Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
124 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C71 | #357 |
| commutator | C16 | C2×C8×C71 |
| Frattini, cosocle | C4×C16 | C22×C71 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C42×C71 ◁ C4×C8×C71 ◁ C4×C16×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. Q32×C71 | C8 | semidirect |
| 53. #7386 | C71 | direct (N × H) |
| 60. C71 | #7386 | direct (N × H) |
| 71. Q32 | C8×C71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×Q32×C71 | 2 |
| C71×#1397 | 2 |
| C71×#1372 | 2 |
| #7386 | 71 |