Finite Group DatabasebetaFinite Group DatabasebetaOrder 5248.
728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 728 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. C2×2−1+4×C41 | C2 | semidirect |
| 2. C2×AGL(1,ℤ/8)×C41 | C2 | semidirect |
| 3. C2×#124×C41 | C2 | semidirect |
| 4. C41×#454 | C2 | direct (N × H) |
| 5. C2×#104×C41 | C2 | semidirect |
| 6. C2×#101×C41 | C2 | semidirect |
| 9. 2−1+4×C41 | C22 | semidirect |
| 17. AGL(1,ℤ/8)×C41 | C22 | semidirect |
| 22. #124×C41 | C22 | semidirect |
| 25. #104×C41 | C22 | semidirect |
| 26. #101×C41 | C22 | semidirect |
| 111. C2×#454 | C41 | direct (N × H) |
| 116. C2×C41 | #454 | direct (N × H) |
| 118. C2×2−1+4 | C2×C41 | semidirect |
| 119. C2×AGL(1,ℤ/8) | C2×C41 | semidirect |
| 120. C2×#124 | C2×C41 | semidirect |
| 121. #454 | C2×C41 | direct (N × H) |
| 122. C2×#104 | C2×C41 | semidirect |
| 123. C2×#101 | C2×C41 | semidirect |
| 124. C41 | C2×#454 | direct (N × H) |
| 127. 2−1+4 | C22×C41 | semidirect |
| 135. AGL(1,ℤ/8) | C22×C41 | semidirect |
| 140. #124 | C22×C41 | semidirect |
| 143. #104 | C22×C41 | semidirect |
| 144. #101 | C22×C41 | semidirect |
| 233. C2 | C41×#454 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×2−1+4×C41 | 2 |
| C2×AGL(1,ℤ/8)×C41 | 2 |
| C2×#124×C41 | 2 |
| C41×#454 | 2 |
| C2×#104×C41 | 2 |
| C2×#101×C41 | 2 |
| C2×#454 | 41 |