Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
186 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 74 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×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C71 |
| prime radical | C71×#2215 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C22×C8×C71 ◁ C22×C16×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#366×C71 | C2 | semidirect |
| 3. C71×#1423 | C2 | semidirect |
| 4. C71×#1415 | C2 | semidirect |
| 5. C71×C22⋊C32 | C2 | semidirect |
| 71. #13711 | C71 | direct (N × H) |
| 78. C2×#366 | C2×C71 | semidirect |
| 79. #1423 | C2×C71 | semidirect |
| 80. #1415 | C2×C71 | semidirect |
| 81. C22⋊C32 | C2×C71 | semidirect |
| 82. C71 | #13711 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#2215 | 2 |
| C2×#366×C71 | 2 |
| C71×#1423 | 2 |
| C71×#1415 | 2 |
| C71×C22⋊C32 | 2 |
| #13711 | 71 |