Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
408 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 408 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 | #23773829 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23765084 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23761569 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #23761566 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C79×#1411 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14579184 | — | 2 | 2 | no | — |
| 7 | #19099114 | C4 | 1 | 1 | yes | characteristic |
| 8 | #19099114 | C22 | 1 | 1 | yes | characteristic |
| 9 | #19098948 | — | 2 | 2 | no | — |
| 10 | #19098940 | C4 | 1 | 1 | yes | characteristic |
| 11 | #19098937 | C4 | 1 | 1 | yes | characteristic |
| 12 | #19098937 | C22 | 1 | 1 | yes | characteristic |
| 13 | #19098546 | — | 2 | 2 | no | — |
| 14 | C4×C8×D79 | C4 | 1 | 1 | yes | characteristic |
| 15 | #19098362 | C22 | 1 | 2 | yes | — |
| 16 | #19098359 | — | 2 | 2 | no | — |
| 17 | #494×C79 | C22 | 1 | 1 | yes | characteristic |
| 18 | #356×C79 | C22 | 1 | 2 | yes | — |
| 19 | C2×#11929625 | — | 2 | 1 | no | — |
| 20 | C2×#11929625 | D4 | 1 | 2 | yes | — |
| 21 | #14579184 | — | 4 | 4 | no | — |
| 22 | #14579184 | — | 4 | 4 | no | — |
| 23 | C2×#11929611 | — | 2 | 2 | no | — |
| 24 | C2×#11929611 | — | 2 | 2 | no | — |
| 25 | C2×D632 | — | 2 | 2 | no | — |
| 26 | M16×D79 | — | 4 | 2 | no | — |
| 27 | C2×C8×D79 | — | 2 | 1 | no | — |
| 28 | #14579158 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #14579158 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | #14579150 | — | 2 | 1 | no | — |
| 31 | #14579150 | D4 | 1 | 2 | yes | — |
| 32 | #14579136 | — | 4 | 1 | no | — |
| 33 | #14579136 | — | 4 | 1 | no | — |
| 34 | #14579123 | — | 4 | 1 | no | — |
| 35 | #14579123 | — | 4 | 1 | no | — |
| 36 | C42×D79 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | #14579080 | — | 2 | 2 | no | — |
| 38 | #14579077 | — | 4 | 2 | no | — |
| 39 | C4×C79⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C8×Dic79 | — | 2 | 1 | no | — |
| 41 | C2×Q16×C79 | — | 2 | 2 | no | — |
| 42 | C4⋊Q8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C4⋊Q8×C79 | C23 | 1 | 1 | yes | characteristic |
| 44 | #101×C79 | — | 2 | 2 | no | — |
| 45 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #11929625 | — | 4 | 2 | no | — |
| 47 | #11929625 | — | 4 | 2 | no | — |
| 48 | #11929625 | — | 4 | 2 | no | — |
| 49 | #11929625 | — | 4 | 2 | no | — |
| 50 | C2×D316 | — | 2 | 2 | no | — |
| 51 | C2×D316 | — | 2 | 1 | no | — |
| 52 | C2×D316 | — | 4 | 1 | no | — |
| 53 | C2×D316 | — | 4 | 1 | no | — |
| 54 | C2×D316 | — | 4 | 1 | no | — |
| 55 | C2×D316 | C22⋊C4 | 1 | 2 | yes | — |
| 56 | C2×C4×D79 | — | 2 | 2 | no | — |
| 57 | C2×C4×D79 | — | 2 | 1 | no | — |
