← #34200057
Subgroups — #34200057
Order 20224.
224 conjugacy classes of subgroups.
The subgroup lattice has 212 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23765309 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #23765232 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #23761464 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Q8⋊(C79⋊C16) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | D4⋊(C79⋊C16) | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C8×C79⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C8×SD16×C79 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14579101 | — | 2 | 1 | no | — |
| 9 | C4⋊(C79⋊C16) | — | 2 | 1 | no | — |
| 10 | C4⋊(C79⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 11 | C4⋊(C79⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 12 | C22⋊(C79⋊C16) | — | 2 | 1 | no | — |
| 13 | #19098339 | — | 2 | 1 | no | — |
| 14 | C4×C79⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×C79⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 16 | C8×Q8×C79 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×SD16×C79 | C4 | 1 | 1 | yes | characteristic |
| 18 | C8×D4×C79 | C22 | 1 | 1 | yes | characteristic |
| 19 | #334×C79 | C4 | 1 | 1 | yes | characteristic |
| 20 | Q8⋊C8×C79 | C4 | 1 | 1 | yes | characteristic |
| 21 | D4⋊C8×C79 | C4 | 1 | 1 | yes | characteristic |
| 22 | C82×C79 | C22 | 1 | 1 | yes | characteristic |
| 23 | #14579101 | — | 4 | 2 | no | — |
| 24 | C2×C79⋊C16 | — | 2 | 1 | no | — |
| 25 | C2×C79⋊C16 | — | 2 | 1 | no | — |
| 26 | C2×C79⋊C16 | — | 2 | 1 | no | — |
| 27 | C2×C79⋊C16 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×C79⋊C16 | D4 | 1 | 1 | yes | characteristic |
| 29 | C2×SD16×C79 | C8 | 1 | 1 | yes | characteristic |
| 30 | C22×C8×C79 | — | 2 | 1 | no | — |
| 31 | C4×Q8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C4×D4×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | #106×C79 | C8 | 1 | 1 | yes | characteristic |
| 34 | C4⋊C8×C79 | — | 2 | 1 | no | — |
| 35 | C4⋊C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C4⋊C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | #103×C79 | C8 | 1 | 1 | yes | characteristic |
| 38 | #102×C79 | C8 | 1 | 1 | yes | characteristic |
| 39 | C22⋊C8×C79 | — | 2 | 1 | no | — |
| 40 | C4×C8×C79 | — | 2 | 1 | no | — |
| 41 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 44 | C79⋊C16 | — | 2 | 2 | no | — |
| 45 | C79⋊C16 | — | 2 | 1 | no | — |
| 46 | C79⋊C16 | — | 4 | 1 | no | — |
| 47 | C79⋊C16 | — | 4 | 1 | no | — |
| 48 | C79⋊C16 | SD16 | 1 | 2 | yes | complemented |
| 49 | C2×Q8×C79 | C2×C8 | 1 | 1 | yes | characteristic |
| 50 | C2×D4×C79 | C2×C8 | 1 | 1 | yes | characteristic |
| 51 | C22×C4×C79 | — | 2 | 1 | no | — |
| 52 | SD16×C79 | — | 2 | 2 | no | — |
| 53 | C2×C8×C79 | — | 2 | 1 | no | — |
| 54 | C2×C8×C79 | — | 2 | 1 | no | — |
| 55 | C2×C8×C79 | — | 2 | 1 | no | — |
| 56 | C2×C8×C79 | — | 4 | 1 | no | — |
| 57 | C2×C8×C79 | — | 4 | 1 | no | — |
| 58 | C2×C8×C79 | C2×C8 | 1 | 1 | yes | characteristic |
| 59 | C2×C8×C79 | C2×C8 | 1 | 1 | yes | characteristic |
| 60 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C8×C79 | #41 | 1 | 1 | yes | characteristic |
| 62 | C4⋊C4×C79 | — | 2 | 1 | no | — |
| 63 | C4⋊C4×C79 | C2×C8 | 1 | 1 | yes | characteristic |
| 64 | C4⋊C4×C79 | C2×C8 | 1 | 1 | yes | characteristic |
