Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
738 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 224 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×C251 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C251 |
| prime radical | #525×C251 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C251 | C2 | semidirect |
| 2. C2×#137×C251 | C2 | semidirect |
| 4. #451×C251 | C2 | semidirect |
| 5. #448×C251 | C2 | semidirect |
| 6. #419×C251 | C2 | semidirect |
| 7. #407×C251 | C2 | semidirect |
| 17. C2×Q16×C251 | C22 | semidirect |
| 18. C2×SD16×C251 | C22 | semidirect |
| 26. #103×C251 | C22 | semidirect |
| 36. C2×Q8×C251 | D4 | semidirect |
| 95. C22×C251 | #137 | semidirect |
| 108. C251 | #3004 | direct (N × H) |
| 109. #3004 | C251 | direct (N × H) |
| 110. C23×Q8 | C2×C251 | semidirect |
| 111. C2×#137 | C2×C251 | semidirect |
| 113. #451 | C2×C251 | semidirect |
| 114. #448 | C2×C251 | semidirect |
| 115. #419 | C2×C251 | semidirect |
| 116. #407 | C2×C251 | semidirect |
| 126. C2×Q16 | C22×C251 | semidirect |
| 127. C2×SD16 | C22×C251 | semidirect |
| 135. #103 | C22×C251 | semidirect |
| 145. C2×Q8 | D4×C251 | semidirect |
| 204. C22 | #137×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C251 | 2 |
| C2×#137×C251 | 2 |
| #525×C251 | 2 |
| #451×C251 | 2 |
| #448×C251 | 2 |
| #419×C251 | 2 |
| #407×C251 | 2 |
| #3004 | 251 |