| 58 | C2×C4×D79 | — | 4 | 1 | no | — |
| 59 | C2×C4×D79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | #11929611 | — | 4 | 4 | no | — |
| 61 | #11929611 | — | 4 | 4 | no | — |
| 62 | D632 | — | 4 | 4 | no | — |
| 63 | #11929608 | — | 4 | 2 | no | — |
| 64 | #11929607 | — | 8 | 2 | no | — |
| 65 | C2×C79⋊C8 | — | 2 | 1 | no | — |
| 66 | C8×D79 | — | 2 | 2 | no | — |
| 67 | C8×D79 | — | 2 | 2 | no | — |
| 68 | C8×D79 | — | 8 | 2 | no | — |
| 69 | (C2×D79)⋊C4 | — | 4 | 2 | no | — |
| 70 | (C2×D79)⋊C4 | — | 4 | 2 | no | — |
| 71 | C4×Dic79 | — | 2 | 1 | no | — |
| 72 | C4×Dic79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C2×Q8×C79 | — | 2 | 1 | no | — |
| 74 | C2×Q8×C79 | C2×D4 | 1 | 2 | yes | — |
| 75 | Q16×C79 | — | 4 | 4 | no | — |
| 76 | M16×C79 | — | 4 | 2 | no | — |
| 77 | C2×C8×C79 | — | 2 | 1 | no | — |
| 78 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 79 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 80 | C42×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C22×D79 | — | 2 | 2 | no | — |
| 82 | C22×D79 | — | 4 | 1 | no | — |
| 83 | C22×D79 | #99 | 1 | 1 | yes | characteristic |
| 84 | Q8×C79 | — | 4 | 2 | no | — |
| 85 | Q8×C79 | — | 4 | 2 | no | — |
| 86 | C2×C4×C79 | — | 2 | 1 | no | — |
| 87 | C2×C4×C79 | — | 2 | 2 | no | — |
| 88 | C2×C4×C79 | — | 4 | 1 | no | — |
| 89 | C2×C4×C79 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C2×Dic79 | — | 2 | 1 | no | — |
| 91 | C2×Dic79 | #99 | 1 | 1 | yes | characteristic |
| 92 | D316 | — | 4 | 2 | no | — |
| 93 | D316 | — | 4 | 2 | no | — |
| 94 | D316 | — | 8 | 2 | no | — |
| 95 | D316 | — | 8 | 2 | no | — |
| 96 | D316 | — | 8 | 2 | no | — |
| 97 | D316 | — | 8 | 2 | no | — |
| 98 | C4×D79 | — | 2 | 1 | no | — |
| 99 | C4×D79 | — | 2 | 1 | no | — |
| 100 | C4×D79 | — | 4 | 2 | no | — |
| 101 | C4×D79 | — | 4 | 2 | no | — |
| 102 | C4×D79 | — | 4 | 2 | no | — |
| 103 | C4×D79 | — | 8 | 1 | no | — |
| 104 | C4×D79 | — | 8 | 1 | no | — |
| 105 | C8×C79 | — | 2 | 2 | no | — |
| 106 | C8×C79 | — | 8 | 2 | no | — |
| 107 | C79⋊C8 | — | 2 | 2 | no | — |
| 108 | C79⋊C8 | — | 8 | 2 | no | — |
| 109 | C22×C79 | C2×#99 | 1 | 1 | yes | characteristic |
| 110 | C2×D79 | — | 2 | 1 | no | — |
| 111 | C2×D79 | — | 4 | 1 | no | — |
| 112 | C2×D79 | — | 4 | 2 | no | — |
| 113 | C2×D79 | — | 8 | 2 | no | — |
| 114 | C2×D79 | — | 8 | 2 | no | — |
| 115 | C2×D79 | — | 8 | 1 | no | — |
| 116 | C4×C79 | — | 2 | 2 | no | — |
| 117 | C4×C79 | — | 2 | 1 | no | — |
| 118 | C4×C79 | — | 4 | 2 | no | — |
| 119 | C4×C79 | — | 8 | 1 | no | — |
| 120 | Dic79 | — | 4 | 1 | no | — |
| 121 | Dic79 | #353 | 1 | 1 | yes | characteristic |
| 122 | Dic79 | #353 | 1 | 1 | yes | characteristic |
| 123 | #13589 | — | 79 | 1 | no | maximal, Sylow |
| 124 | C2×C79 | — | 2 | 1 | no | — |
