Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
190 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 74 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×C71 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C71 |
| prime radical | C23×C71 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×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 |
|---|---|---|
| 1. C2×Q64×C71 | C2 | semidirect |
| 4. C71×#1424 | C2 | semidirect |
| 5. C71×#1416 | C2 | semidirect |
| 62. #13698 | C71 | direct (N × H) |
| 66. C2×Q64 | C2×C71 | semidirect |
| 69. #1424 | C2×C71 | semidirect |
| 70. #1416 | C2×C71 | semidirect |
| 71. C71 | #13698 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q64×C71 | 2 |
| C2×M64×C71 | 2 |
| C71×#2216 | 2 |
| C71×#1424 | 2 |
| C71×#1416 | 2 |
| #13698 | 71 |