Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
492 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 49 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 | #11756782 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | D5×#131⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11752126 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×#5403 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11751620 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10612674 | — | 3 | 1 | no | maximal, Hall |
| 7 | D5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | #91956 | C22 | 1 | 2 | yes | — |
| 9 | C5×#984 | C22 | 1 | 2 | yes | — |
| 10 | #91897 | — | 2 | 2 | no | — |
| 11 | #91893 | — | 2 | 2 | no | — |
| 12 | C5×#131⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 13 | #91271 | — | 4 | 1 | no | maximal |
| 14 | #4094⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 15 | #85451 | — | 5 | 1 | no | maximal, Hall |
| 16 | #66111 | — | 3 | 2 | no | — |
| 17 | C2×#4065 | — | 3 | 1 | no | — |
| 18 | D5×#131 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C5×#473 | — | 3 | 1 | no | — |
| 20 | #65492 | — | 3 | 2 | no | — |
| 21 | #65481 | — | 3 | 1 | no | — |
| 22 | #65431 | — | 3 | 1 | no | — |
| 23 | #65186 | — | 3 | 1 | no | — |
| 24 | D5×#108 | — | 3 | 1 | no | — |
| 25 | #65043 | — | 3 | 2 | no | — |
| 26 | #65040 | — | 3 | 2 | no | — |
| 27 | C5×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 28 | #6773 | — | 4 | 2 | no | — |
| 29 | D5×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 30 | C5⋊(#41⋊C3) | D4 | 1 | 1 | yes | characteristic |
| 31 | C5⋊GL(2,𝔽3) | — | 2 | 2 | no | — |
| 32 | C5×#226 | — | 2 | 2 | no | — |
| 33 | D120 | — | 4 | 1 | no | — |
| 34 | C5×Dic12 | — | 4 | 1 | no | — |
| 35 | C3×C8×D5 | — | 4 | 1 | no | — |
| 36 | #6660 | — | 4 | 2 | no | — |
| 37 | #5923 | — | 5 | 2 | no | — |
| 38 | C2×#131⋊C3 | — | 5 | 1 | no | — |
| 39 | #5407 | — | 5 | 1 | no | — |
| 40 | #5406 | — | 5 | 2 | no | — |
| 41 | #5403 | D5 | 1 | 1 | yes | characteristic, complemented |
| 42 | D5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | C2×#713 | — | 3 | 2 | no | — |
| 44 | C5×#137 | — | 3 | 2 | no | — |
| 45 | C2×C5×Q16 | — | 3 | 1 | no | — |
| 46 | C5×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | #4095 | — | 3 | 2 | no | — |
| 48 | #4094 | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | #4088 | — | 3 | 2 | no | — |
| 50 | C2×#688 | — | 3 | 2 | no | — |
| 51 | #4065 | — | 6 | 1 | no | — |
| 52 | #4065 | — | 6 | 1 | no | — |
| 53 | #4065 | — | 6 | 1 | no | — |
| 54 | #4065 | — | 6 | 2 | no | — |
| 55 | #4065 | — | 6 | 1 | no | — |
| 56 | #4064 | — | 6 | 2 | no | — |
| 57 | #4062 | — | 6 | 2 | no | — |
| 58 | #4060 | — | 6 | 2 | no | — |
| 59 | #4054 | — | 3 | 2 | no | — |
| 60 | #4053 | — | 3 | 1 | no | — |
