Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
750 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 318 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C127 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C127×#1710 | C2 | direct (N × H) |
| 69. C2×#1710 | C127 | direct (N × H) |
| 70. C2×C127 | #1710 | direct (N × H) |
| 75. #1710 | C2×C127 | direct (N × H) |
| 77. C127 | C2×#1710 | direct (N × H) |
| 142. C2 | C127×#1710 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#500×C127 | 2 |
| C2×#487×C127 | 2 |
| C127×#1710 | 2 |
| C2×C82×C127 | 2 |
| C2×#1710 | 127 |