Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
166 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C71 | #357 |
| commutator | C16 | C42×C71 |
| Frattini, cosocle | C22×C16 | C22×C71 |
| socle | C2×C71 | #1380 |
| prime radical | C23×C71 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C71 ◁ C22×C4×C71 ◁ C22×C8×C71 ◁ C22×C16×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. M64×C71 | C4 | semidirect |
| 11. #368×C71 | C4 | semidirect |
| 67. #7480 | C71 | direct (N × H) |
| 75. C71 | #7480 | direct (N × H) |
| 80. M64 | C4×C71 | semidirect |
| 83. #368 | C4×C71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M64×C71 | 2 |
| C2×#368×C71 | 2 |
| C71×#2158 | 2 |
| #7480 | 71 |