Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
398 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 50 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×D5×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#4133⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10617543 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 4 | C2×C5×#41⋊C3 | C4 | 1 | 1 | yes | characteristic |
| 5 | D5×#41⋊C3 | — | 2 | 2 | no | — |
| 6 | C2×D5×SL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×C5⋊(#41⋊C3) | C22 | 1 | 1 | yes | characteristic |
| 8 | #4133⋊C3 | — | 2 | 2 | no | — |
| 9 | C3×#4003 | — | 4 | 1 | no | maximal |
| 10 | #2880⋊C3 | — | 5 | 1 | no | maximal, Hall |
| 11 | C2×D5×#41 | C2×C3 | 1 | 1 | yes | characteristic |
| 12 | C2×#4133 | C2×C3 | 1 | 2 | yes | complemented |
| 13 | #65776 | — | 3 | 2 | no | — |
| 14 | #65756 | — | 3 | 1 | no | — |
| 15 | D5⋊(C22⋊C8) | — | 3 | 1 | no | — |
| 16 | C2×C3×C4×D5 | — | 4 | 1 | no | — |
| 17 | C5×#41⋊C3 | — | 2 | 1 | no | — |
| 18 | C2×C5×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×#170 | — | 4 | 2 | no | — |
| 20 | D5×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 21 | C5⋊(#41⋊C3) | D4 | 1 | 2 | yes | — |
| 22 | C22×#41⋊C3 | — | 5 | 1 | no | — |
| 23 | C2×#131⋊C3 | — | 5 | 2 | no | — |
| 24 | C2×C5×#41 | C3×C4 | 1 | 1 | yes | characteristic |
| 25 | D5×#41 | — | 2 | 2 | no | — |
| 26 | D5×#41 | — | 6 | 2 | no | — |
| 27 | C2×#713 | C22×C3 | 1 | 1 | yes | characteristic |
| 28 | C2×Q8×D5 | C3×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×#711 | — | 3 | 1 | no | — |
| 30 | C2×D4×D5 | — | 3 | 1 | no | — |
| 31 | C2×#709 | — | 3 | 1 | no | — |
| 32 | C22×C4×D5 | — | 3 | 1 | no | — |
| 33 | #4133 | — | 2 | 2 | no | — |
| 34 | #4133 | — | 6 | 2 | no | — |
| 35 | C2×#700 | — | 3 | 2 | no | — |
| 36 | C2×D5⋊C8 | — | 3 | 2 | no | — |
| 37 | C22⋊#170 | — | 3 | 2 | no | — |
| 38 | C22⋊#170 | — | 3 | 2 | no | — |
| 39 | #4003 | — | 3 | 1 | no | — |
| 40 | #4003 | A4 | 1 | 1 | yes | characteristic |
| 41 | #2880 | — | 5 | 1 | no | Sylow |
| 42 | C22×C3×D5 | — | 4 | 1 | no | — |
| 43 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 44 | C2×C3×Dic5 | — | 4 | 1 | no | — |
| 45 | C3×C4×D5 | — | 8 | 2 | no | — |
| 46 | C5×SL(2,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | C3×#170 | — | 8 | 2 | no | — |
| 48 | C2×#41⋊C3 | — | 5 | 1 | no | — |
| 49 | C2×#41⋊C3 | — | 10 | 2 | no | — |
| 50 | C2×#41⋊C3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 51 | C22×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 52 | #131⋊C3 | — | 10 | 2 | no | — |
| 53 | C3×C22⋊C8 | — | 20 | 1 | no | — |
| 54 | C23×D5 | — | 3 | 1 | no | — |
| 55 | C5×#41 | — | 2 | 1 | no | — |
| 56 | C5×#41 | — | 6 | 1 | no | — |
| 57 | C2×C5×Q8 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 58 | C2×C5×D4 | — | 3 | 1 | no | — |
| 59 | C22×C4×C5 | — | 3 | 1 | no | — |
| 60 | C2×C22⋊D5 | — | 3 | 2 | no | — |
| 61 | C22×Dic5 | — | 3 | 1 | no | — |
| 62 | #713 | — | 3 | 2 | no | — |
