Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
126 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C71 | #1404 |
| commutator | #125 | C2×C4×C71 |
| Frattini, cosocle | #491 | C22×C71 |
| prime radical | C22×C71 | AΣL(1,𝔽16) |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C71 ◁ C22×C71 ◁ C2×C4×C71 ◁ C42×C71 ◁ #491×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 58. #7526 | C71 | direct (N × H) |
| 64. C71 | #7526 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#1663 | 2 |
| C71×#1413 | 2 |
| C71×#1356 | 2 |
| #7526 | 71 |