← C2×C42×ΓL(1,𝔽16) #14319324
Order 1920.
2544 conjugacy classes of subgroups.
The subgroup lattice has 192 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C4×ΓL(1,𝔽16) | C2 | 1 | 6 | yes | maximal |
| 2 | C2×C3×C42×D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C42×ΓL(1,𝔽16) | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×C42×AGL(1,𝔽5) | — | 3 | 1 | no | maximal, Hall |
| 5 | C23×ΓL(1,𝔽16) | C22 | 1 | 4 | yes | — |
| 6 | C22×C3×C4×D5 | C22 | 1 | 3 | yes | — |
| 7 | C2×C4×ΓL(1,𝔽16) | C4 | 1 | 48 | yes | direct product factor |
| 8 | C2×C4×ΓL(1,𝔽16) | C22 | 1 | 24 | yes | — |
| 9 | C2×C3×C42×C5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C3×C4×Dic5 | C4 | 1 | 3 | yes | complemented |
| 11 | C3×C42×D5 | C4 | 1 | 4 | yes | complemented |
| 12 | C3×C42×D5 | C22 | 1 | 4 | yes | — |
| 13 | C2×C42×Dic3 | — | 5 | 1 | no | maximal, Hall |
| 14 | C22×C4×AGL(1,𝔽5) | — | 3 | 6 | no | — |
| 15 | C2×C42×D5 | D3 | 1 | 1 | yes | characteristic |
| 16 | C42×AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 17 | C23×C3×D5 | C23 | 1 | 1 | yes | characteristic |
| 18 | C22×ΓL(1,𝔽16) | C2×C4 | 1 | 48 | yes | — |
| 19 | C22×ΓL(1,𝔽16) | C23 | 1 | 8 | yes | — |
| 20 | C22×C3×C4×C5 | C2×C4 | 1 | 3 | yes | — |
| 21 | C22×C3×Dic5 | C2×C4 | 1 | 3 | yes | — |
| 22 | C2×C3×C4×D5 | C2×C4 | 1 | 6 | yes | — |
| 23 | C2×C3×C4×D5 | C2×C4 | 1 | 24 | yes | — |
| 24 | C2×C3×C4×D5 | C23 | 1 | 6 | yes | — |
| 25 | C4×ΓL(1,𝔽16) | C2×C4 | 1 | 96 | yes | direct product factor |
| 26 | C3×C42×C5 | C2×C4 | 1 | 4 | yes | complemented |
| 27 | C3×C4×Dic5 | C2×C4 | 1 | 12 | yes | complemented |
| 28 | C22×C3×C42 | — | 5 | 1 | no | — |
| 29 | C22×C4×Dic3 | — | 5 | 6 | no | — |
| 30 | C42×Dic3 | — | 5 | 8 | no | — |
| 31 | C23×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 32 | C22×C4×D5 | C2×D3 | 1 | 3 | yes | — |
| 33 | C2×C4×AGL(1,𝔽5) | — | 3 | 48 | no | — |
| 34 | C2×C4×AGL(1,𝔽5) | — | 3 | 24 | no | — |
| 35 | C2×C42×C5 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×C4×Dic5 | Dic3 | 1 | 3 | yes | complemented |
| 37 | C42×D5 | Dic3 | 1 | 4 | yes | complemented |
| 38 | C42×D5 | C2×D3 | 1 | 4 | yes | — |
| 39 | C2×C43 | — | 15 | 1 | no | Sylow |
| 40 | C23×C3×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 41 | C22×C3×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 42 | C22×C3×D5 | C22×C4 | 1 | 12 | yes | — |
| 43 | C22×C3×D5 | C24 | 1 | 1 | yes | characteristic |
| 44 | C2×ΓL(1,𝔽16) | C42 | 1 | 64 | yes | direct product factor |
| 45 | C2×ΓL(1,𝔽16) | C22×C4 | 1 | 48 | yes | — |
| 46 | C2×C3×C4×C5 | C42 | 1 | 12 | yes | complemented |
| 47 | C2×C3×C4×C5 | C22×C4 | 1 | 6 | yes | — |
| 48 | C2×C3×Dic5 | C42 | 1 | 12 | yes | complemented |
| 49 | C2×C3×Dic5 | C22×C4 | 1 | 6 | yes | — |
| 50 | C3×C4×D5 | C42 | 1 | 24 | yes | complemented |