| 125 | C2×C79 | C2×#353 | 1 | 1 | yes | characteristic |
| 126 | D79 | — | 4 | 1 | no | — |
| 127 | D79 | — | 8 | 2 | no | — |
| 128 | D79 | — | 16 | 1 | no | — |
| 129 | #3147 | — | 79 | 1 | no | — |
| 130 | #2132 | — | 79 | 2 | no | — |
| 131 | #1411 | — | 79 | 1 | no | — |
| 132 | #1411 | D79 | 1 | 1 | yes | characteristic, complemented |
| 133 | #1410 | — | 79 | 2 | no | — |
| 134 | C79 | #13589 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 135 | C2×#135 | — | 158 | 2 | no | — |
| 136 | #532 | — | 79 | 1 | no | — |
| 137 | #532 | — | 79 | 1 | no | — |
| 138 | #495 | — | 158 | 2 | no | — |
| 139 | #494 | C2×D79 | 1 | 1 | yes | characteristic |
| 140 | #493 | — | 79 | 1 | no | — |
| 141 | #492 | — | 79 | 1 | no | — |
| 142 | #492 | — | 79 | 1 | no | — |
| 143 | #413 | — | 158 | 2 | no | — |
| 144 | C2×C4×C8 | — | 79 | 1 | no | — |
| 145 | #356 | — | 79 | 2 | no | — |
| 146 | #356 | C2×D79 | 1 | 2 | yes | — |
| 147 | #355 | — | 158 | 2 | no | — |
| 148 | C2×#41 | — | 79 | 2 | no | — |
| 149 | C2×#41 | — | 158 | 1 | no | — |
| 150 | #135 | — | 316 | 4 | no | — |
| 151 | #135 | — | 316 | 4 | no | — |
| 152 | C2×Q16 | — | 2 | 2 | no | — |
| 153 | C2×SD16 | — | 158 | 2 | no | — |
| 154 | C2×SD16 | — | 158 | 2 | no | — |
| 155 | C2×D8 | — | 158 | 2 | no | — |
| 156 | C2×M16 | — | 316 | 2 | no | — |
| 157 | C22×C8 | — | 158 | 1 | no | — |
| 158 | C4⋊Q8 | C4×D79 | 1 | 1 | yes | characteristic |
| 159 | C4⋊Q8 | C22×D79 | 1 | 1 | yes | characteristic |
| 160 | #127 | — | 79 | 1 | no | — |
| 161 | #127 | — | 79 | 1 | no | — |
| 162 | #124 | — | 79 | 2 | no | — |
| 163 | #124 | — | 158 | 1 | no | — |
| 164 | #121 | — | 316 | 1 | no | — |
| 165 | #121 | — | 316 | 1 | no | — |
| 166 | C4×D4 | — | 316 | 1 | no | — |
| 167 | C4×D4 | — | 316 | 1 | no | — |
| 168 | C2×C42 | — | 79 | 1 | no | — |
| 169 | #101 | — | 2 | 2 | no | — |
| 170 | #101 | — | 158 | 2 | no | — |
| 171 | #100 | — | 316 | 2 | no | — |
| 172 | C4×C8 | — | 79 | 1 | no | — |
| 173 | C4×C8 | — | 158 | 1 | no | — |
| 174 | C4×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 175 | #41 | — | 316 | 2 | no | — |
| 176 | #41 | — | 316 | 2 | no | — |
| 177 | #41 | — | 316 | 2 | no | — |
| 178 | #41 | — | 316 | 2 | no | — |
| 179 | C2×Q8 | — | 2 | 1 | no | — |
| 180 | C2×Q8 | D4×D79 | 1 | 2 | yes | — |
| 181 | C2×D4 | — | 316 | 1 | no | — |
| 182 | C2×D4 | — | 316 | 1 | no | — |
| 183 | C2×D4 | — | 316 | 1 | no | — |
| 184 | C2×D4 | — | 79 | 2 | no | — |
| 185 | C2×D4 | — | 158 | 2 | no | — |
| 186 | C2×D4 | — | 158 | 1 | no | — |
| 187 | C22×C4 | — | 316 | 1 | no | — |
| 188 | C22×C4 | — | 79 | 1 | no | — |
| 189 | C22×C4 | — | 158 | 2 | no | — |
| 190 | C22×C4 | — | 158 | 1 | no | — |
| 191 | Q16 | — | 4 | 4 | no | — |
| 192 | SD16 | — | 316 | 4 | no | — |