| 65 | C22⋊C4×C79 | — | 2 | 1 | no | — |
| 66 | C42×C79 | — | 2 | 1 | no | — |
| 67 | C42×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C23×C79 | — | 2 | 1 | no | — |
| 69 | Q8×C79 | — | 2 | 1 | no | — |
| 70 | Q8×C79 | — | 2 | 1 | no | — |
| 71 | D4×C79 | — | 2 | 1 | no | — |
| 72 | D4×C79 | — | 2 | 1 | no | — |
| 73 | C2×C4×C79 | — | 2 | 1 | no | — |
| 74 | C2×C4×C79 | — | 2 | 1 | no | — |
| 75 | C2×C4×C79 | — | 2 | 1 | no | — |
| 76 | C2×C4×C79 | — | 4 | 1 | no | — |
| 77 | C2×C4×C79 | — | 4 | 1 | no | — |
| 78 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C79 | C22×C8 | 1 | 1 | yes | characteristic |
| 80 | C2×C4×C79 | #131 | 1 | 1 | yes | characteristic |
| 81 | C8×C79 | — | 2 | 2 | no | — |
| 82 | C8×C79 | — | 2 | 1 | no | — |
| 83 | C8×C79 | — | 2 | 1 | no | — |
| 84 | C8×C79 | — | 4 | 1 | no | — |
| 85 | C8×C79 | — | 4 | 1 | no | — |
| 86 | C8×C79 | M32 | 1 | 2 | yes | complemented |
| 87 | C8×C79 | C2×SD16 | 1 | 1 | yes | characteristic |
| 88 | C8×C79 | #135 | 1 | 1 | yes | characteristic |
| 89 | C22×C79 | — | 4 | 1 | no | — |
| 90 | C22×C79 | — | 4 | 1 | no | — |
| 91 | C22×C79 | — | 4 | 1 | no | — |
| 92 | C22×C79 | C8×D4 | 1 | 1 | yes | characteristic |
| 93 | C4×C79 | — | 2 | 1 | no | — |
| 94 | C4×C79 | — | 4 | 1 | no | — |
| 95 | C4×C79 | — | 4 | 1 | no | — |
| 96 | C4×C79 | — | 4 | 1 | no | — |
| 97 | C4×C79 | C4×SD16 | 1 | 1 | yes | characteristic |
| 98 | C4×C79 | #443 | 1 | 1 | yes | characteristic |
| 99 | C4×C79 | C2×M32 | 1 | 1 | yes | characteristic |
| 100 | C4×C79 | #504 | 1 | 1 | yes | characteristic |
| 101 | #12051 | — | 79 | 1 | no | maximal, Sylow |
| 102 | C2×C79 | — | 8 | 1 | no | — |
| 103 | C2×C79 | C8×SD16 | 1 | 1 | yes | characteristic |
| 104 | C2×C79 | #2169 | 1 | 1 | yes | characteristic |
| 105 | C2×C79 | #2170 | 1 | 1 | yes | characteristic |
| 106 | #2183 | — | 79 | 1 | no | — |
| 107 | #2169 | — | 79 | 1 | no | — |
| 108 | C8×SD16 | D79 | 1 | 1 | yes | characteristic |
| 109 | #1367 | — | 79 | 1 | no | — |
| 110 | Q8⋊C16 | — | 79 | 1 | no | — |
| 111 | D4⋊C16 | — | 79 | 1 | no | — |
| 112 | C8×C16 | — | 79 | 1 | no | — |
| 113 | C79 | #12051 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 114 | C2×M32 | — | 158 | 1 | no | — |
| 115 | C8×Q8 | C2×D79 | 1 | 1 | yes | characteristic |
| 116 | C4×SD16 | Dic79 | 1 | 1 | yes | characteristic |
| 117 | C8×D4 | C2×D79 | 1 | 1 | yes | characteristic |
| 118 | C4⋊C16 | — | 79 | 1 | no | — |
| 119 | C4⋊C16 | — | 79 | 1 | no | — |
| 120 | C4⋊C16 | — | 158 | 1 | no | — |
| 121 | C22⋊C16 | — | 158 | 1 | no | — |
| 122 | #346 | — | 158 | 1 | no | — |
| 123 | C4×C16 | — | 79 | 1 | no | — |
| 124 | C4×C16 | — | 79 | 1 | no | — |
| 125 | #334 | Dic79 | 1 | 1 | yes | characteristic |
| 126 | Q8⋊C8 | Dic79 | 1 | 1 | yes | characteristic |
| 127 | D4⋊C8 | Dic79 | 1 | 1 | yes | characteristic |
| 128 | C82 | C2×D79 | 1 | 1 | yes | characteristic |
| 129 | C2×SD16 | C79⋊C8 | 1 | 1 | yes | characteristic |
| 130 | C22×C8 | — | 2 | 1 | no | — |
| 131 | C4×Q8 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 132 | C4×D4 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 133 | M32 | — | 316 | 2 | no | — |
| 134 | C2×C16 | — | 79 | 1 | no | — |
| 135 | C2×C16 | — | 79 | 1 | no | — |
| 136 | C2×C16 | — | 158 | 1 | no | — |
| 137 | C2×C16 | — | 158 | 1 | no | — |
