Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
638 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 638 conjugacy classes of subgroups — too many to draw. Only its 59 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 | C2×C5⋊#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#91956 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#5424 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11751602 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #10610565 | — | 3 | 1 | no | maximal, Hall |
| 6 | C2×C5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C5⋊#994 | — | 2 | 2 | no | — |
| 8 | C2×C5⋊(#41⋊C3) | C22 | 1 | 2 | yes | complemented |
| 9 | #91956 | — | 2 | 2 | no | — |
| 10 | C2×C5⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 11 | #91374 | — | 4 | 1 | no | maximal |
| 12 | C5×#859 | C22 | 1 | 2 | yes | complemented |
| 13 | #85530 | — | 5 | 1 | no | maximal, Hall |
| 14 | C2×#4149 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×#4095 | — | 3 | 1 | no | — |
| 16 | C22×D40 | — | 3 | 1 | no | — |
| 17 | #65859 | — | 3 | 1 | no | — |
| 18 | C5×#416 | — | 3 | 1 | no | — |
| 19 | #65399 | — | 3 | 1 | no | — |
| 20 | #65290 | — | 3 | 1 | no | — |
| 21 | #65100 | — | 3 | 4 | no | — |
| 22 | #65062 | — | 3 | 4 | no | — |
| 23 | C2×D60 | — | 4 | 1 | no | — |
| 24 | C2×C3×Dic10 | — | 4 | 1 | no | — |
| 25 | C5×#41⋊C3 | D4 | 1 | 2 | yes | complemented |
| 26 | C2×C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 27 | C5⋊(#41⋊C3) | — | 2 | 2 | no | — |
| 28 | C5⋊GL(2,𝔽3) | — | 2 | 2 | no | — |
| 29 | C4×C5×Dic3 | — | 4 | 1 | no | — |
| 30 | #6667 | — | 4 | 4 | no | — |
| 31 | C2×#994 | — | 5 | 1 | no | — |
| 32 | C2×#986 | — | 5 | 1 | no | — |
| 33 | #5427 | — | 5 | 4 | no | — |
| 34 | #5424 | D5 | 1 | 1 | yes | characteristic, complemented |
| 35 | C2×C5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 36 | #4149 | — | 2 | 2 | no | — |
| 37 | #4149 | — | 6 | 2 | no | — |
| 38 | C2×#713 | C2×D3 | 1 | 2 | yes | — |
| 39 | C2×D4×D5 | — | 3 | 2 | no | — |
| 40 | C2×#709 | — | 3 | 1 | no | — |
| 41 | C22×D20 | — | 3 | 1 | no | — |
| 42 | C22×D20 | — | 3 | 1 | no | — |
| 43 | C22×C5×C8 | — | 3 | 1 | no | — |
| 44 | C5×#117 | — | 3 | 1 | no | — |
| 45 | #4095 | — | 6 | 2 | no | — |
| 46 | #4095 | — | 6 | 2 | no | — |
| 47 | C2×#688 | — | 3 | 2 | no | — |
| 48 | C2×#686 | — | 3 | 2 | no | — |
| 49 | C2×#681 | — | 3 | 1 | no | — |
| 50 | C2×D40 | — | 3 | 4 | no | — |
| 51 | C2×D40 | — | 6 | 4 | no | — |
| 52 | #4041 | — | 3 | 4 | no | — |
| 53 | #4028 | — | 6 | 2 | no | — |
| 54 | #4021 | — | 6 | 1 | no | — |
| 55 | #4020 | — | 6 | 1 | no | — |
| 56 | C5×#103 | — | 3 | 2 | no | — |
| 57 | C5×#102 | — | 3 | 2 | no | — |
