Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
496 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#66017 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11751163 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11749388 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C5×#4542 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11746012 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10599953 | — | 3 | 1 | no | maximal, Hall |
| 7 | #92098 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3×C5×#124 | C22 | 1 | 1 | yes | characteristic |
| 9 | #91821 | — | 2 | 1 | no | — |
| 10 | C3×#4093 | C22 | 1 | 2 | yes | — |
| 11 | C3×#4083 | — | 2 | 1 | no | — |
| 12 | C3×#4031 | — | 2 | 2 | no | — |
| 13 | C3×#4028 | — | 2 | 2 | no | — |
| 14 | C3×#4021 | — | 2 | 1 | no | — |
| 15 | #91312 | — | 2 | 2 | no | — |
| 16 | #91120 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #91089 | — | 2 | 1 | no | — |
| 18 | C5×#836 | C22 | 1 | 2 | yes | complemented |
| 19 | #90981 | — | 2 | 2 | no | — |
| 20 | #71951 | — | 5 | 1 | no | maximal, Hall |
| 21 | #66017 | D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | #65351 | — | 3 | 1 | no | — |
| 23 | #65042 | — | 3 | 2 | no | — |
| 24 | C5×#355 | — | 3 | 1 | no | — |
| 25 | #64697 | — | 3 | 2 | no | — |
| 26 | C23×C3×D5 | — | 2 | 1 | no | — |
| 27 | C2×C3×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×C3×C5×D4 | D4 | 1 | 1 | yes | characteristic |
| 29 | #6821 | — | 4 | 1 | no | — |
| 30 | Q8×D15 | — | 4 | 1 | no | — |
| 31 | #6817 | — | 2 | 2 | no | — |
| 32 | #6817 | — | 2 | 1 | no | — |
| 33 | C2×C3×C22⋊D5 | — | 2 | 2 | no | — |
| 34 | C2×C3×D20 | — | 2 | 1 | no | — |
| 35 | C2×C3×D20 | D4 | 1 | 2 | yes | — |
| 36 | C2×C3×Dic10 | — | 2 | 1 | no | — |
| 37 | C3×C5×C22⋊C4 | — | 2 | 2 | no | — |
| 38 | C3×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 39 | #6716 | — | 4 | 1 | no | — |
| 40 | #6715 | — | 4 | 1 | no | — |
| 41 | #6710 | — | 2 | 1 | no | — |
| 42 | C5×#208 | — | 2 | 2 | no | — |
| 43 | C3×C22⋊Dic5 | — | 2 | 1 | no | — |
| 44 | C3×#685 | — | 2 | 2 | no | — |
| 45 | C3×#685 | — | 2 | 2 | no | — |
| 46 | C3×#685 | — | 2 | 2 | no | — |
| 47 | C3×C4×Dic5 | D4 | 1 | 2 | yes | — |
| 48 | #6657 | — | 4 | 2 | no | — |
| 49 | #6653 | — | 4 | 2 | no | — |
| 50 | C3×H(3,𝔽4) | — | 5 | 1 | no | — |
| 51 | #5062 | — | 5 | 1 | no | — |
| 52 | #5062 | — | 5 | 2 | no | — |
| 53 | #4542 | — | 5 | 2 | no | — |
| 54 | #4542 | D5 | 1 | 1 | yes | characteristic, complemented |
| 55 | #4147 | — | 3 | 1 | no | — |
| 56 | C5×#124 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | #4093 | C2×D3 | 1 | 2 | yes | — |
| 58 | #4083 | — | 2 | 1 | no | — |
| 59 | #4078 | — | 6 | 1 | no | — |
| 60 | #4054 | — | 6 | 2 | no | — |
| 61 | #4031 | — | 2 | 2 | no | — |
| 62 | #4028 | — | 2 | 2 | no | — |
| 63 | #4021 | — | 2 | 1 | no | — |
| 64 | C5×#101 | — | 3 | 2 | no | — |
