Finite Group DatabasebetaFinite Group DatabasebetaOrder 21376.
360 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C167 | C2×D4 |
| commutator | C2×C4 | C22×C4×C167 |
| Frattini, cosocle | C22×C4 | C23×C167 |
| prime radical | C24×C167 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C167 ◁ C23×C4×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#103×C167 | C2 | semidirect |
| 5. #340×C167 | C2 | semidirect |
| 12. C22⋊Q8×C167 | C4 | semidirect |
| 95. C167 | #1894 | direct (N × H) |
| 96. #1894 | C167 | direct (N × H) |
| 99. C2×#103 | C2×C167 | semidirect |
| 101. #340 | C2×C167 | semidirect |
| 108. C22⋊Q8 | C4×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C167 | 2 |
| C2×C22⋊Q8×C167 | 2 |
| C2×#103×C167 | 2 |
| #380×C167 | 2 |
| #340×C167 | 2 |
| #1894 | 167 |