| 58 | #3952 | — | 3 | 4 | no | — |
| 59 | #3941 | — | 3 | 4 | no | — |
| 60 | #3008 | — | 15 | 1 | no | Sylow |
| 61 | C22×D15 | — | 4 | 2 | no | — |
| 62 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 63 | D60 | — | 8 | 2 | no | — |
| 64 | C2×C5×Dic3 | — | 4 | 2 | no | — |
| 65 | C2×C3×Dic5 | — | 4 | 2 | no | — |
| 66 | C3×Dic10 | — | 8 | 2 | no | — |
| 67 | C5×SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | #994 | — | 10 | 2 | no | — |
| 69 | C2×#41⋊C3 | — | 5 | 2 | no | — |
| 70 | C2×#41⋊C3 | C2×D5 | 1 | 1 | yes | characteristic |
| 71 | #986 | — | 10 | 2 | no | — |
| 72 | C2×GL(2,𝔽3) | — | 5 | 2 | no | — |
| 73 | #947 | — | 20 | 1 | no | — |
| 74 | #859 | C2×D5 | 1 | 2 | yes | complemented |
| 75 | C23×D5 | — | 6 | 1 | no | — |
| 76 | C23×D5 | — | 6 | 1 | no | — |
| 77 | C5×#41 | — | 3 | 2 | no | — |
| 78 | C5×#41 | #67 | 1 | 2 | yes | — |
| 79 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 80 | C2×C5×D4 | — | 3 | 1 | no | — |
| 81 | C22×C4×C5 | — | 3 | 1 | no | — |
| 82 | C2×C22⋊D5 | — | 3 | 2 | no | — |
| 83 | #713 | — | 2 | 2 | no | — |
| 84 | #713 | — | 6 | 2 | no | — |
| 85 | D4×D5 | — | 6 | 4 | no | — |
| 86 | D4×D5 | — | 6 | 4 | no | — |
| 87 | #709 | — | 6 | 2 | no | — |
| 88 | #709 | — | 6 | 2 | no | — |
| 89 | C2×D20 | — | 3 | 2 | no | — |
| 90 | C2×D20 | — | 3 | 4 | no | — |
| 91 | C2×D20 | — | 3 | 2 | no | — |
| 92 | C2×D20 | — | 3 | 1 | no | — |
| 93 | C2×D20 | — | 6 | 2 | no | — |
| 94 | C2×D20 | — | 6 | 2 | no | — |
| 95 | C2×D20 | — | 6 | 2 | no | — |
| 96 | C2×D20 | — | 6 | 1 | no | — |
| 97 | C2×D20 | — | 6 | 4 | no | — |
| 98 | C2×C4×D5 | — | 3 | 2 | no | — |
| 99 | C2×Dic10 | S4 | 1 | 1 | yes | characteristic |
| 100 | C2×C5×C8 | — | 3 | 2 | no | — |
| 101 | C2×C5×C8 | — | 6 | 2 | no | — |
| 102 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 103 | C5×C22⋊C4 | — | 6 | 1 | no | — |
| 104 | C42×C5 | — | 6 | 1 | no | — |
| 105 | #688 | — | 6 | 2 | no | — |
| 106 | #688 | — | 6 | 2 | no | — |
| 107 | #686 | — | 6 | 4 | no | — |
| 108 | #685 | — | 6 | 4 | no | — |
| 109 | #681 | — | 6 | 2 | no | — |
| 110 | C2×#168 | — | 3 | 2 | no | — |
| 111 | D40 | — | 6 | 8 | no | — |
| 112 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 113 | C2×AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 114 | C22×D8 | — | 15 | 1 | no | — |
| 115 | #535 | — | 15 | 1 | no | — |
| 116 | #449 | — | 15 | 4 | no | — |
| 117 | #447 | — | 15 | 4 | no | — |
| 118 | #418 | — | 15 | 1 | no | — |
| 119 | #416 | — | 3 | 1 | no | — |
| 120 | #408 | — | 15 | 1 | no | — |
| 121 | C22×C3×C5 | — | 4 | 1 | no | — |
| 122 | C2×D15 | — | 8 | 2 | no | — |
| 123 | C2×D15 | — | 8 | 4 | no | — |
| 124 | C3×C4×C5 | — | 4 | 2 | no | — |
| 125 | C3×Dic5 | — | 8 | 2 | no | — |