| 65 | #3968 | — | 3 | 1 | no | — |
| 66 | #3957 | — | 6 | 2 | no | — |
| 67 | #3937 | — | 6 | 1 | no | — |
| 68 | #2202 | — | 15 | 1 | no | Sylow |
| 69 | C23×C3×C5 | — | 2 | 1 | no | — |
| 70 | C22×C3×D5 | — | 2 | 2 | no | — |
| 71 | C22×C3×D5 | — | 8 | 2 | no | — |
| 72 | C22×C3×D5 | — | 8 | 1 | no | — |
| 73 | C3×C5×Q8 | — | 4 | 1 | no | — |
| 74 | C3×C5×D4 | — | 4 | 1 | no | — |
| 75 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 76 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 78 | #1220 | — | 4 | 1 | no | — |
| 79 | D60 | — | 2 | 1 | no | — |
| 80 | D60 | — | 2 | 2 | no | — |
| 81 | C4×D15 | — | 4 | 2 | no | — |
| 82 | C4×D15 | — | 4 | 1 | no | — |
| 83 | Dic30 | — | 2 | 2 | no | — |
| 84 | Dic30 | — | 2 | 1 | no | — |
| 85 | C3×C22⋊D5 | — | 8 | 2 | no | — |
| 86 | C2×C3×Dic5 | — | 2 | 2 | no | — |
| 87 | C3×D20 | — | 4 | 2 | no | — |
| 88 | C3×D20 | — | 8 | 1 | no | — |
| 89 | C3×Dic10 | — | 8 | 1 | no | — |
| 90 | C3⋊#168 | — | 4 | 1 | no | — |
| 91 | C5×C3⋊C8 | — | 4 | 2 | no | — |
| 92 | #1007 | — | 5 | 1 | no | — |
| 93 | C3×#124 | — | 5 | 2 | no | — |
| 94 | C3×#124 | — | 10 | 2 | no | — |
| 95 | C3×#124 | — | 10 | 1 | no | — |
| 96 | C3×#124 | C2×D5 | 1 | 1 | yes | characteristic |
| 97 | C3×#120 | — | 10 | 2 | no | — |
| 98 | C3×#120 | — | 10 | 1 | no | — |
| 99 | #932 | — | 10 | 2 | no | — |
| 100 | #932 | — | 10 | 1 | no | — |
| 101 | #836 | — | 5 | 1 | no | — |
| 102 | #836 | C2×D5 | 1 | 2 | yes | complemented |
| 103 | #805 | — | 10 | 2 | no | — |
| 104 | #805 | — | 10 | 1 | no | — |
| 105 | C23×D5 | — | 2 | 1 | no | — |
| 106 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 107 | C2×C5×D4 | #67 | 1 | 1 | yes | characteristic |
| 108 | C2×C22⋊D5 | — | 2 | 2 | no | — |
| 109 | #713 | — | 12 | 1 | no | — |
| 110 | Q8×D5 | — | 12 | 1 | no | — |
| 111 | #709 | — | 6 | 1 | no | — |
| 112 | #709 | — | 6 | 2 | no | — |
| 113 | C2×D20 | — | 2 | 1 | no | — |
| 114 | C2×D20 | #67 | 1 | 2 | yes | — |
| 115 | C2×Dic10 | — | 2 | 1 | no | — |
| 116 | C5×M16 | — | 6 | 2 | no | — |
| 117 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 118 | C42×C5 | #67 | 1 | 1 | yes | characteristic |
| 119 | C22⋊Dic5 | — | 2 | 1 | no | — |
| 120 | #687 | — | 12 | 1 | no | — |
| 121 | #686 | — | 12 | 1 | no | — |
| 122 | #685 | — | 2 | 2 | no | — |
| 123 | #685 | — | 2 | 2 | no | — |
| 124 | #685 | — | 2 | 2 | no | — |
| 125 | C4×Dic5 | #67 | 1 | 2 | yes | — |
| 126 | #681 | — | 6 | 1 | no | — |
| 127 | D40 | — | 12 | 2 | no | — |
| 128 | #677 | — | 12 | 2 | no | — |
| 129 | H(3,𝔽4) | — | 5 | 1 | no | — |
| 130 | #454 | — | 15 | 2 | no | — |
| 131 | #454 | — | 15 | 1 | no | — |
| 132 | #355 | — | 3 | 1 | no | — |
| 133 | #355 | — | 15 | 2 | no | — |
| 134 | C22×C3×C5 | — | 4 | 2 | no | — |
| 135 | C22×C3×C5 | — | 4 | 1 | no | — |
| 136 | C22×C3×C5 | #120 | 1 | 1 | yes | characteristic |
| 137 | C2×D15 | — | 4 | 1 | no | — |
