Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
126 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 58 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 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C23×C127 |
| prime radical | C127×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×C8×C127 ◁ C2×C16×C127 ◁ C2×C32×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q64×C127 | C2 | semidirect |
| 3. C127×Q128 | C2 | semidirect |
| 4. C127×SD128 | C2 | semidirect |
| 5. C127×M128 | C2 | semidirect |
| 51. #13743 | C127 | direct (N × H) |
| 56. C2×Q64 | C2×C127 | semidirect |
| 57. Q128 | C2×C127 | semidirect |
| 58. SD128 | C2×C127 | semidirect |
| 59. M128 | C2×C127 | semidirect |
| 60. C127 | #13743 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2262 | 2 |
| C2×Q64×C127 | 2 |
| C127×Q128 | 2 |
| C127×SD128 | 2 |
| C127×M128 | 2 |
| #13743 | 127 |