| 126 | C5×Dic3 | — | 8 | 2 | no | — |
| 127 | C2×C3×Q8 | — | 20 | 1 | no | — |
| 128 | C2×D12 | — | 20 | 1 | no | — |
| 129 | #41⋊C3 | — | 10 | 2 | no | — |
| 130 | #41⋊C3 | D20 | 1 | 2 | yes | complemented |
| 131 | C2×SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic |
| 132 | GL(2,𝔽3) | — | 10 | 2 | no | — |
| 133 | (C2×D3)⋊C4 | — | 20 | 4 | no | — |
| 134 | C4×Dic3 | — | 4 | 1 | no | — |
| 135 | C23×C5 | — | 3 | 1 | no | — |
| 136 | C22×D5 | — | 3 | 2 | no | — |
| 137 | C22×D5 | — | 6 | 4 | no | — |
| 138 | C22×D5 | — | 6 | 2 | no | — |
| 139 | C22×D5 | — | 12 | 2 | no | — |
| 140 | C22×D5 | — | 24 | 2 | no | — |
| 141 | C22×D5 | — | 24 | 2 | no | — |
| 142 | C5×Q8 | — | 3 | 1 | no | — |
| 143 | C5×Q8 | C2×#67 | 1 | 1 | yes | characteristic |
| 144 | C5×D4 | — | 3 | 2 | no | — |
| 145 | C5×D4 | — | 6 | 1 | no | — |
| 146 | C2×C4×C5 | — | 3 | 1 | no | — |
| 147 | C2×C4×C5 | — | 3 | 2 | no | — |
| 148 | C2×C4×C5 | — | 3 | 2 | no | — |
| 149 | C2×C4×C5 | — | 6 | 2 | no | — |
| 150 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 151 | C22⋊D5 | — | 6 | 4 | no | — |
| 152 | C2×Dic5 | C2×S4 | 1 | 2 | yes | — |
| 153 | D20 | — | 3 | 4 | no | — |
| 154 | D20 | — | 6 | 2 | no | — |
| 155 | D20 | — | 6 | 2 | no | — |
| 156 | D20 | — | 6 | 2 | no | — |
| 157 | D20 | — | 6 | 2 | no | — |
| 158 | D20 | — | 12 | 2 | no | — |
| 159 | D20 | — | 24 | 4 | no | — |
| 160 | C4×D5 | — | 6 | 2 | no | — |
| 161 | C4×D5 | — | 6 | 2 | no | — |
| 162 | Dic10 | — | 2 | 2 | no | — |
| 163 | C5×C8 | — | 6 | 2 | no | — |
| 164 | #168 | — | 6 | 2 | no | — |
| 165 | H(5,𝔽2) | — | 10 | 2 | no | — |
| 166 | H(5,𝔽2) | — | 30 | 2 | no | — |
| 167 | C2×#41 | — | 5 | 2 | no | — |
| 168 | C2×#41 | — | 15 | 1 | no | — |
| 169 | C2×#41 | D3×D5 | 1 | 1 | yes | characteristic |
| 170 | C22×D4 | — | 15 | 1 | no | — |
| 171 | C22×D4 | — | 15 | 1 | no | — |
| 172 | C22×D4 | — | 15 | 2 | no | — |
| 173 | AGL(1,ℤ/8) | — | 30 | 2 | no | — |
| 174 | AGL(1,ℤ/8) | — | 30 | 2 | no | — |
| 175 | C2×SD16 | — | 15 | 2 | no | — |
| 176 | C2×D8 | — | 15 | 4 | no | — |
| 177 | C2×D8 | — | 15 | 2 | no | — |
| 178 | C2×D8 | — | 30 | 4 | no | — |
| 179 | C2×M16 | — | 15 | 1 | no | — |
| 180 | C22×C8 | — | 3 | 1 | no | — |
| 181 | #127 | — | 30 | 1 | no | — |
| 182 | #124 | — | 30 | 1 | no | — |
| 183 | #121 | — | 15 | 4 | no | — |
| 184 | #120 | — | 30 | 2 | no | — |
| 185 | #117 | — | 3 | 1 | no | — |
| 186 | #103 | — | 3 | 2 | no | — |
| 187 | #102 | — | 3 | 2 | no | — |
| 188 | #102 | — | 15 | 4 | no | — |
| 189 | C22⋊C8 | — | 15 | 4 | no | — |
| 190 | C2×C3×C5 | — | 4 | 2 | no | — |
| 191 | C2×C3×C5 | — | 4 | 1 | no | — |
| 192 | D15 | — | 16 | 2 | no | — |
| 193 | C22×D3 | — | 20 | 2 | no | — |
