Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
760 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 216 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×C163 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C163 |
| prime radical | C2×H(5,𝔽2)×C163 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C163 ◁ C22×C4×C163 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#137×C163 | C2 | semidirect |
| 3. C2×#124×C163 | C2 | semidirect |
| 4. #455×C163 | C2 | direct (N × H) |
| 5. C2×#104×C163 | C2 | semidirect |
| 6. C2×#100×C163 | C2 | semidirect |
| 12. C22×Q8×C163 | C22 | semidirect |
| 17. #137×C163 | C22 | semidirect |
| 20. C2×M16×C163 | C22 | semidirect |
| 22. #124×C163 | C22 | semidirect |
| 24. C2×C42×C163 | C22 | semidirect |
| 25. #104×C163 | C22 | semidirect |
| 26. #100×C163 | C22 | semidirect |
| 39. C2×Q8×C163 | C23 | semidirect |
| 59. M16×C163 | C23 | semidirect |
| 63. C42×C163 | C23 | semidirect |
| 117. C2×C163 | #455 | direct (N × H) |
| 119. C163 | C2×#455 | direct (N × H) |
| 120. C2×#455 | C163 | direct (N × H) |
| 122. C2×#137 | C2×C163 | semidirect |
| 123. C2×#124 | C2×C163 | semidirect |
| 124. #455 | C2×C163 | direct (N × H) |
| 125. C2×#104 | C2×C163 | semidirect |
| 126. C2×#100 | C2×C163 | semidirect |
| 132. C22×Q8 | C22×C163 | semidirect |
| 137. #137 | C22×C163 | semidirect |
| 140. C2×M16 | C22×C163 | semidirect |
| 142. #124 | C22×C163 | semidirect |
| 144. C2×C42 | C22×C163 | semidirect |
| 145. #104 | C22×C163 | semidirect |
| 146. #100 | C22×C163 | semidirect |
| 159. C2×Q8 | C23×C163 | semidirect |
| 179. M16 | C23×C163 | semidirect |
| 183. C42 | C23×C163 | semidirect |
| 237. C2 | #455×C163 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2)×C163 | 2 |
| C2×#137×C163 | 2 |
| C2×#124×C163 | 2 |
| #455×C163 | 2 |
| C2×#104×C163 | 2 |
| C2×#100×C163 | 2 |
| C2×#455 | 163 |