| 51 | C3×C4×D5 | C22×C4 | 1 | 24 | yes | — |
| 52 | C23×C3×C4 | — | 5 | 3 | no | — |
| 53 | C23×Dic3 | — | 5 | 4 | no | — |
| 54 | C2×C3×C42 | — | 5 | 3 | no | — |
| 55 | C2×C3×C42 | — | 5 | 4 | no | — |
| 56 | C2×C3×C42 | — | 5 | 4 | no | — |
| 57 | C2×C3×C42 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 58 | C2×C4×Dic3 | — | 5 | 24 | no | — |
| 59 | C2×C4×Dic3 | — | 5 | 48 | no | — |
| 60 | C23×D5 | C22×D3 | 1 | 1 | yes | characteristic |
| 61 | C22×AGL(1,𝔽5) | — | 3 | 48 | no | — |
| 62 | C22×AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 63 | C22×C4×C5 | C2×Dic3 | 1 | 3 | yes | — |
| 64 | C22×Dic5 | C2×Dic3 | 1 | 3 | yes | — |
| 65 | C2×C4×D5 | C4×D3 | 1 | 24 | yes | — |
| 66 | C2×C4×D5 | C2×Dic3 | 1 | 6 | yes | — |
| 67 | C2×C4×D5 | C22×D3 | 1 | 6 | yes | — |
| 68 | C4×AGL(1,𝔽5) | — | 3 | 96 | no | — |
| 69 | C42×C5 | C2×Dic3 | 1 | 4 | yes | complemented |
| 70 | C4×Dic5 | C2×Dic3 | 1 | 12 | yes | complemented |
| 71 | C22×C42 | — | 5 | 1 | no | — |
| 72 | C22×C42 | — | 15 | 6 | no | — |
| 73 | C43 | — | 15 | 8 | no | — |
| 74 | C22×C3×C5 | C2×C42 | 1 | 6 | yes | — |
| 75 | C22×C3×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 76 | C2×C3×D5 | C2×C42 | 1 | 16 | yes | — |
| 77 | C2×C3×D5 | C2×C42 | 1 | 6 | yes | — |
| 78 | C2×C3×D5 | C23×C4 | 1 | 6 | yes | — |
| 79 | ΓL(1,𝔽16) | C2×C42 | 1 | 32 | yes | direct product factor |
| 80 | C3×C4×C5 | C2×C42 | 1 | 12 | yes | complemented |
| 81 | C3×Dic5 | C2×C42 | 1 | 12 | yes | complemented |
| 82 | C24×C3 | — | 5 | 1 | no | — |
| 83 | C22×C3×C4 | — | 5 | 3 | no | — |
| 84 | C22×C3×C4 | — | 5 | 24 | no | — |
| 85 | C22×C3×C4 | — | 5 | 6 | no | — |
| 86 | C22×C3×C4 | — | 5 | 6 | no | — |
| 87 | C22×C3×C4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 88 | C22×Dic3 | — | 5 | 8 | no | — |
| 89 | C22×Dic3 | — | 5 | 48 | no | — |
| 90 | C3×C42 | — | 5 | 12 | no | — |
| 91 | C3×C42 | C2×AGL(1,𝔽5) | 1 | 4 | yes | complemented |
| 92 | C4×Dic3 | — | 5 | 96 | no | — |
| 93 | C23×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 94 | C22×D5 | C2×C4×D3 | 1 | 12 | yes | — |
| 95 | C22×D5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 96 | C22×D5 | C23×D3 | 1 | 1 | yes | characteristic |
| 97 | C2×AGL(1,𝔽5) | — | 3 | 64 | no | — |
| 98 | C2×AGL(1,𝔽5) | — | 3 | 48 | no | — |
| 99 | C2×C4×C5 | C4×Dic3 | 1 | 12 | yes | complemented |
| 100 | C2×C4×C5 | C22×Dic3 | 1 | 6 | yes | — |
| 101 | C2×Dic5 | C4×Dic3 | 1 | 12 | yes | complemented |
| 102 | C2×Dic5 | C22×Dic3 | 1 | 6 | yes | — |
| 103 | C4×D5 | C4×Dic3 | 1 | 24 | yes | complemented |
| 104 | C4×D5 | C2×C4×D3 | 1 | 24 | yes | — |
| 105 | C23×C4 | — | 5 | 3 | no | — |
| 106 | C23×C4 | — | 15 | 4 | no | — |
| 107 | C2×C42 | — | 5 | 3 | no | — |
| 108 | C2×C42 | — | 5 | 4 | no | — |
| 109 | C2×C42 | — | 5 | 4 | no | — |
| 110 | C2×C42 | — | 15 | 24 | no | — |