| 61 | D5×M16 | — | 3 | 1 | no | — |
| 62 | D5×M16 | — | 3 | 2 | no | — |
| 63 | C2×D40 | — | 3 | 1 | no | — |
| 64 | #4047 | — | 3 | 1 | no | — |
| 65 | C2×C8×D5 | — | 3 | 1 | no | — |
| 66 | C5×#108 | — | 3 | 1 | no | — |
| 67 | C5×#101 | — | 3 | 2 | no | — |
| 68 | #3968 | — | 3 | 2 | no | — |
| 69 | #3955 | — | 3 | 2 | no | — |
| 70 | #3955 | — | 3 | 2 | no | — |
| 71 | #3954 | — | 3 | 2 | no | — |
| 72 | #3951 | — | 3 | 1 | no | — |
| 73 | #3067 | — | 15 | 1 | no | Sylow |
| 74 | D60 | — | 4 | 2 | no | — |
| 75 | C5×Dic6 | — | 4 | 2 | no | — |
| 76 | C3×C4×D5 | — | 4 | 1 | no | — |
| 77 | C5×SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | #1202 | — | 8 | 2 | no | — |
| 79 | #1200 | — | 8 | 2 | no | — |
| 80 | C3×C5×C8 | — | 4 | 1 | no | — |
| 81 | C3×#168 | — | 4 | 1 | no | — |
| 82 | C2×#41⋊C3 | — | 5 | 1 | no | — |
| 83 | #986 | — | 5 | 2 | no | — |
| 84 | #985 | — | 10 | 2 | no | — |
| 85 | #984 | C2×D5 | 1 | 2 | yes | — |
| 86 | #983 | — | 10 | 2 | no | — |
| 87 | #904 | — | 20 | 1 | no | — |
| 88 | #131⋊C3 | — | 5 | 1 | no | — |
| 89 | #131⋊C3 | C2×D5 | 1 | 1 | yes | characteristic |
| 90 | C5×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 91 | C2×C5×Q8 | — | 3 | 2 | no | — |
| 92 | #713 | — | 6 | 2 | no | — |
| 93 | #713 | — | 6 | 2 | no | — |
| 94 | #713 | — | 6 | 2 | no | — |
| 95 | #713 | — | 6 | 2 | no | — |
| 96 | #713 | D12 | 1 | 1 | yes | characteristic |
| 97 | Q8×D5 | D12 | 1 | 1 | yes | characteristic |
| 98 | #711 | — | 3 | 1 | no | — |
| 99 | D4×D5 | — | 3 | 1 | no | — |
| 100 | #709 | — | 3 | 1 | no | — |
| 101 | C2×D20 | — | 3 | 2 | no | — |
| 102 | C2×D20 | — | 6 | 2 | no | — |
| 103 | C2×C4×D5 | — | 3 | 1 | no | — |
| 104 | C2×C4×D5 | — | 6 | 2 | no | — |
| 105 | C5×Q16 | — | 6 | 1 | no | — |
| 106 | C5×Q16 | — | 6 | 2 | no | — |
| 107 | C5×Q16 | — | 6 | 1 | no | — |
| 108 | C5×SD16 | — | 6 | 2 | no | — |
| 109 | C5×M16 | — | 3 | 1 | no | — |
| 110 | C5×M16 | — | 3 | 2 | no | — |
| 111 | C2×C5×C8 | — | 3 | 1 | no | — |
| 112 | #689 | — | 6 | 2 | no | — |
| 113 | #688 | — | 6 | 2 | no | — |
| 114 | #688 | — | 6 | 2 | no | — |
| 115 | #688 | — | 6 | 2 | no | — |
| 116 | #688 | — | 6 | 2 | no | — |
| 117 | #686 | — | 6 | 2 | no | — |
| 118 | #681 | — | 3 | 1 | no | — |
| 119 | #681 | — | 3 | 2 | no | — |
| 120 | C2×#168 | — | 3 | 1 | no | — |
| 121 | D40 | — | 6 | 1 | no | — |
| 122 | D40 | — | 6 | 2 | no | — |
| 123 | D40 | — | 6 | 1 | no | — |
| 124 | #677 | — | 6 | 2 | no | — |
| 125 | #676 | — | 6 | 1 | no | — |
| 126 | #676 | — | 6 | 2 | no | — |
| 127 | C8×D5 | — | 3 | 1 | no | — |
| 128 | C8×D5 | — | 3 | 1 | no | — |
| 129 | C8×D5 | — | 3 | 1 | no | — |
| 130 | C8×D5 | — | 6 | 2 | no | — |
| 131 | C8×D5 | S4 | 1 | 1 | yes | characteristic |
| 132 | #575 | — | 15 | 2 | no | — |
| 133 | C2×#135 | — | 15 | 1 | no | — |
| 134 | C2×#131 | — | 5 | 1 | no | — |