| 138 | C2×C3×D5 | — | 4 | 2 | no | — |
| 139 | C2×C3×D5 | — | 4 | 2 | no | — |
| 140 | C2×C3×D5 | — | 4 | 2 | no | — |
| 141 | C2×C3×D5 | — | 8 | 4 | no | — |
| 142 | C3×C4×C5 | — | 2 | 2 | no | — |
| 143 | C3×C4×C5 | — | 2 | 1 | no | — |
| 144 | C3×C4×C5 | — | 4 | 1 | no | — |
| 145 | Dic15 | — | 4 | 1 | no | — |
| 146 | C3×Dic5 | — | 4 | 2 | no | — |
| 147 | C24×C3 | — | 10 | 1 | no | — |
| 148 | C2×C3×Q8 | — | 10 | 1 | no | — |
| 149 | C2×C3×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×D4 | — | 5 | 2 | no | — |
| 151 | C2×C3×D4 | — | 10 | 1 | no | — |
| 152 | C2×C3×D4 | — | 10 | 2 | no | — |
| 153 | C2×C3×D4 | D20 | 1 | 1 | yes | characteristic |
| 154 | #239 | — | 20 | 1 | no | — |
| 155 | D3×Q8 | — | 20 | 1 | no | — |
| 156 | #235 | — | 10 | 1 | no | — |
| 157 | #235 | — | 10 | 2 | no | — |
| 158 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 159 | C3×C22⋊C4 | — | 10 | 2 | no | — |
| 160 | C3×C22⋊C4 | — | 10 | 1 | no | — |
| 161 | C3×C22⋊C4 | — | 10 | 2 | no | — |
| 162 | C3×C22⋊C4 | — | 10 | 2 | no | — |
| 163 | C3×C42 | — | 5 | 2 | no | — |
| 164 | C3×C42 | D20 | 1 | 1 | yes | characteristic |
| 165 | #214 | — | 20 | 2 | no | — |
| 166 | #214 | — | 20 | 1 | no | — |
| 167 | #213 | — | 20 | 2 | no | — |
| 168 | #213 | — | 20 | 1 | no | — |
| 169 | #208 | — | 2 | 2 | no | — |
| 170 | #208 | — | 10 | 1 | no | — |
| 171 | C23×C5 | — | 2 | 1 | no | — |
| 172 | C22×D5 | — | 2 | 2 | no | — |
| 173 | C22×D5 | — | 8 | 2 | no | — |
| 174 | C22×D5 | — | 8 | 1 | no | — |
| 175 | C5×Q8 | — | 4 | 1 | no | — |
| 176 | C5×D4 | — | 4 | 1 | no | — |
| 177 | C2×C4×C5 | — | 2 | 1 | no | — |
| 178 | C2×C4×C5 | C2×#67 | 1 | 1 | yes | characteristic |
| 179 | C2×C4×C5 | C2×#67 | 1 | 2 | yes | — |
| 180 | C22⋊D5 | — | 8 | 2 | no | — |
| 181 | C22⋊D5 | — | 12 | 1 | no | — |
| 182 | C2×Dic5 | — | 2 | 2 | no | — |
| 183 | D20 | — | 4 | 2 | no | — |
| 184 | D20 | — | 6 | 2 | no | — |
| 185 | D20 | — | 6 | 1 | no | — |
| 186 | D20 | — | 8 | 1 | no | — |
| 187 | C4×D5 | — | 12 | 1 | no | — |
| 188 | C4×D5 | — | 12 | 2 | no | — |
| 189 | Dic10 | — | 6 | 2 | no | — |
| 190 | Dic10 | — | 6 | 1 | no | — |
| 191 | Dic10 | — | 8 | 1 | no | — |
| 192 | C5×C8 | — | 12 | 2 | no | — |
| 193 | #168 | — | 12 | 1 | no | — |
| 194 | 2−1+4 | — | 15 | 1 | no | — |
| 195 | AGL(1,ℤ/8) | — | 30 | 2 | no | — |
| 196 | AGL(1,ℤ/8) | — | 30 | 1 | no | — |
| 197 | #124 | — | 5 | 2 | no | — |
| 198 | #124 | — | 10 | 2 | no | — |
| 199 | #124 | — | 10 | 1 | no | — |
| 200 | #124 | D3×D5 | 1 | 1 | yes | characteristic |
| 201 | #120 | — | 10 | 2 | no | — |
| 202 | #120 | — | 10 | 1 | no | — |
| 203 | #104 | — | 30 | 2 | no | — |
| 204 | #104 | — | 30 | 1 | no | — |
| 205 | #101 | — | 3 | 2 | no | — |
| 206 | #101 | — | 15 | 1 | no | — |
| 207 | C2×C3×C5 | — | 2 | 1 | no | — |
| 208 | C2×C3×C5 | — | 8 | 1 | no | — |