| 193 | SD16 | — | 316 | 4 | no | — |
| 194 | D8 | — | 316 | 4 | no | — |
| 195 | M16 | — | 4 | 2 | no | — |
| 196 | M16 | — | 316 | 2 | no | — |
| 197 | M16 | — | 632 | 2 | no | — |
| 198 | C2×C8 | — | 2 | 1 | no | — |
| 199 | C2×C8 | — | 632 | 2 | no | — |
| 200 | C2×C8 | — | 158 | 2 | no | — |
| 201 | C2×C8 | — | 158 | 2 | no | — |
| 202 | C2×C8 | — | 158 | 1 | no | — |
| 203 | C4⋊C4 | — | 4 | 1 | no | — |
| 204 | C4⋊C4 | — | 4 | 1 | no | — |
| 205 | C22⋊C4 | — | 316 | 2 | no | — |
| 206 | C22⋊C4 | — | 316 | 2 | no | — |
| 207 | C42 | — | 79 | 1 | no | — |
| 208 | C42 | — | 158 | 1 | no | — |
| 209 | C42 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 210 | C23 | — | 316 | 1 | no | — |
| 211 | C23 | — | 79 | 1 | no | — |
| 212 | C23 | — | 158 | 2 | no | — |
| 213 | Q8 | — | 4 | 2 | no | — |
| 214 | Q8 | — | 4 | 2 | no | — |
| 215 | D4 | — | 316 | 2 | no | — |
| 216 | D4 | — | 316 | 2 | no | — |
| 217 | D4 | — | 632 | 2 | no | — |
| 218 | D4 | — | 632 | 2 | no | — |
| 219 | D4 | — | 632 | 2 | no | — |
| 220 | D4 | — | 632 | 2 | no | — |
| 221 | C2×C4 | — | 2 | 1 | no | — |
| 222 | C2×C4 | — | 2 | 2 | no | — |
| 223 | C2×C4 | — | 4 | 1 | no | — |
| 224 | C2×C4 | — | 316 | 2 | no | — |
| 225 | C2×C4 | — | 316 | 2 | no | — |
| 226 | C2×C4 | — | 316 | 2 | no | — |
| 227 | C2×C4 | — | 79 | 1 | no | — |
| 228 | C2×C4 | — | 632 | 1 | no | — |
| 229 | C2×C4 | — | 632 | 1 | no | — |
| 230 | C2×C4 | — | 158 | 1 | no | — |
| 231 | C2×C4 | — | 158 | 1 | no | — |
| 232 | C2×C4 | — | 158 | 1 | no | — |
| 233 | C2×C4 | C22⋊C4×D79 | 1 | 1 | yes | characteristic |
| 234 | C8 | — | 2 | 2 | no | — |
| 235 | C8 | — | 8 | 2 | no | — |
| 236 | C8 | — | 632 | 2 | no | — |
| 237 | C8 | — | 158 | 2 | no | — |
| 238 | C22 | — | 316 | 1 | no | — |
| 239 | C22 | — | 316 | 2 | no | — |
| 240 | C22 | — | 632 | 2 | no | — |
| 241 | C22 | — | 632 | 2 | no | — |
| 242 | C22 | — | 632 | 1 | no | — |
| 243 | C22 | — | 158 | 1 | no | — |
| 244 | C22 | #99×D79 | 1 | 1 | yes | characteristic |
| 245 | C4 | — | 2 | 2 | no | — |
| 246 | C4 | — | 2 | 1 | no | — |
| 247 | C4 | — | 4 | 2 | no | — |
| 248 | C4 | — | 8 | 1 | no | — |
| 249 | C4 | — | 316 | 1 | no | — |
| 250 | C4 | — | 79 | 1 | no | — |
| 251 | C4 | — | 79 | 1 | no | — |
| 252 | C2 | — | 2 | 1 | no | — |
| 253 | C2 | — | 316 | 1 | no | — |
| 254 | C2 | — | 632 | 2 | no | — |
| 255 | C2 | — | 1264 | 1 | no | — |
| 256 | C2 | #353×D79 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23765084 | C2 | semidirect |
| 3. #23761569 | C2 | semidirect |
| 4. #23761566 | C2 | semidirect |
| 5. C79×#1411 | C2 | semidirect |
| 132. #1411 | D79 | semidirect |
| 134. C79 | #13589 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23773829 | 2 |
| #23765084 | 2 |
| #23761569 | 2 |
| #23761566 | 2 |
| C79×#1411 | 2 |
| #13589 | 79 |