| 135 | #473 | — | 3 | 1 | no | — |
| 136 | #473 | — | 15 | 1 | no | — |
| 137 | #472 | — | 15 | 1 | no | — |
| 138 | #472 | — | 15 | 1 | no | — |
| 139 | #471 | — | 15 | 2 | no | — |
| 140 | #471 | — | 15 | 2 | no | — |
| 141 | C2×#108 | — | 15 | 1 | no | — |
| 142 | #413 | — | 15 | 2 | no | — |
| 143 | C2×D15 | — | 8 | 2 | no | — |
| 144 | C2×C3×D5 | — | 4 | 1 | no | — |
| 145 | C3×C4×C5 | — | 4 | 1 | no | — |
| 146 | C3×Dic5 | — | 4 | 1 | no | — |
| 147 | C5×Dic3 | — | 8 | 2 | no | — |
| 148 | #235 | — | 20 | 2 | no | — |
| 149 | #41⋊C3 | — | 5 | 1 | no | — |
| 150 | #41⋊C3 | C22×D5 | 1 | 1 | yes | characteristic |
| 151 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 152 | GL(2,𝔽3) | — | 10 | 2 | no | — |
| 153 | #226 | — | 2 | 2 | no | — |
| 154 | C2×C3×C8 | — | 20 | 1 | no | — |
| 155 | Dic12 | — | 4 | 1 | no | — |
| 156 | D24 | — | 20 | 1 | no | — |
| 157 | #204 | — | 20 | 2 | no | — |
| 158 | C22×D5 | — | 3 | 1 | no | — |
| 159 | C22×D5 | — | 6 | 2 | no | — |
| 160 | C5×Q8 | — | 6 | 2 | no | — |
| 161 | C5×Q8 | — | 6 | 2 | no | — |
| 162 | C5×Q8 | C2×D12 | 1 | 1 | yes | characteristic |
| 163 | C5×D4 | — | 3 | 1 | no | — |
| 164 | C2×C4×C5 | — | 3 | 1 | no | — |
| 165 | C2×C4×C5 | — | 6 | 2 | no | — |
| 166 | C22⋊D5 | — | 6 | 1 | no | — |
| 167 | C2×Dic5 | — | 3 | 1 | no | — |
| 168 | D20 | — | 3 | 1 | no | — |
| 169 | D20 | — | 6 | 2 | no | — |
| 170 | D20 | — | 6 | 2 | no | — |
| 171 | D20 | — | 12 | 2 | no | — |
| 172 | D20 | — | 12 | 2 | no | — |
| 173 | C4×D5 | — | 3 | 1 | no | — |
| 174 | C4×D5 | — | 3 | 1 | no | — |
| 175 | C4×D5 | — | 3 | 1 | no | — |
| 176 | C4×D5 | — | 12 | 2 | no | — |
| 177 | C4×D5 | — | 12 | 2 | no | — |
| 178 | C4×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 179 | Dic10 | — | 3 | 1 | no | — |
| 180 | C5×C8 | — | 3 | 1 | no | — |
| 181 | C5×C8 | — | 6 | 2 | no | — |
| 182 | C5×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 183 | #168 | — | 3 | 1 | no | — |
| 184 | #168 | — | 6 | 2 | no | — |
| 185 | #168 | C2×S4 | 1 | 1 | yes | characteristic |
| 186 | C2×#41 | — | 5 | 1 | no | — |
| 187 | C2×#41 | — | 15 | 2 | no | — |
| 188 | #137 | — | 3 | 2 | no | — |
| 189 | #137 | — | 15 | 2 | no | — |
| 190 | #137 | — | 30 | 2 | no | — |
| 191 | AGL(1,ℤ/8) | — | 15 | 2 | no | — |
| 192 | AGL(1,ℤ/8) | — | 15 | 2 | no | — |
| 193 | AGL(1,ℤ/8) | — | 30 | 2 | no | — |
| 194 | #135 | — | 30 | 1 | no | — |
| 195 | #135 | — | 30 | 1 | no | — |
| 196 | #135 | — | 30 | 1 | no | — |
| 197 | #135 | — | 30 | 1 | no | — |
| 198 | #135 | — | 30 | 2 | no | — |
| 199 | #135 | — | 30 | 2 | no | — |
| 200 | C2×Q16 | — | 3 | 1 | no | — |
| 201 | C2×SD16 | — | 15 | 2 | no | — |
| 202 | C2×D8 | — | 15 | 1 | no | — |
| 203 | #131 | — | 5 | 1 | no | — |
| 204 | #131 | — | 15 | 1 | no | — |
| 205 | #131 | — | 15 | 1 | no | — |
