Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
180 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | #41×C127 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ #131×C127 ◁ #504×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q64×C127 | C2 | semidirect |
| 4. C127×#1425 | C2 | semidirect |
| 5. C127×#1420 | C2 | semidirect |
| 63. #13701 | C127 | direct (N × H) |
| 67. C2×Q64 | C2×C127 | semidirect |
| 70. #1425 | C2×C127 | semidirect |
| 71. #1420 | C2×C127 | semidirect |
| 72. C127 | #13701 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q64×C127 | 2 |
| C127×#2258 | 2 |
| C127×#2222 | 2 |
| C127×#1425 | 2 |
| C127×#1420 | 2 |
| #13701 | 127 |