| 111 | C2×C42 | — | 15 | 48 | no | — |
| 112 | C2×C42 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, direct product factor |
| 113 | C2×C3×C5 | C43 | 1 | 4 | yes | complemented |
| 114 | C2×C3×C5 | C22×C42 | 1 | 3 | yes | — |
| 115 | C3×D5 | C43 | 1 | 4 | yes | complemented |
| 116 | C3×D5 | C22×C42 | 1 | 4 | yes | — |
| 117 | C23×C3 | — | 5 | 12 | no | — |
| 118 | C23×C3 | — | 5 | 1 | no | — |
| 119 | C23×C3 | — | 5 | 1 | no | — |
| 120 | C23×C3 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 121 | C2×C3×C4 | — | 5 | 12 | no | — |
| 122 | C2×C3×C4 | — | 5 | 6 | no | — |
| 123 | C2×C3×C4 | — | 5 | 24 | no | — |
| 124 | C2×C3×C4 | — | 5 | 24 | no | — |
| 125 | C2×C3×C4 | C4×AGL(1,𝔽5) | 1 | 12 | yes | complemented |
| 126 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 127 | C2×Dic3 | — | 5 | 64 | no | — |
| 128 | C2×Dic3 | — | 5 | 48 | no | — |
| 129 | C22×C5 | C2×C4×Dic3 | 1 | 6 | yes | — |
| 130 | C22×C5 | C23×Dic3 | 1 | 1 | yes | characteristic |
| 131 | C2×D5 | C42×D3 | 1 | 16 | yes | — |
| 132 | C2×D5 | C2×C4×Dic3 | 1 | 6 | yes | — |
| 133 | C2×D5 | C22×C4×D3 | 1 | 6 | yes | — |
| 134 | AGL(1,𝔽5) | — | 3 | 32 | no | — |
| 135 | C4×C5 | C2×C4×Dic3 | 1 | 12 | yes | complemented |
| 136 | Dic5 | C2×C4×Dic3 | 1 | 12 | yes | complemented |
| 137 | C24 | — | 5 | 1 | no | — |
| 138 | C22×C4 | — | 5 | 3 | no | — |
| 139 | C22×C4 | — | 5 | 24 | no | — |
| 140 | C22×C4 | — | 5 | 6 | no | — |
| 141 | C22×C4 | — | 5 | 6 | no | — |
| 142 | C22×C4 | — | 15 | 8 | no | — |
| 143 | C22×C4 | — | 15 | 48 | no | — |
| 144 | C22×C4 | C2×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 145 | C42 | — | 5 | 12 | no | — |
| 146 | C42 | — | 15 | 96 | no | — |
| 147 | C42 | C2×ΓL(1,𝔽16) | 1 | 4 | yes | direct product factor |
| 148 | C3×C5 | C2×C43 | 1 | 1 | yes | characteristic, complemented, Hall |
| 149 | C22×C3 | — | 5 | 16 | no | — |
| 150 | C22×C3 | — | 5 | 6 | no | — |
| 151 | C22×C3 | — | 5 | 6 | no | — |
| 152 | C22×C3 | C2×C4×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 153 | C22×C3 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 154 | C3×C4 | — | 5 | 12 | no | — |
| 155 | C3×C4 | C2×C4×AGL(1,𝔽5) | 1 | 12 | yes | complemented |
| 156 | Dic3 | — | 5 | 32 | no | — |
| 157 | C2×C5 | C42×Dic3 | 1 | 4 | yes | complemented |
| 158 | C2×C5 | C22×C4×Dic3 | 1 | 3 | yes | — |
| 159 | D5 | C42×Dic3 | 1 | 4 | yes | complemented |
| 160 | D5 | C2×C42×D3 | 1 | 4 | yes | — |
| 161 | C23 | — | 5 | 12 | no | — |
| 162 | C23 | — | 5 | 1 | no | — |
| 163 | C23 | — | 5 | 1 | no | — |
| 164 | C23 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 165 | C2×C4 | — | 5 | 12 | no | — |
| 166 | C2×C4 | — | 5 | 6 | no | — |
| 167 | C2×C4 | — | 5 | 24 | no | — |
| 168 | C2×C4 | — | 5 | 24 | no | — |
| 169 | C2×C4 | — | 15 | 48 | no | — |
| 170 | C2×C4 | — | 15 | 64 | no | — |