| 63 | #713 | C3×D4 | 1 | 2 | yes | — |
| 64 | Q8×D5 | — | 2 | 1 | no | — |
| 65 | Q8×D5 | — | 6 | 1 | no | — |
| 66 | #711 | — | 6 | 2 | no | — |
| 67 | #711 | — | 6 | 2 | no | — |
| 68 | D4×D5 | — | 6 | 2 | no | — |
| 69 | D4×D5 | — | 6 | 2 | no | — |
| 70 | #709 | — | 6 | 2 | no | — |
| 71 | #709 | — | 6 | 2 | no | — |
| 72 | C2×D20 | — | 3 | 1 | no | — |
| 73 | C2×C4×D5 | — | 3 | 1 | no | — |
| 74 | C2×C4×D5 | — | 3 | 1 | no | — |
| 75 | C2×C4×D5 | — | 3 | 1 | no | — |
| 76 | C2×C4×D5 | — | 6 | 2 | no | — |
| 77 | C2×C4×D5 | — | 6 | 2 | no | — |
| 78 | C2×C4×D5 | C2×A4 | 1 | 1 | yes | characteristic |
| 79 | C2×Dic10 | — | 3 | 1 | no | — |
| 80 | C2×#170 | — | 3 | 2 | no | — |
| 81 | C2×#170 | C2×A4 | 1 | 2 | yes | — |
| 82 | #700 | — | 6 | 2 | no | — |
| 83 | #700 | — | 6 | 2 | no | — |
| 84 | D5⋊C8 | — | 6 | 2 | no | — |
| 85 | D5⋊C8 | — | 6 | 2 | no | — |
| 86 | C22×#41 | — | 5 | 1 | no | — |
| 87 | C2×#131 | — | 5 | 2 | no | — |
| 88 | #408 | — | 15 | 2 | no | — |
| 89 | #407 | — | 15 | 1 | no | — |
| 90 | C2×C22⋊C8 | — | 15 | 1 | no | — |
| 91 | C22×C3×C5 | — | 4 | 1 | no | — |
| 92 | C2×C3×D5 | — | 4 | 2 | no | — |
| 93 | C2×C3×D5 | — | 8 | 1 | no | — |
| 94 | C2×C3×D5 | — | 8 | 1 | no | — |
| 95 | C3×C4×C5 | — | 8 | 1 | no | — |
| 96 | C3×Dic5 | — | 4 | 2 | no | — |
| 97 | C22×C3×C4 | — | 20 | 1 | no | — |
| 98 | #41⋊C3 | — | 2 | 1 | no | — |
| 99 | #41⋊C3 | — | 5 | 2 | no | — |
| 100 | C2×SL(2,𝔽3) | — | 10 | 1 | no | — |
| 101 | C2×SL(2,𝔽3) | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 102 | C2×C3×C8 | — | 20 | 2 | no | — |
| 103 | C23×C5 | — | 3 | 1 | no | — |
| 104 | C22×D5 | — | 3 | 1 | no | — |
| 105 | C22×D5 | — | 6 | 2 | no | — |
| 106 | C22×D5 | — | 6 | 2 | no | — |
| 107 | C22×D5 | — | 12 | 1 | no | — |
| 108 | C22×D5 | C4×A4 | 1 | 1 | yes | characteristic |
| 109 | C5×Q8 | — | 3 | 1 | no | — |
| 110 | C5×Q8 | C3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C5×D4 | — | 6 | 1 | no | — |
| 112 | C5×D4 | — | 6 | 1 | no | — |
| 113 | C2×C4×C5 | — | 3 | 1 | no | — |
| 114 | C2×C4×C5 | — | 6 | 1 | no | — |
| 115 | C2×C4×C5 | — | 6 | 1 | no | — |
| 116 | C2×C4×C5 | C4×A4 | 1 | 1 | yes | characteristic |
| 117 | C22⋊D5 | — | 6 | 4 | no | — |
| 118 | C2×Dic5 | — | 3 | 1 | no | — |
| 119 | C2×Dic5 | — | 6 | 2 | no | — |
| 120 | C2×Dic5 | C22×A4 | 1 | 1 | yes | characteristic |
| 121 | D20 | — | 3 | 2 | no | — |
| 122 | D20 | — | 3 | 2 | no | — |
| 123 | C4×D5 | — | 2 | 2 | no | — |
| 124 | C4×D5 | — | 3 | 2 | no | — |
| 125 | C4×D5 | — | 3 | 2 | no | — |
| 126 | C4×D5 | — | 6 | 2 | no | — |
| 127 | C4×D5 | — | 6 | 1 | no | — |
| 128 | C4×D5 | — | 6 | 1 | no | — |
| 129 | Dic10 | — | 6 | 1 | no | — |
| 130 | Dic10 | — | 6 | 1 | no | — |
| 131 | #170 | — | 2 | 2 | no | — |
| 132 | #170 | — | 6 | 2 | no | — |
| 133 | C2×#41 | — | 5 | 1 | no | — |
| 134 | C2×#41 | — | 10 | 2 | no | — |
| 135 | C2×#41 | — | 15 | 1 | no | — |
| 136 | C2×#41 | — | 15 | 1 | no | — |
| 137 | C2×#41 | — | 30 | 2 | no | — |
