Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C71 |
| prime radical | C2×#41×C71 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#131×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q32×C71 | C2 | semidirect |
| 3. C2×#362×C71 | C2 | semidirect |
| 5. C71×#1381 | C2 | semidirect |
| 114. #13315 | C71 | direct (N × H) |
| 119. C22×Q32 | C2×C71 | semidirect |
| 121. C2×#362 | C2×C71 | semidirect |
| 123. #1381 | C2×C71 | semidirect |
| 124. C71 | #13315 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q32×C71 | 2 |
| C2×#473×C71 | 2 |
| C2×#362×C71 | 2 |
| C71×#2113 | 2 |
| C71×#1381 | 2 |
| #13315 | 71 |