Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
812 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 812 conjugacy classes of subgroups — too many to draw. Only its 163 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 | #23777895 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23774794 | C2 | 1 | 2 | yes | maximal |
| 3 | #23773910 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #23762678 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #23762643 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #23761854 | C2 | 1 | 2 | yes | maximal |
| 7 | C79×#1561 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C79⋊#564 | — | 2 | 2 | no | — |
| 9 | #19099292 | — | 2 | 2 | no | — |
| 10 | #19099291 | — | 2 | 2 | no | — |
| 11 | #19099288 | C22 | 1 | 4 | yes | — |
| 12 | #19099175 | — | 2 | 2 | no | — |
| 13 | #19099173 | — | 2 | 4 | no | — |
| 14 | #19099120 | C22 | 1 | 1 | yes | characteristic |
| 15 | #19099120 | C22 | 1 | 2 | yes | — |
| 16 | #19098966 | C22 | 1 | 2 | yes | — |
| 17 | #19098962 | — | 2 | 2 | no | — |
| 18 | #19098959 | C22 | 1 | 2 | yes | — |
| 19 | #19098849 | — | 2 | 2 | no | — |
| 20 | #19098849 | C22 | 1 | 4 | yes | — |
| 21 | #19098821 | — | 2 | 4 | no | — |
| 22 | #19098774 | — | 2 | 2 | no | — |
| 23 | #19098774 | C22 | 1 | 4 | yes | — |
| 24 | #19098599 | — | 2 | 2 | no | — |
| 25 | C2×C4×C79⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 26 | #19098293 | C22 | 1 | 4 | yes | — |
| 27 | #19098293 | C22 | 1 | 4 | yes | — |
| 28 | #533×C79 | C22 | 1 | 1 | yes | characteristic |
| 29 | #423×C79 | C22 | 1 | 2 | yes | — |
| 30 | #332×C79 | C22 | 1 | 4 | yes | — |
| 31 | C2×Dic316 | — | 2 | 4 | no | — |
| 32 | C22×C79⋊C8 | — | 2 | 1 | no | — |
| 33 | #14579161 | — | 2 | 2 | no | — |
| 34 | #14579161 | D4 | 1 | 2 | yes | — |
| 35 | #14579161 | D4 | 1 | 2 | yes | — |
| 36 | #14579161 | C23 | 1 | 2 | yes | — |
| 37 | #14579159 | — | 2 | 4 | no | — |
| 38 | #14579154 | — | 2 | 8 | no | — |
| 39 | #14579153 | — | 2 | 2 | no | — |
| 40 | #14579153 | — | 2 | 2 | no | — |
| 41 | #14579153 | — | 2 | 1 | no | — |
| 42 | #14579152 | — | 2 | 8 | no | — |
| 43 | #14579152 | — | 2 | 4 | no | — |
| 44 | #14579151 | — | 2 | 8 | no | — |
| 45 | #14579151 | — | 2 | 8 | no | — |
| 46 | #14579140 | — | 2 | 2 | no | — |
| 47 | #14579140 | — | 2 | 2 | no | — |
| 48 | #14579140 | D4 | 1 | 4 | yes | — |
| 49 | Q8×Dic79 | — | 2 | 4 | no | — |
| 50 | C4×Dic158 | — | 2 | 4 | no | — |
| 51 | C4×Dic158 | — | 2 | 2 | no | — |
| 52 | C4×Dic158 | — | 2 | 2 | no | — |
| 53 | C4×Dic158 | C23 | 1 | 4 | yes | — |
| 54 | #14579126 | — | 2 | 4 | no | — |
| 55 | #14579120 | — | 2 | 2 | no | — |
| 56 | #14579120 | — | 2 | 1 | no | — |
| 57 | #14579094 | — | 2 | 4 | no | — |
| 58 | #14579093 | — | 2 | 4 | no | — |
| 59 | #14579092 | — | 2 | 4 | no | — |
| 60 | #14579086 | — | 2 | 4 | no | — |
| 61 | #14579086 | — | 2 | 4 | no | — |
| 62 | C4×C79⋊C8 | — | 2 | 1 | no | — |
| 63 | C4×C79⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 64 | C4×C79⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×M16×C79 | — | 2 | 2 | no | — |
| 66 | C4⋊Q8×C79 | C23 | 1 | 2 | yes | — |
| 67 | #125×C79 | D4 | 1 | 2 | yes | — |
| 68 | C22⋊Q8×C79 | — | 2 | 2 | no | — |
| 69 | C4×Q8×C79 | — | 2 | 2 | no | — |
| 70 | #117×C79 | D4 | 1 | 2 | yes | — |
| 71 | C2×C42×C79 | C23 | 1 | 1 | yes | characteristic |
| 72 | C4⋊C8×C79 | — | 2 | 2 | no | — |
| 73 | C4⋊C8×C79 | C23 | 1 | 4 | yes | — |
| 74 | C2×Dic158 | — | 2 | 2 | no | — |
| 75 | C2×Dic158 | — | 2 | 2 | no | — |
| 76 | C2×Dic158 | C2×D4 | 1 | 4 | yes | — |
| 77 | Dic316 | — | 8 | 4 | no | — |
| 78 | C2×C79⋊C8 | — | 2 | 4 | no | — |
| 79 | C2×C79⋊C8 | — | 2 | 1 | no | — |
| 80 | C2×C79⋊C8 | — | 2 | 1 | no | — |
| 81 | C4⋊Dic79 | — | 2 | 8 | no | — |
| 82 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 83 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 84 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 85 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 86 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 87 | C4⋊Dic79 | C2×D4 | 1 | 4 | yes | — |
| 88 | Dic79⋊C4 | — | 2 | 8 | no | — |
| 89 | Dic79⋊C4 | — | 2 | 8 | no | — |
| 90 | Dic79⋊C4 | — | 2 | 4 | no | — |
| 91 | Dic79⋊C4 | — | 2 | 4 | no | — |
| 92 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 93 | C22⋊Dic79 | — | 2 | 4 | no | — |
| 94 | C22⋊Dic79 | — | 4 | 1 | no | — |
| 95 | C4×Dic79 | — | 2 | 8 | no | — |
| 96 | C4×Dic79 | — | 2 | 4 | no | — |
| 97 | C4×Dic79 | — | 4 | 1 | no | — |
| 98 | C2×Q8×C79 | — | 2 | 2 | no | — |
| 99 | C22×C4×C79 | C2×D4 | 1 | 2 | yes | — |
| 100 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | M16×C79 | — | 8 | 2 | no | — |
| 102 | C2×C8×C79 | — | 2 | 4 | no | — |
| 103 | C4⋊C4×C79 | — | 2 | 4 | no | — |
| 104 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 105 | C4⋊C4×C79 | C2×D4 | 1 | 4 | yes | — |
| 106 | C22⋊C4×C79 | — | 2 | 2 | no | — |
| 107 | C42×C79 | — | 2 | 2 | no | — |
| 108 | C42×C79 | C2×D4 | 1 | 2 | yes | — |
| 109 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C79 | C24 | 1 | 1 | yes | characteristic |
| 111 | C23×C79 | #120 | 1 | 1 | yes | characteristic |
| 112 | Q8×C79 | — | 4 | 2 | no | — |
| 113 | Q8×C79 | — | 4 | 2 | no | — |
| 114 | C2×C4×C79 | — | 2 | 2 | no | — |
| 115 | C2×C4×C79 | — | 2 | 2 | no | — |
| 116 | C2×C4×C79 | — | 2 | 2 | no | — |
| 117 | C2×C4×C79 | — | 2 | 4 | no | — |
| 118 | C2×C4×C79 | #120 | 1 | 2 | yes | — |
| 119 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C79 | C22×D4 | 1 | 2 | yes | — |
| 121 | C2×C4×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×Dic79 | — | 2 | 4 | no | — |
| 123 | C2×Dic79 | — | 2 | 4 | no | — |
| 124 | C2×Dic79 | — | 4 | 2 | no | — |
| 125 | Dic158 | — | 2 | 4 | no | — |
| 126 | Dic158 | — | 2 | 4 | no | — |
| 127 | Dic158 | — | 4 | 4 | no | — |
| 128 | Dic158 | — | 4 | 2 | no | — |
| 129 | Dic158 | — | 4 | 2 | no | — |
| 130 | C8×C79 | — | 4 | 4 | no | — |
| 131 | C79⋊C8 | — | 2 | 2 | no | — |
| 132 | C79⋊C8 | — | 2 | 2 | no | — |
| 133 | C22×C79 | — | 2 | 2 | no | — |
| 134 | C22×C79 | — | 2 | 1 | no | — |
| 135 | C22×C79 | C2×#120 | 1 | 1 | yes | characteristic |
| 136 | C4×C79 | — | 2 | 2 | no | — |
| 137 | C4×C79 | — | 2 | 1 | no | — |
| 138 | C4×C79 | — | 4 | 4 | no | — |
| 139 | C4×C79 | #456 | 1 | 2 | yes | — |
| 140 | C4×C79 | #577 | 1 | 2 | yes | — |
| 141 | C4×C79 | #578 | 1 | 2 | yes | — |
| 142 | Dic79 | — | 4 | 4 | no | — |
| 143 | Dic79 | — | 4 | 4 | no | — |
| 144 | Dic79 | — | 8 | 2 | no | — |
| 145 | #18703 | — | 79 | 1 | no | maximal, Sylow |
| 146 | C2×C79 | — | 4 | 1 | no | — |
| 147 | C2×C79 | #3013 | 1 | 2 | yes | — |
| 148 | C2×C79 | C2×#456 | 1 | 1 | yes | characteristic |
| 149 | #3556 | — | 79 | 1 | no | — |
| 150 | #3255 | — | 79 | 2 | no | — |
| 151 | #3160 | — | 79 | 1 | no | — |
| 152 | #1653 | — | 79 | 4 | no | — |
| 153 | #1648 | — | 79 | 4 | no | — |
| 154 | #1561 | D79 | 1 | 1 | yes | characteristic |
| 155 | #1493 | — | 79 | 2 | no | — |
| 156 | C79 | #18703 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 157 | #564 | — | 158 | 2 | no | — |
| 158 | #557 | — | 79 | 4 | no | — |
| 159 | #557 | — | 158 | 2 | no | — |
| 160 | #557 | — | 158 | 2 | no | — |
| 161 | #541 | — | 158 | 4 | no | — |
| 162 | #541 | — | 158 | 2 | no | — |
| 163 | #533 | — | 79 | 2 | no | — |
| 164 | #533 | — | 79 | 1 | no | — |
| 165 | #533 | C2×D79 | 1 | 1 | yes | characteristic |
| 166 | #500 | — | 79 | 2 | no | — |
| 167 | #499 | — | 158 | 2 | no | — |
| 168 | #498 | — | 79 | 2 | no | — |
| 169 | #475 | — | 79 | 4 | no | — |
| 170 | #475 | — | 158 | 2 | no | — |
| 171 | #470 | — | 158 | 4 | no | — |
| 172 | #462 | — | 79 | 4 | no | — |
| 173 | #462 | — | 158 | 2 | no | — |
| 174 | #427 | — | 158 | 2 | no | — |
| 175 | #423 | C2×D79 | 1 | 2 | yes | — |
| 176 | C2×C4×C8 | — | 79 | 1 | no | — |
| 177 | #332 | C2×D79 | 1 | 4 | yes | — |
| 178 | Q8⋊C8 | — | 79 | 4 | no | — |
| 179 | Q8⋊C8 | — | 79 | 4 | no | — |
| 180 | C2×Q16 | — | 158 | 4 | no | — |
| 181 | C2×M16 | — | 2 | 2 | no | — |
| 182 | C22×C8 | — | 158 | 1 | no | — |
| 183 | C4⋊Q8 | — | 79 | 2 | no | — |
| 184 | C4⋊Q8 | — | 79 | 2 | no | — |
| 185 | C4⋊Q8 | — | 79 | 2 | no | — |
| 186 | C4⋊Q8 | — | 158 | 2 | no | — |
| 187 | C4⋊Q8 | — | 158 | 4 | no | — |
| 188 | C4⋊Q8 | C22×D79 | 1 | 2 | yes | — |
| 189 | #126 | — | 158 | 8 | no | — |
| 190 | #125 | — | 158 | 2 | no | — |
| 191 | #125 | — | 158 | 4 | no | — |
| 192 | #125 | — | 158 | 8 | no | — |
| 193 | #125 | — | 158 | 8 | no | — |
| 194 | #125 | — | 158 | 8 | no | — |
| 195 | #125 | — | 158 | 1 | no | — |
| 196 | #125 | — | 158 | 2 | no | — |
| 197 | #125 | D316 | 1 | 2 | yes | — |
| 198 | C22⋊Q8 | — | 2 | 2 | no | — |
| 199 | C22⋊Q8 | — | 79 | 4 | no | — |
| 200 | C22⋊Q8 | — | 158 | 2 | no | — |
| 201 | C22⋊Q8 | — | 158 | 2 | no | — |
| 202 | C4×Q8 | — | 2 | 2 | no | — |
| 203 | C4×Q8 | — | 79 | 4 | no | — |
| 204 | C4×Q8 | — | 158 | 4 | no | — |
| 205 | C4×Q8 | — | 158 | 2 | no | — |
| 206 | C4×Q8 | — | 158 | 4 | no | — |
| 207 | C4×Q8 | — | 158 | 4 | no | — |
| 208 | C4×Q8 | — | 158 | 2 | no | — |
| 209 | #117 | — | 158 | 1 | no | — |
| 210 | #117 | — | 158 | 2 | no | — |
| 211 | #117 | D316 | 1 | 2 | yes | — |
| 212 | C2×C42 | C22×D79 | 1 | 1 | yes | characteristic |
| 213 | #107 | — | 158 | 4 | no | — |
| 214 | #106 | — | 158 | 4 | no | — |
| 215 | #106 | — | 158 | 4 | no | — |
| 216 | C4⋊C8 | — | 2 | 2 | no | — |
| 217 | C4⋊C8 | C22×D79 | 1 | 4 | yes | — |
| 218 | #103 | — | 158 | 4 | no | — |
| 219 | #103 | — | 158 | 4 | no | — |
| 220 | C4×C8 | — | 79 | 1 | no | — |
| 221 | C4×C8 | — | 79 | 1 | no | — |
| 222 | C4×C8 | — | 158 | 1 | no | — |
| 223 | C2×Q8 | — | 2 | 2 | no | — |
| 224 | C2×Q8 | — | 79 | 4 | no | — |
| 225 | C2×Q8 | — | 158 | 2 | no | — |
| 226 | C2×Q8 | — | 158 | 2 | no | — |
| 227 | C22×C4 | D4×D79 | 1 | 2 | yes | — |
| 228 | C22×C4 | C2×D316 | 1 | 1 | yes | characteristic |
| 229 | Q16 | — | 632 | 4 | no | — |
| 230 | M16 | — | 8 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 4 | no | — |
| 232 | C2×C8 | — | 158 | 4 | no | — |
| 233 | C2×C8 | — | 158 | 1 | no | — |
| 234 | C2×C8 | — | 158 | 1 | no | — |
| 235 | C4⋊C4 | — | 2 | 4 | no | — |
| 236 | C4⋊C4 | — | 2 | 2 | no | — |
| 237 | C4⋊C4 | — | 316 | 2 | no | — |
| 238 | C4⋊C4 | — | 79 | 4 | no | — |
| 239 | C4⋊C4 | — | 158 | 8 | no | — |
| 240 | C4⋊C4 | — | 158 | 8 | no | — |
| 241 | C4⋊C4 | — | 158 | 8 | no | — |
| 242 | C4⋊C4 | — | 158 | 2 | no | — |
| 243 | C4⋊C4 | — | 158 | 2 | no | — |
| 244 | C4⋊C4 | — | 158 | 4 | no | — |
| 245 | C4⋊C4 | — | 158 | 2 | no | — |
| 246 | C4⋊C4 | — | 158 | 2 | no | — |
| 247 | C4⋊C4 | — | 158 | 2 | no | — |
| 248 | C4⋊C4 | — | 158 | 4 | no | — |
| 249 | C4⋊C4 | C2×D316 | 1 | 4 | yes | — |
| 250 | C22⋊C4 | — | 2 | 2 | no | — |
| 251 | C22⋊C4 | — | 316 | 1 | no | — |
| 252 | C22⋊C4 | — | 158 | 4 | no | — |
| 253 | C42 | — | 2 | 2 | no | — |
| 254 | C42 | — | 316 | 1 | no | — |
| 255 | C42 | — | 158 | 8 | no | — |
| 256 | C42 | — | 158 | 4 | no | — |
| 257 | C42 | D4×D79 | 1 | 2 | yes | — |
| 258 | C42 | C2×D316 | 1 | 1 | yes | characteristic |
| 259 | C42 | C23×D79 | 1 | 1 | yes | characteristic |
| 260 | C23 | #14579131 | 1 | 1 | yes | characteristic |
| 261 | Q8 | — | 4 | 2 | no | — |
| 262 | Q8 | — | 4 | 2 | no | — |
| 263 | Q8 | — | 316 | 4 | no | — |
| 264 | Q8 | — | 316 | 2 | no | — |
| 265 | Q8 | — | 316 | 2 | no | — |
| 266 | Q8 | — | 158 | 4 | no | — |
| 267 | Q8 | — | 158 | 4 | no | — |
| 268 | C2×C4 | — | 2 | 2 | no | — |
| 269 | C2×C4 | — | 2 | 2 | no | — |
| 270 | C2×C4 | — | 2 | 2 | no | — |
| 271 | C2×C4 | — | 2 | 4 | no | — |
| 272 | C2×C4 | — | 316 | 2 | no | — |
| 273 | C2×C4 | — | 158 | 4 | no | — |
| 274 | C2×C4 | — | 158 | 4 | no | — |
| 275 | C2×C4 | #14579131 | 1 | 2 | yes | — |
| 276 | C2×C4 | #14579131 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C2×D4×D79 | 1 | 2 | yes | — |
| 278 | C2×C4 | C22×D316 | 1 | 1 | yes | characteristic |
| 279 | C8 | — | 4 | 4 | no | — |
| 280 | C8 | — | 158 | 2 | no | — |
| 281 | C8 | — | 158 | 2 | no | — |
| 282 | C22 | — | 2 | 2 | no | — |
| 283 | C22 | — | 2 | 1 | no | — |
| 284 | C22 | C2×#14579131 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 2 | 1 | no | — |
| 287 | C4 | — | 4 | 4 | no | — |
| 288 | C4 | — | 316 | 4 | no | — |
| 289 | C4 | — | 316 | 4 | no | — |
| 290 | C4 | — | 632 | 2 | no | — |
| 291 | C4 | #19098748 | 1 | 2 | yes | — |
| 292 | C4 | #19099373 | 1 | 2 | yes | — |
| 293 | C4 | #19099379 | 1 | 2 | yes | — |
| 294 | C2 | — | 4 | 1 | no | — |
| 295 | C2 | #23772717 | 1 | 2 | yes | — |
| 296 | C2 | C2×#19098748 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #23762678 | C2 | semidirect |
| 5. #23762643 | C2 | semidirect |
| 156. C79 | #18703 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23777895 | 2 |
| #23774794 | 2 |
| #23773910 | 2 |
| #23762678 | 2 |
| #23762643 | 2 |
| #23761854 | 2 |
| C79×#1561 | 2 |
| #18703 | 79 |