Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
196 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #1677 |
| commutator, Frattini, cosocle | C4×C8 | C23×C71 |
| prime radical | C71×#2213 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C71 ◁ C2×C4×C71 ◁ C4×C8×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C71×#2244 | C2 | semidirect |
| 2. C71×#2231 | C2 | semidirect |
| 4. C71×#2171 | C2 | semidirect |
| 5. C71×#1369 | C2 | semidirect |
| 6. C71×#1366 | C2 | semidirect |
| 7. C71×#1357 | C2 | semidirect |
| 91. #12465 | C71 | direct (N × H) |
| 96. #2244 | C2×C71 | semidirect |
| 97. #2231 | C2×C71 | semidirect |
| 99. #2171 | C2×C71 | semidirect |
| 100. #1369 | C2×C71 | semidirect |
| 101. #1366 | C2×C71 | semidirect |
| 102. #1357 | C2×C71 | semidirect |
| 103. C71 | #12465 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#2244 | 2 |
| C71×#2231 | 2 |
| C71×#2213 | 2 |
| C71×#2171 | 2 |
| C71×#1369 | 2 |
| C71×#1366 | 2 |
| C71×#1357 | 2 |
| #12465 | 71 |