| 194 | C3×Q8 | — | 40 | 2 | no | — |
| 195 | C2×C3×C4 | — | 4 | 1 | no | — |
| 196 | C2×C3×C4 | — | 20 | 2 | no | — |
| 197 | C2×Dic3 | — | 4 | 2 | no | — |
| 198 | D12 | — | 40 | 2 | no | — |
| 199 | SL(2,𝔽3) | C2×D20 | 1 | 1 | yes | characteristic |
| 200 | C22×C5 | — | 3 | 2 | no | — |
| 201 | C22×C5 | — | 6 | 2 | no | — |
| 202 | C22×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 203 | C2×D5 | — | 6 | 4 | no | — |
| 204 | C2×D5 | — | 6 | 2 | no | — |
| 205 | C2×D5 | — | 12 | 4 | no | — |
| 206 | C2×D5 | — | 12 | 2 | no | — |
| 207 | C2×D5 | — | 24 | 4 | no | — |
| 208 | C2×D5 | — | 24 | 4 | no | — |
| 209 | C4×C5 | — | 3 | 1 | no | — |
| 210 | C4×C5 | — | 3 | 1 | no | — |
| 211 | C4×C5 | — | 12 | 2 | no | — |
| 212 | C4×C5 | GL(2,ℤ/4) | 1 | 2 | yes | — |
| 213 | Dic5 | — | 2 | 2 | no | — |
| 214 | C24 | — | 30 | 1 | no | — |
| 215 | C24 | — | 30 | 1 | no | — |
| 216 | #41 | — | 3 | 2 | no | — |
| 217 | #41 | — | 10 | 2 | no | — |
| 218 | #41 | — | 30 | 2 | no | — |
| 219 | #41 | — | 30 | 2 | no | — |
| 220 | #41 | — | 30 | 2 | no | — |
| 221 | #41 | #1202 | 1 | 2 | yes | — |
| 222 | C2×Q8 | — | 5 | 1 | no | — |
| 223 | C2×Q8 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 224 | C2×D4 | — | 3 | 1 | no | — |
| 225 | C2×D4 | — | 15 | 2 | no | — |
| 226 | C2×D4 | — | 15 | 4 | no | — |
| 227 | C2×D4 | — | 15 | 2 | no | — |
| 228 | C2×D4 | — | 15 | 2 | no | — |
| 229 | C2×D4 | — | 15 | 1 | no | — |
| 230 | C2×D4 | — | 30 | 2 | no | — |
| 231 | C2×D4 | — | 30 | 2 | no | — |
| 232 | C2×D4 | — | 30 | 2 | no | — |
| 233 | C2×D4 | — | 30 | 4 | no | — |
| 234 | C2×D4 | — | 30 | 4 | no | — |
| 235 | C2×D4 | — | 30 | 4 | no | — |
| 236 | C2×D4 | — | 30 | 1 | no | — |
| 237 | C22×C4 | — | 3 | 1 | no | — |
| 238 | C22×C4 | — | 15 | 2 | no | — |
| 239 | SD16 | — | 30 | 2 | no | — |
| 240 | SD16 | — | 30 | 2 | no | — |
| 241 | D8 | — | 30 | 8 | no | — |
| 242 | D8 | — | 30 | 4 | no | — |
| 243 | M16 | — | 30 | 2 | no | — |
| 244 | C2×C8 | — | 3 | 2 | no | — |
| 245 | C2×C8 | — | 6 | 2 | no | — |
| 246 | C2×C8 | — | 15 | 2 | no | — |
| 247 | C4⋊C4 | — | 3 | 2 | no | — |
| 248 | C22⋊C4 | — | 6 | 1 | no | — |
| 249 | C22⋊C4 | — | 30 | 4 | no | — |
| 250 | C42 | — | 6 | 1 | no | — |
| 251 | C3×C5 | — | 4 | 1 | no | Hall |
| 252 | C22×C3 | — | 4 | 1 | no | — |
| 253 | C2×D3 | — | 40 | 2 | no | — |
| 254 | C2×D3 | — | 40 | 4 | no | — |
| 255 | C3×C4 | — | 4 | 2 | no | — |
| 256 | C3×C4 | — | 40 | 2 | no | — |
| 257 | Dic3 | — | 8 | 2 | no | — |
| 258 | C2×C5 | — | 6 | 2 | no | — |
| 259 | C2×C5 | #5925 | 1 | 2 | yes | — |
| 260 | C2×C5 | C2×GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 261 | D5 | — | 12 | 2 | no | — |
| 262 | D5 | — | 24 | 2 | no | — |