| 209 | C2×C3×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 210 | D15 | — | 8 | 1 | no | — |
| 211 | C3×D5 | — | 8 | 2 | no | — |
| 212 | C23×C3 | — | 2 | 1 | no | — |
| 213 | C23×C3 | — | 10 | 2 | no | — |
| 214 | C23×C3 | — | 40 | 2 | no | — |
| 215 | C23×C3 | — | 40 | 1 | no | — |
| 216 | C3×Q8 | — | 4 | 1 | no | — |
| 217 | C3×Q8 | — | 40 | 1 | no | — |
| 218 | C3×D4 | — | 4 | 1 | no | — |
| 219 | C3×D4 | — | 20 | 2 | no | — |
| 220 | C3×D4 | — | 40 | 1 | no | — |
| 221 | C3×D4 | — | 40 | 2 | no | — |
| 222 | C2×C3×C4 | — | 2 | 1 | no | — |
| 223 | C2×C3×C4 | — | 10 | 2 | no | — |
| 224 | C2×C3×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 225 | C2×C3×C4 | D4×D5 | 1 | 2 | yes | — |
| 226 | #67 | — | 20 | 1 | no | — |
| 227 | D12 | — | 10 | 2 | no | — |
| 228 | D12 | — | 10 | 1 | no | — |
| 229 | C4×D3 | — | 20 | 2 | no | — |
| 230 | C4×D3 | — | 20 | 1 | no | — |
| 231 | Dic6 | — | 10 | 2 | no | — |
| 232 | Dic6 | — | 10 | 1 | no | — |
| 233 | C3⋊C8 | — | 4 | 2 | no | — |
| 234 | C3⋊C8 | — | 20 | 1 | no | — |
| 235 | C22×C5 | — | 4 | 2 | no | — |
| 236 | C22×C5 | — | 4 | 1 | no | — |
| 237 | C22×C5 | #953 | 1 | 1 | yes | characteristic |
| 238 | C2×D5 | — | 4 | 2 | no | — |
| 239 | C2×D5 | — | 4 | 2 | no | — |
| 240 | C2×D5 | — | 4 | 2 | no | — |
| 241 | C2×D5 | — | 8 | 4 | no | — |
| 242 | C2×D5 | — | 12 | 1 | no | — |
| 243 | C4×C5 | — | 2 | 1 | no | — |
| 244 | C4×C5 | — | 2 | 2 | no | — |
| 245 | C4×C5 | — | 4 | 1 | no | — |
| 246 | Dic5 | — | 4 | 2 | no | — |
| 247 | Dic5 | — | 12 | 1 | no | — |
| 248 | C24 | — | 10 | 1 | no | — |
| 249 | #41 | — | 30 | 1 | no | — |
| 250 | #41 | — | 30 | 2 | no | — |
| 251 | #41 | — | 60 | 1 | no | — |
| 252 | C2×Q8 | — | 10 | 1 | no | — |
| 253 | C2×Q8 | — | 60 | 1 | no | — |
| 254 | C2×Q8 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 255 | C2×D4 | — | 5 | 2 | no | — |
| 256 | C2×D4 | — | 10 | 1 | no | — |
| 257 | C2×D4 | — | 10 | 2 | no | — |
| 258 | C2×D4 | #1202 | 1 | 1 | yes | characteristic |
| 259 | SD16 | — | 60 | 2 | no | — |
| 260 | SD16 | — | 60 | 1 | no | — |
| 261 | D8 | — | 60 | 2 | no | — |
| 262 | D8 | — | 60 | 1 | no | — |
| 263 | M16 | — | 6 | 2 | no | — |
| 264 | M16 | — | 30 | 1 | no | — |
| 265 | C22⋊C4 | — | 2 | 2 | no | — |
| 266 | C22⋊C4 | — | 10 | 2 | no | — |
| 267 | C22⋊C4 | — | 10 | 1 | no | — |
| 268 | C22⋊C4 | — | 10 | 2 | no | — |
| 269 | C22⋊C4 | — | 10 | 2 | no | — |
| 270 | C42 | — | 5 | 2 | no | — |
| 271 | C42 | #1202 | 1 | 1 | yes | characteristic |
| 272 | C3×C5 | #2202 | 1 | 1 | yes | characteristic, complemented, Hall |
| 273 | C22×C3 | — | 4 | 2 | no | — |
| 274 | C22×C3 | — | 4 | 1 | no | — |
| 275 | C22×C3 | — | 20 | 2 | no | — |
| 276 | C22×C3 | — | 20 | 2 | no | — |
| 277 | C22×C3 | — | 20 | 2 | no | — |
| 278 | C22×C3 | — | 40 | 4 | no | — |
| 279 | C22×C3 | #4028 | 1 | 1 | yes | characteristic |
| 280 | C2×D3 | — | 20 | 1 | no | — |
| 281 | C3×C4 | — | 2 | 2 | no | — |