| 138 | C2×C16 | — | 158 | 1 | no | — |
| 139 | #106 | C79⋊C8 | 1 | 1 | yes | characteristic |
| 140 | C4⋊C8 | — | 2 | 1 | no | — |
| 141 | C4⋊C8 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 142 | C4⋊C8 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 143 | #103 | C79⋊C8 | 1 | 1 | yes | characteristic |
| 144 | #102 | C79⋊C8 | 1 | 1 | yes | characteristic |
| 145 | C22⋊C8 | — | 2 | 1 | no | — |
| 146 | C4×C8 | — | 2 | 1 | no | — |
| 147 | C4×C8 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 148 | C4×C8 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 149 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 150 | C2×Q8 | C2×C79⋊C8 | 1 | 1 | yes | characteristic |
| 151 | C2×D4 | C2×C79⋊C8 | 1 | 1 | yes | characteristic |
| 152 | C22×C4 | — | 2 | 1 | no | — |
| 153 | SD16 | — | 2 | 2 | no | — |
| 154 | C2×C8 | — | 2 | 1 | no | — |
| 155 | C2×C8 | — | 2 | 1 | no | — |
| 156 | C2×C8 | — | 2 | 1 | no | — |
| 157 | C2×C8 | — | 4 | 1 | no | — |
| 158 | C2×C8 | — | 4 | 1 | no | — |
| 159 | C2×C8 | C2×C79⋊C8 | 1 | 1 | yes | characteristic |
| 160 | C2×C8 | C2×C79⋊C8 | 1 | 1 | yes | characteristic |
| 161 | C2×C8 | D4×D79 | 1 | 1 | yes | characteristic |
| 162 | C2×C8 | #11929623 | 1 | 1 | yes | characteristic |
| 163 | C4⋊C4 | — | 2 | 1 | no | — |
| 164 | C4⋊C4 | C2×C79⋊C8 | 1 | 1 | yes | characteristic |
| 165 | C4⋊C4 | C2×C79⋊C8 | 1 | 1 | yes | characteristic |
| 166 | C22⋊C4 | — | 2 | 1 | no | — |
| 167 | C42 | — | 2 | 1 | no | — |
| 168 | C42 | C22×Dic79 | 1 | 1 | yes | characteristic |
| 169 | C16 | — | 316 | 1 | no | — |
| 170 | C16 | — | 316 | 1 | no | — |
| 171 | C16 | — | 79 | 2 | no | — |
| 172 | C16 | — | 158 | 2 | no | — |
| 173 | C16 | — | 158 | 1 | no | — |
| 174 | C23 | — | 2 | 1 | no | — |
| 175 | Q8 | — | 2 | 1 | no | — |
| 176 | Q8 | — | 2 | 1 | no | — |
| 177 | D4 | — | 2 | 1 | no | — |
| 178 | D4 | — | 2 | 1 | no | — |
| 179 | C2×C4 | — | 2 | 1 | no | — |
| 180 | C2×C4 | — | 2 | 1 | no | — |
| 181 | C2×C4 | — | 2 | 1 | no | — |
| 182 | C2×C4 | — | 4 | 1 | no | — |
| 183 | C2×C4 | — | 4 | 1 | no | — |
| 184 | C2×C4 | D4×Dic79 | 1 | 1 | yes | characteristic |
| 185 | C2×C4 | C22×C79⋊C8 | 1 | 1 | yes | characteristic |
| 186 | C2×C4 | #14579168 | 1 | 1 | yes | characteristic |
| 187 | C8 | — | 2 | 2 | no | — |
| 188 | C8 | — | 2 | 1 | no | — |
| 189 | C8 | — | 2 | 1 | no | — |
| 190 | C8 | — | 4 | 1 | no | — |
| 191 | C8 | — | 4 | 1 | no | — |
| 192 | C8 | #14579101 | 1 | 2 | yes | complemented |
| 193 | C8 | SD16×D79 | 1 | 1 | yes | characteristic |
| 194 | C8 | #14579182 | 1 | 1 | yes | characteristic |
| 195 | C22 | — | 4 | 1 | no | — |
| 196 | C22 | — | 4 | 1 | no | — |
| 197 | C22 | — | 4 | 1 | no | — |
| 198 | C22 | D4×C79⋊C8 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 2 | 1 | no | — |
| 200 | C4 | — | 4 | 1 | no | — |
| 201 | C4 | — | 4 | 1 | no | — |
| 202 | C4 | — | 4 | 1 | no | — |
| 203 | C4 | SD16×Dic79 | 1 | 1 | yes | characteristic |
| 204 | C4 | #19098680 | 1 | 1 | yes | characteristic |
| 205 | C4 | C2×#14579101 | 1 | 1 | yes | characteristic |
| 206 | C4 | #19098980 | 1 | 1 | yes | characteristic |
| 207 | C2 | — | 8 | 1 | no | — |
| 208 | C2 | SD16×C79⋊C8 | 1 | 1 | yes | characteristic |
| 209 | C2 | #23765232 | 1 | 1 | yes | characteristic |
| 210 | C2 | #23765237 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta