Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #427 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1835 |
| prime radical | C23×C127 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ #117×C127 ◁ #405×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. #346×C127 | C4 | semidirect |
| 12. C4×C16×C127 | C4 | semidirect |
| 70. #12893 | C127 | direct (N × H) |
| 79. C127 | #12893 | direct (N × H) |
| 85. #346 | C4×C127 | semidirect |
| 86. C4×C16 | C4×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2108 | 2 |
| C127×#1833 | 2 |
| C127×#1385 | 2 |
| C127×#1384 | 2 |
| C127×#1382 | 2 |
| #12893 | 127 |