Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
316 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 110 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 |
|---|---|---|
| 2. C127×#2247 | C2 | semidirect |
| 3. C127×#1713 | C2 | semidirect |
| 5. C127×#1311 | C2 | semidirect |
| 62. #12520 | C127 | direct (N × H) |
| 67. #2247 | C2×C127 | semidirect |
| 68. #1713 | C2×C127 | semidirect |
| 70. #1311 | C2×C127 | semidirect |
| 71. C127 | #12520 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2248 | 2 |
| C127×#2247 | 2 |
| C127×#1713 | 2 |
| C127×#1709 | 2 |
| C127×#1311 | 2 |
| #12520 | 127 |