| 282 | C3×C4 | — | 2 | 1 | no | — |
| 283 | C3×C4 | — | 4 | 1 | no | — |
| 284 | C3×C4 | — | 20 | 2 | no | — |
| 285 | Dic3 | — | 20 | 1 | no | — |
| 286 | C2×C5 | — | 2 | 1 | no | — |
| 287 | C2×C5 | — | 8 | 1 | no | — |
| 288 | C2×C5 | #5244 | 1 | 1 | yes | characteristic |
| 289 | D5 | — | 8 | 2 | no | — |
| 290 | D5 | — | 24 | 1 | no | — |
| 291 | C23 | — | 2 | 1 | no | — |
| 292 | C23 | — | 10 | 2 | no | — |
| 293 | C23 | — | 40 | 2 | no | — |
| 294 | C23 | — | 40 | 1 | no | — |
| 295 | Q8 | — | 4 | 1 | no | — |
| 296 | Q8 | — | 30 | 2 | no | — |
| 297 | Q8 | — | 30 | 1 | no | — |
| 298 | Q8 | — | 40 | 1 | no | — |
| 299 | D4 | — | 4 | 1 | no | — |
| 300 | D4 | — | 20 | 2 | no | — |
| 301 | D4 | — | 30 | 2 | no | — |
| 302 | D4 | — | 30 | 1 | no | — |
| 303 | D4 | — | 40 | 1 | no | — |
| 304 | D4 | — | 40 | 2 | no | — |
| 305 | D4 | — | 60 | 1 | no | — |
| 306 | C2×C4 | — | 2 | 1 | no | — |
| 307 | C2×C4 | — | 10 | 2 | no | — |
| 308 | C2×C4 | — | 60 | 2 | no | — |
| 309 | C2×C4 | — | 60 | 1 | no | — |
| 310 | C2×C4 | C2×#1202 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | D5×#67 | 1 | 2 | yes | — |
| 312 | C8 | — | 12 | 2 | no | — |
| 313 | C8 | — | 60 | 1 | no | — |
| 314 | C2×C3 | — | 2 | 1 | no | — |
| 315 | C2×C3 | — | 8 | 1 | no | — |
| 316 | C2×C3 | — | 40 | 2 | no | — |
| 317 | C2×C3 | #65030 | 1 | 1 | yes | characteristic |
| 318 | D3 | — | 40 | 1 | no | — |
| 319 | C5 | #71951 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 320 | C22 | — | 4 | 2 | no | — |
| 321 | C22 | — | 4 | 1 | no | — |
| 322 | C22 | — | 20 | 2 | no | — |
| 323 | C22 | — | 20 | 2 | no | — |
| 324 | C22 | — | 20 | 2 | no | — |
| 325 | C22 | — | 40 | 4 | no | — |
| 326 | C22 | — | 60 | 1 | no | — |
| 327 | C22 | #91564 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 2 | 2 | no | — |
| 329 | C4 | — | 2 | 1 | no | — |
| 330 | C4 | — | 4 | 1 | no | — |
| 331 | C4 | — | 20 | 2 | no | — |
| 332 | C4 | — | 60 | 1 | no | — |
| 333 | C3 | #10599953 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 334 | C2 | — | 2 | 1 | no | — |
| 335 | C2 | — | 8 | 1 | no | — |
| 336 | C2 | — | 40 | 2 | no | — |
| 337 | C2 | — | 120 | 1 | no | — |
| 338 | C2 | #11749176 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#66017 | C2 | semidirect |
| 2. #11751163 | C2 | semidirect |
| 3. #11749388 | C2 | semidirect |
| 4. C5×#4542 | C2 | semidirect |
| 5. #11746012 | C2 | semidirect |
| 7. #92098 | C22 | semidirect |
| 16. #91120 | C22 | semidirect |
| 18. C5×#836 | C22 | semidirect |
| 21. #66017 | D3 | semidirect |
| 54. #4542 | D5 | semidirect |
| 102. #836 | C2×D5 | semidirect |
| 272. C3×C5 | #2202 | semidirect |
| 319. C5 | #71951 | semidirect |
| 333. C3 | #10599953 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#66017 | 2 |
| #11751163 | 2 |
| #11749388 | 2 |
| C5×#4542 | 2 |
| #11746012 | 2 |
| #10599953 | 3 |
| #71951 | 5 |