| 206 | #131 | D3×D5 | 1 | 1 | yes | characteristic |
| 207 | C2×M16 | — | 15 | 1 | no | — |
| 208 | C2×M16 | — | 15 | 2 | no | — |
| 209 | C22×C8 | — | 15 | 1 | no | — |
| 210 | #108 | — | 3 | 1 | no | — |
| 211 | #108 | — | 15 | 1 | no | — |
| 212 | #108 | — | 15 | 2 | no | — |
| 213 | #101 | — | 3 | 2 | no | — |
| 214 | #101 | — | 15 | 2 | no | — |
| 215 | #100 | — | 15 | 2 | no | — |
| 216 | #100 | — | 15 | 2 | no | — |
| 217 | C2×C3×C5 | — | 4 | 1 | no | — |
| 218 | D15 | — | 16 | 2 | no | — |
| 219 | C3×D5 | — | 8 | 1 | no | — |
| 220 | C2×C3×C4 | — | 20 | 1 | no | — |
| 221 | #67 | — | 40 | 2 | no | — |
| 222 | D12 | — | 20 | 2 | no | — |
| 223 | C4×D3 | — | 40 | 2 | no | — |
| 224 | Dic6 | — | 4 | 2 | no | — |
| 225 | SL(2,𝔽3) | D4×D5 | 1 | 1 | yes | characteristic |
| 226 | C3×C8 | — | 4 | 1 | no | — |
| 227 | C3×C8 | — | 20 | 1 | no | — |
| 228 | C22×C5 | — | 3 | 1 | no | — |
| 229 | C2×D5 | — | 3 | 1 | no | — |
| 230 | C2×D5 | — | 12 | 1 | no | — |
| 231 | C2×D5 | — | 12 | 2 | no | — |
| 232 | C2×D5 | — | 24 | 2 | no | — |
| 233 | C2×D5 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 234 | C4×C5 | — | 3 | 1 | no | — |
| 235 | C4×C5 | — | 12 | 2 | no | — |
| 236 | C4×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 237 | Dic5 | — | 3 | 1 | no | — |
| 238 | Dic5 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 239 | #41 | — | 5 | 1 | no | — |
| 240 | #41 | — | 15 | 1 | no | — |
| 241 | #41 | — | 15 | 1 | no | — |
| 242 | #41 | — | 30 | 2 | no | — |
| 243 | #41 | — | 30 | 2 | no | — |
| 244 | #41 | — | 30 | 2 | no | — |
| 245 | #41 | — | 30 | 2 | no | — |
| 246 | #41 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 247 | C2×Q8 | — | 3 | 2 | no | — |
| 248 | C2×Q8 | — | 5 | 1 | no | — |
| 249 | C2×D4 | — | 15 | 2 | no | — |
| 250 | C2×D4 | — | 15 | 1 | no | — |
| 251 | C2×D4 | — | 30 | 2 | no | — |
| 252 | C22×C4 | — | 15 | 1 | no | — |
| 253 | C22×C4 | — | 30 | 2 | no | — |
| 254 | Q16 | — | 6 | 1 | no | — |
| 255 | Q16 | — | 6 | 2 | no | — |
| 256 | Q16 | — | 6 | 1 | no | — |
| 257 | Q16 | — | 30 | 2 | no | — |
| 258 | SD16 | — | 6 | 2 | no | — |
| 259 | SD16 | — | 30 | 2 | no | — |
| 260 | SD16 | — | 30 | 2 | no | — |
| 261 | SD16 | — | 30 | 2 | no | — |
| 262 | SD16 | — | 30 | 2 | no | — |
| 263 | SD16 | — | 30 | 2 | no | — |
| 264 | D8 | — | 30 | 1 | no | — |
| 265 | D8 | — | 30 | 2 | no | — |
| 266 | D8 | — | 30 | 2 | no | — |
| 267 | D8 | — | 30 | 1 | no | — |
| 268 | M16 | — | 3 | 1 | no | — |
| 269 | M16 | — | 3 | 2 | no | — |
| 270 | M16 | — | 15 | 1 | no | — |
| 271 | M16 | — | 15 | 2 | no | — |
| 272 | M16 | — | 30 | 1 | no | — |
| 273 | M16 | — | 30 | 2 | no | — |
| 274 | C2×C8 | — | 3 | 1 | no | — |
| 275 | C2×C8 | — | 5 | 1 | no | — |
| 276 | C2×C8 | — | 15 | 1 | no | — |
| 277 | C2×C8 | — | 15 | 1 | no | — |
| 278 | C2×C8 | — | 15 | 1 | no | — |
