Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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×#106 |
| commutator | C2×C8 | C22×C4×C71 |
| Frattini, cosocle | C4×C8 | C23×C71 |
| prime radical | C23×C71 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C2×C42×C71 ◁ C2×C4×C8×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C71×#1368 | C2 | semidirect |
| 73. #12031 | C71 | direct (N × H) |
| 81. #1368 | C2×C71 | semidirect |
| 82. C71 | #12031 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#2163 | 2 |
| C71×#1568 | 2 |
| C71×#1564 | 2 |
| C71×#1368 | 2 |
| #12031 | 71 |