| 171 | C2×C4 | C4×ΓL(1,𝔽16) | 1 | 12 | yes | direct product factor |
| 172 | C2×C4 | C22×ΓL(1,𝔽16) | 1 | 6 | yes | — |
| 173 | C2×C3 | — | 5 | 4 | no | — |
| 174 | C2×C3 | — | 5 | 4 | no | — |
| 175 | C2×C3 | C42×AGL(1,𝔽5) | 1 | 4 | yes | complemented |
| 176 | C2×C3 | C22×C4×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 177 | C5 | C2×C42×Dic3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 178 | C22 | — | 5 | 16 | no | — |
| 179 | C22 | — | 5 | 6 | no | — |
| 180 | C22 | — | 5 | 6 | no | — |
| 181 | C22 | C2×C4×ΓL(1,𝔽16) | 1 | 6 | yes | — |
| 182 | C22 | C23×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 183 | C4 | — | 5 | 12 | no | — |
| 184 | C4 | — | 15 | 32 | no | — |
| 185 | C4 | C2×C4×ΓL(1,𝔽16) | 1 | 12 | yes | direct product factor |
| 186 | C3 | C2×C42×AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 187 | C2 | — | 5 | 4 | no | — |
| 188 | C2 | — | 5 | 4 | no | — |
| 189 | C2 | C42×ΓL(1,𝔽16) | 1 | 4 | yes | direct product factor |
| 190 | C2 | C22×C4×ΓL(1,𝔽16) | 1 | 3 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 3. C42×ΓL(1,𝔽16) | C2 | direct (N × H) |
| 7. C2×C4×ΓL(1,𝔽16) | C4 | direct (N × H) |
| 9. C2×C3×C42×C5 | C4 | semidirect |
| 10. C2×C3×C4×Dic5 | C4 | semidirect |
| 11. C3×C42×D5 | C4 | semidirect |
| 25. C4×ΓL(1,𝔽16) | C2×C4 | direct (N × H) |
| 26. C3×C42×C5 | C2×C4 | semidirect |
| 27. C3×C4×Dic5 | C2×C4 | semidirect |
| 35. C2×C42×C5 | Dic3 | semidirect |
| 36. C2×C4×Dic5 | Dic3 | semidirect |
| 37. C42×D5 | Dic3 | semidirect |
| 44. C2×ΓL(1,𝔽16) | C42 | direct (N × H) |
| 46. C2×C3×C4×C5 | C42 | semidirect |
| 48. C2×C3×Dic5 | C42 | semidirect |
| 50. C3×C4×D5 | C42 | semidirect |
| 57. C2×C3×C42 | AGL(1,𝔽5) | semidirect |
| 69. C42×C5 | C2×Dic3 | semidirect |
| 70. C4×Dic5 | C2×Dic3 | semidirect |
| 79. ΓL(1,𝔽16) | C2×C42 | direct (N × H) |
| 80. C3×C4×C5 | C2×C42 | semidirect |
| 81. C3×Dic5 | C2×C42 | semidirect |
| 91. C3×C42 | C2×AGL(1,𝔽5) | semidirect |
| 99. C2×C4×C5 | C4×Dic3 | semidirect |
| 101. C2×Dic5 | C4×Dic3 | semidirect |
| 103. C4×D5 | C4×Dic3 | semidirect |
| 112. C2×C42 | ΓL(1,𝔽16) | direct (N × H) |
| 113. C2×C3×C5 | C43 | semidirect |
| 115. C3×D5 | C43 | semidirect |
| 125. C2×C3×C4 | C4×AGL(1,𝔽5) | semidirect |
| 135. C4×C5 | C2×C4×Dic3 | semidirect |
| 136. Dic5 | C2×C4×Dic3 | semidirect |
| 147. C42 | C2×ΓL(1,𝔽16) | direct (N × H) |
| 148. C3×C5 | C2×C43 | semidirect |
| 155. C3×C4 | C2×C4×AGL(1,𝔽5) | semidirect |
| 157. C2×C5 | C42×Dic3 | semidirect |
| 159. D5 | C42×Dic3 | semidirect |
| 171. C2×C4 | C4×ΓL(1,𝔽16) | direct (N × H) |
| 175. C2×C3 | C42×AGL(1,𝔽5) | semidirect |
| 177. C5 | C2×C42×Dic3 | semidirect |
| 185. C4 | C2×C4×ΓL(1,𝔽16) | direct (N × H) |
| 186. C3 | C2×C42×AGL(1,𝔽5) | semidirect |
| 189. C2 | C42×ΓL(1,𝔽16) | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta