Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
272 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C127 |
| prime radical | #493×C127 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C127×#2252 | C2 | semidirect |
| 3. C127×#1713 | C2 | semidirect |
| 5. C127×#1312 | C2 | semidirect |
| 72. #12531 | C127 | direct (N × H) |
| 77. #2252 | C2×C127 | semidirect |
| 79. #1713 | C2×C127 | semidirect |
| 81. #1312 | C2×C127 | semidirect |
| 82. C127 | #12531 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2252 | 2 |
| C127×#2240 | 2 |
| C127×#1713 | 2 |
| C127×#1706 | 2 |
| C127×#1312 | 2 |
| #12531 | 127 |