 | semidirect |
| 161. #41 | D4×C41 | semidirect |
| 163. C2×Q8 | D4×C41 | semidirect |
| 178. C2×D4 | D4×C41 | semidirect |
| 180. C2×D4 | C23×C41 | semidirect |
| 188. M16 | C23×C41 | semidirect |
| 191. C42 | C23×C41 | semidirect |
| 205. Q8 | C2×D4×C41 | semidirect |
| 215. D4 | C2×D4×C41 | semidirect |
| 251. C2 | C41×#453 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2)×C41 | 2 |
| C2×AGL(1,ℤ/8)×C41 | 2 |
| C2×#127×C41 | 2 |
| C41×#453 | 2 |
| C2×#104×C41 | 2 |
| C2×#100×C41 | 2 |
| C2×#453 | 41 |