| 263 | C23 | — | 3 | 1 | no | — |
| 264 | C23 | — | 15 | 2 | no | — |
| 265 | C23 | — | 30 | 4 | no | — |
| 266 | C23 | — | 30 | 2 | no | — |
| 267 | C23 | — | 60 | 2 | no | — |
| 268 | C23 | — | 120 | 2 | no | — |
| 269 | C23 | — | 120 | 2 | no | — |
| 270 | Q8 | — | 3 | 1 | no | — |
| 271 | Q8 | — | 10 | 2 | no | — |
| 272 | Q8 | C2×#1202 | 1 | 1 | yes | characteristic |
| 273 | D4 | — | 3 | 2 | no | — |
| 274 | D4 | — | 6 | 1 | no | — |
| 275 | D4 | — | 15 | 4 | no | — |
| 276 | D4 | — | 30 | 2 | no | — |
| 277 | D4 | — | 30 | 2 | no | — |
| 278 | D4 | — | 30 | 2 | no | — |
| 279 | D4 | — | 30 | 4 | no | — |
| 280 | D4 | — | 30 | 2 | no | — |
| 281 | D4 | — | 60 | 2 | no | — |
| 282 | D4 | — | 120 | 4 | no | — |
| 283 | C2×C4 | — | 3 | 1 | no | — |
| 284 | C2×C4 | — | 3 | 2 | no | — |
| 285 | C2×C4 | — | 3 | 2 | no | — |
| 286 | C2×C4 | — | 5 | 2 | no | — |
| 287 | C2×C4 | — | 6 | 2 | no | — |
| 288 | C2×C4 | — | 30 | 2 | no | — |
| 289 | C2×C4 | — | 30 | 2 | no | — |
| 290 | C2×C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 291 | C8 | — | 6 | 2 | no | — |
| 292 | C8 | — | 30 | 2 | no | — |
| 293 | C2×C3 | — | 4 | 2 | no | — |
| 294 | C2×C3 | — | 4 | 1 | no | — |
| 295 | D3 | — | 80 | 2 | no | — |
| 296 | C5 | #85530 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 297 | C22 | — | 3 | 2 | no | — |
| 298 | C22 | — | 6 | 2 | no | — |
| 299 | C22 | — | 30 | 4 | no | — |
| 300 | C22 | — | 30 | 2 | no | — |
| 301 | C22 | — | 60 | 4 | no | — |
| 302 | C22 | — | 60 | 2 | no | — |
| 303 | C22 | — | 120 | 4 | no | — |
| 304 | C22 | — | 120 | 4 | no | — |
| 305 | C22 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| 306 | C4 | — | 3 | 1 | no | — |
| 307 | C4 | — | 3 | 1 | no | — |
| 308 | C4 | — | 10 | 2 | no | — |
| 309 | C4 | — | 12 | 2 | no | — |
| 310 | C4 | #91905 | 1 | 2 | yes | — |
| 311 | C3 | — | 4 | 1 | no | Sylow |
| 312 | C2 | — | 6 | 2 | no | — |
| 313 | C2 | — | 60 | 2 | no | — |
| 314 | C2 | — | 120 | 2 | no | — |
| 315 | C2 | #11756775 | 1 | 2 | yes | — |
| 316 | C2 | C2×#91905 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5⋊#994 | C2 | semidirect |
| 2. C2×#91956 | C2 | semidirect |
| 3. C5×#5424 | C2 | semidirect |
| 4. #11751602 | C2 | semidirect |
| 8. C2×C5⋊(#41⋊C3) | C22 | semidirect |
| 10. C2×C5⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C5×#859 | C22 | semidirect |
| 14. C2×#4149 | D3 | semidirect |
| 25. C5×#41⋊C3 | D4 | semidirect |
| 34. #5424 | D5 | semidirect |
| 74. #859 | C2×D5 | semidirect |
| 130. #41⋊C3 | D20 | semidirect |
| 296. C5 | #85530 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5⋊#994 | 2 |
| C2×#91956 | 2 |
| C5×#5424 | 2 |
| #11751602 | 2 |
| #10610565 | 3 |
| #91374 | 4 |
| #85530 | 5 |