Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
574 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 122 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 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C71 |
| prime radical | C71×#3150 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C71×#3226 | C2 | semidirect |
| 3. C71×#1960 | C2 | semidirect |
| 4. C71×#1517 | C2 | semidirect |
| 5. C71×#1281 | C2 | semidirect |
| 122. #10225 | C71 | direct (N × H) |
| 128. #3226 | C2×C71 | semidirect |
| 130. #1960 | C2×C71 | semidirect |
| 131. #1517 | C2×C71 | semidirect |
| 132. #1281 | C2×C71 | semidirect |
| 133. C71 | #10225 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#3226 | 2 |
| C71×#3150 | 2 |
| C71×#1960 | 2 |
| C71×#1517 | 2 |
| C71×#1281 | 2 |
| #10225 | 71 |