| 138 | C2×#41 | C3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 139 | C22×Q8 | — | 5 | 1 | no | — |
| 140 | C22×D4 | — | 15 | 1 | no | — |
| 141 | C23×C4 | — | 15 | 1 | no | — |
| 142 | #131 | — | 10 | 2 | no | — |
| 143 | #131 | — | 30 | 2 | no | — |
| 144 | C2×M16 | — | 15 | 2 | no | — |
| 145 | C22×C8 | — | 15 | 2 | no | — |
| 146 | C22⋊C8 | — | 5 | 1 | no | — |
| 147 | C22⋊C8 | — | 15 | 2 | no | — |
| 148 | C22⋊C8 | — | 15 | 1 | no | — |
| 149 | C22⋊C8 | — | 15 | 2 | no | — |
| 150 | C2×C3×C5 | — | 4 | 1 | no | — |
| 151 | C2×C3×C5 | — | 4 | 1 | no | — |
| 152 | C2×C3×C5 | — | 4 | 1 | no | — |
| 153 | C3×D5 | — | 8 | 2 | no | — |
| 154 | C23×C3 | — | 20 | 1 | no | — |
| 155 | C2×C3×C4 | — | 4 | 1 | no | — |
| 156 | C2×C3×C4 | — | 20 | 1 | no | — |
| 157 | C2×C3×C4 | — | 40 | 2 | no | — |
| 158 | SL(2,𝔽3) | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 159 | C3×C8 | — | 40 | 2 | no | — |
| 160 | C22×C5 | — | 6 | 1 | no | — |
| 161 | C22×C5 | — | 6 | 1 | no | — |
| 162 | C22×C5 | — | 6 | 1 | no | — |
| 163 | C22×C5 | C2×C4×A4 | 1 | 1 | yes | characteristic |
| 164 | C2×D5 | — | 2 | 1 | no | — |
| 165 | C2×D5 | — | 2 | 1 | no | — |
| 166 | C2×D5 | — | 3 | 2 | no | — |
| 167 | C2×D5 | — | 6 | 1 | no | — |
| 168 | C2×D5 | — | 6 | 1 | no | — |
| 169 | C2×D5 | — | 12 | 4 | no | — |
| 170 | C2×D5 | #131⋊C3 | 1 | 2 | yes | complemented |
| 171 | C4×C5 | — | 2 | 1 | no | — |
| 172 | C4×C5 | — | 3 | 1 | no | — |
| 173 | C4×C5 | — | 3 | 1 | no | — |
| 174 | Dic5 | — | 6 | 1 | no | — |
| 175 | Dic5 | D4×A4 | 1 | 2 | yes | — |
| 176 | C24 | — | 15 | 1 | no | — |
| 177 | #41 | — | 2 | 1 | no | — |
| 178 | #41 | — | 5 | 2 | no | — |
| 179 | #41 | — | 6 | 1 | no | — |
| 180 | #41 | — | 15 | 2 | no | — |
| 181 | #41 | — | 30 | 2 | no | — |
| 182 | #41 | — | 30 | 2 | no | — |
| 183 | #41 | — | 30 | 2 | no | — |
| 184 | #41 | — | 30 | 2 | no | — |
| 185 | C2×Q8 | — | 10 | 1 | no | — |
| 186 | C2×Q8 | — | 15 | 1 | no | — |
| 187 | C2×Q8 | — | 30 | 1 | no | — |
| 188 | C2×Q8 | C2×C3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 189 | C2×D4 | — | 3 | 1 | no | — |
| 190 | C2×D4 | — | 15 | 1 | no | — |
| 191 | C2×D4 | — | 15 | 2 | no | — |
| 192 | C2×D4 | — | 30 | 2 | no | — |
| 193 | C2×D4 | — | 30 | 2 | no | — |
| 194 | C22×C4 | — | 3 | 1 | no | — |
| 195 | C22×C4 | — | 5 | 1 | no | — |
| 196 | C22×C4 | — | 15 | 1 | no | — |
| 197 | C22×C4 | — | 15 | 1 | no | — |
| 198 | C22×C4 | — | 15 | 1 | no | — |
| 199 | C22×C4 | — | 15 | 1 | no | — |
| 200 | C22×C4 | — | 30 | 2 | no | — |
| 201 | C22×C4 | — | 30 | 2 | no | — |
| 202 | M16 | — | 30 | 2 | no | — |
| 203 | M16 | — | 30 | 2 | no | — |
| 204 | C2×C8 | — | 5 | 2 | no | — |
| 205 | C2×C8 | — | 15 | 2 | no | — |
| 206 | C2×C8 | — | 30 | 2 | no | — |
| 207 | C2×C8 | — | 30 | 2 | no | — |
| 208 | C3×C5 | — | 4 | 1 | no | Hall |
| 209 | C22×C3 | — | 4 | 1 | no | — |
| 210 | C22×C3 | — | 20 | 2 | no | — |
| 211 | C22×C3 | — | 40 | 1 | no | — |
| 212 | C22×C3 | — | 40 | 1 | no | — |
