Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
592 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 192 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×C251 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C251 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#124×C251 | C2 | semidirect |
| 2. #394×C251 | C2 | semidirect |
| 3. #392×C251 | C2 | semidirect |
| 4. #391×C251 | C2 | semidirect |
| 5. #390×C251 | C2 | semidirect |
| 6. #388×C251 | C2 | semidirect |
| 25. C2×Q8×C251 | D4 | semidirect |
| 96. C251 | #2733 | direct (N × H) |
| 97. #2733 | C251 | direct (N × H) |
| 98. C2×#124 | C2×C251 | semidirect |
| 99. #394 | C2×C251 | semidirect |
| 100. #392 | C2×C251 | semidirect |
| 101. #391 | C2×C251 | semidirect |
| 102. #390 | C2×C251 | semidirect |
| 103. #388 | C2×C251 | semidirect |
| 122. C2×Q8 | D4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#124×C251 | 2 |
| #394×C251 | 2 |
| #392×C251 | 2 |
| #391×C251 | 2 |
| #390×C251 | 2 |
| #388×C251 | 2 |
| #2733 | 251 |