| 279 | C2×C8 | — | 15 | 1 | no | — |
| 280 | C2×C8 | — | 30 | 2 | no | — |
| 281 | C3×C5 | — | 4 | 1 | no | Hall |
| 282 | C22×C3 | — | 20 | 1 | no | — |
| 283 | C2×D3 | — | 40 | 2 | no | — |
| 284 | C3×C4 | — | 4 | 1 | no | — |
| 285 | C3×C4 | — | 20 | 1 | no | — |
| 286 | Dic3 | — | 8 | 2 | no | — |
| 287 | C2×C5 | — | 6 | 1 | no | — |
| 288 | C2×C5 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 289 | D5 | — | 2 | 1 | no | — |
| 290 | D5 | — | 6 | 1 | no | — |
| 291 | D5 | — | 24 | 2 | no | — |
| 292 | C23 | — | 15 | 1 | no | — |
| 293 | C23 | — | 30 | 2 | no | — |
| 294 | Q8 | — | 6 | 2 | no | — |
| 295 | Q8 | — | 6 | 2 | no | — |
| 296 | Q8 | — | 15 | 1 | no | — |
| 297 | Q8 | D5×D12 | 1 | 1 | yes | characteristic |
| 298 | D4 | — | 3 | 1 | no | — |
| 299 | D4 | — | 15 | 1 | no | — |
| 300 | D4 | — | 30 | 2 | no | — |
| 301 | D4 | — | 30 | 1 | no | — |
| 302 | D4 | — | 30 | 2 | no | — |
| 303 | D4 | — | 60 | 2 | no | — |
| 304 | D4 | — | 60 | 2 | no | — |
| 305 | C2×C4 | — | 3 | 1 | no | — |
| 306 | C2×C4 | — | 5 | 1 | no | — |
| 307 | C2×C4 | — | 6 | 2 | no | — |
| 308 | C2×C4 | — | 15 | 1 | no | — |
| 309 | C2×C4 | — | 15 | 1 | no | — |
| 310 | C2×C4 | — | 15 | 1 | no | — |
| 311 | C2×C4 | — | 15 | 1 | no | — |
| 312 | C2×C4 | — | 60 | 2 | no | — |
| 313 | C2×C4 | — | 60 | 2 | no | — |
| 314 | C8 | — | 3 | 1 | no | — |
| 315 | C8 | — | 5 | 1 | no | — |
| 316 | C8 | — | 6 | 2 | no | — |
| 317 | C8 | — | 15 | 1 | no | — |
| 318 | C8 | — | 30 | 2 | no | — |
| 319 | C8 | D5×S4 | 1 | 1 | yes | characteristic |
| 320 | C2×C3 | — | 4 | 1 | no | — |
| 321 | C2×C3 | — | 40 | 1 | no | — |
| 322 | D3 | — | 80 | 2 | no | — |
| 323 | C5 | #85451 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 324 | C22 | — | 3 | 1 | no | — |
| 325 | C22 | — | 5 | 1 | no | — |
| 326 | C22 | — | 15 | 1 | no | — |
| 327 | C22 | — | 60 | 1 | no | — |
| 328 | C22 | — | 60 | 2 | no | — |
| 329 | C22 | — | 120 | 2 | no | — |
| 330 | C4 | — | 3 | 1 | no | — |
| 331 | C4 | — | 5 | 1 | no | — |
| 332 | C4 | — | 12 | 2 | no | — |
| 333 | C4 | — | 15 | 1 | no | — |
| 334 | C4 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| 335 | C3 | — | 4 | 1 | no | Sylow |
| 336 | C2 | — | 6 | 1 | no | — |
| 337 | C2 | — | 10 | 1 | no | — |
| 338 | C2 | — | 30 | 1 | no | — |
| 339 | C2 | — | 120 | 2 | no | — |
| 340 | C2 | D5×C4⋊S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11756782 | C2 | semidirect |
| 2. D5×#131⋊C3 | C2 | semidirect |
| 3. #11752126 | C2 | semidirect |
| 4. C5×#5403 | C2 | semidirect |
| 5. #11751620 | C2 | semidirect |
| 18. D5×#131 | D3 | semidirect |
| 41. #5403 | D5 | semidirect |
| 323. C5 | #85451 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11756782 | 2 |
| D5×#131⋊C3 | 2 |
| #11752126 | 2 |
| C5×#5403 | 2 |
| #11751620 | 2 |
| #10612674 | 3 |
| #91271 | 4 |
| #85451 | 5 |