| 213 | C3×C4 | — | 8 | 1 | no | — |
| 214 | C3×C4 | — | 20 | 2 | no | — |
| 215 | C2×C5 | — | 12 | 1 | no | — |
| 216 | C2×C5 | A4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 217 | C2×C5 | C2×#131⋊C3 | 1 | 1 | yes | characteristic |
| 218 | C2×C5 | #568⋊C3 | 1 | 1 | yes | characteristic |
| 219 | D5 | — | 2 | 2 | no | — |
| 220 | D5 | — | 6 | 2 | no | — |
| 221 | C23 | — | 3 | 1 | no | — |
| 222 | C23 | — | 5 | 1 | no | — |
| 223 | C23 | — | 15 | 1 | no | — |
| 224 | C23 | — | 30 | 2 | no | — |
| 225 | C23 | — | 30 | 2 | no | — |
| 226 | C23 | — | 60 | 1 | no | — |
| 227 | Q8 | — | 3 | 1 | no | — |
| 228 | Q8 | — | 30 | 1 | no | — |
| 229 | Q8 | — | 30 | 1 | no | — |
| 230 | Q8 | C3×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 231 | D4 | — | 6 | 1 | no | — |
| 232 | D4 | — | 6 | 1 | no | — |
| 233 | D4 | — | 15 | 2 | no | — |
| 234 | D4 | — | 15 | 2 | no | — |
| 235 | D4 | — | 30 | 4 | no | — |
| 236 | C2×C4 | — | 3 | 1 | no | — |
| 237 | C2×C4 | — | 5 | 1 | no | — |
| 238 | C2×C4 | — | 6 | 1 | no | — |
| 239 | C2×C4 | — | 6 | 1 | no | — |
| 240 | C2×C4 | — | 10 | 2 | no | — |
| 241 | C2×C4 | — | 15 | 2 | no | — |
| 242 | C2×C4 | — | 15 | 2 | no | — |
| 243 | C2×C4 | — | 15 | 1 | no | — |
| 244 | C2×C4 | — | 30 | 2 | no | — |
| 245 | C2×C4 | — | 30 | 1 | no | — |
| 246 | C2×C4 | — | 30 | 1 | no | — |
| 247 | C2×C4 | — | 30 | 2 | no | — |
| 248 | C2×C4 | A4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 249 | C8 | — | 10 | 2 | no | — |
| 250 | C8 | — | 30 | 2 | no | — |
| 251 | C2×C3 | — | 4 | 1 | no | — |
| 252 | C2×C3 | — | 4 | 1 | no | — |
| 253 | C2×C3 | — | 4 | 1 | no | — |
| 254 | C2×C3 | — | 40 | 2 | no | — |
| 255 | C5 | #2880⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 256 | C22 | — | 5 | 2 | no | — |
| 257 | C22 | — | 6 | 1 | no | — |
| 258 | C22 | — | 6 | 1 | no | — |
| 259 | C22 | — | 6 | 1 | no | — |
| 260 | C22 | — | 10 | 1 | no | — |
| 261 | C22 | — | 10 | 1 | no | — |
| 262 | C22 | — | 15 | 2 | no | — |
| 263 | C22 | — | 30 | 1 | no | — |
| 264 | C22 | — | 30 | 1 | no | — |
| 265 | C22 | — | 60 | 4 | no | — |
| 266 | C22 | C2×A4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 267 | C4 | — | 2 | 1 | no | — |
| 268 | C4 | — | 3 | 1 | no | — |
| 269 | C4 | — | 3 | 1 | no | — |
| 270 | C4 | — | 5 | 2 | no | — |
| 271 | C4 | — | 30 | 1 | no | — |
| 272 | C3 | — | 4 | 1 | no | Sylow |
| 273 | C2 | — | 10 | 2 | no | — |
| 274 | C2 | — | 12 | 1 | no | — |
| 275 | C2 | — | 30 | 2 | no | — |
| 276 | C2 | D5⋊(#131⋊C3) | 1 | 1 | yes | characteristic |
| 277 | C2 | #66264⋊C3 | 1 | 1 | yes | characteristic |
| 278 | C2 | A4×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4133⋊C3 | C2 | semidirect |
| 3. #10617543 | C3 | semidirect |
| 12. C2×#4133 | C2×C3 | semidirect |
| 170. C2×D5 | #131⋊C3 | semidirect |
| 255. C5 | #2880⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D5×#41⋊C3 | 2 |
| C2×#4133⋊C3 | 2 |
| #10617543 | 3 |
| C3×#4003 | 4 |
| #2880⋊C3 | 5 |