← C23×#13625 #19161251
Subgroups — C23×#13625 #19161251
Order 5888.
290 conjugacy classes of subgroups.
The subgroup lattice has 226 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C23×#508 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C23×M64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×C23×#368 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23×#1420 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C23×#1419 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×C23×#135 | — | 2 | 1 | no | — |
| 7 | C23×#508 | — | 2 | 1 | no | — |
| 8 | C23×#508 | — | 2 | 1 | no | — |
| 9 | C23×#508 | C4 | 1 | 2 | yes | — |
| 10 | C23×#508 | C4 | 1 | 2 | yes | — |
| 11 | C2×C23×Q32 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C23×SD32 | — | 2 | 1 | no | — |
| 13 | C2×C23×D16 | C22 | 1 | 1 | yes | characteristic |
| 14 | C22×C16×C23 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C23×#108 | — | 2 | 1 | no | — |
| 16 | C23×M64 | — | 2 | 1 | no | — |
| 17 | C23×M64 | C22 | 1 | 2 | yes | — |
| 18 | C2×C23×C32 | — | 2 | 1 | no | — |
| 19 | C23×#368 | — | 2 | 1 | no | — |
| 20 | C23×#368 | C22 | 1 | 2 | yes | — |
| 21 | C2×#41×C23 | — | 4 | 1 | no | — |
| 22 | C23×#135 | — | 2 | 2 | no | — |
| 23 | C23×#135 | — | 2 | 2 | no | — |
| 24 | C23×#135 | — | 4 | 1 | no | — |
| 25 | C23×#135 | — | 4 | 1 | no | — |
| 26 | C2×Q16×C23 | — | 2 | 1 | no | — |
| 27 | C2×SD16×C23 | — | 4 | 1 | no | — |
| 28 | C2×D8×C23 | — | 2 | 1 | no | — |
| 29 | C2×M16×C23 | — | 4 | 1 | no | — |
| 30 | C22×C8×C23 | D4 | 1 | 1 | yes | characteristic |
| 31 | C23×Q32 | — | 2 | 1 | no | — |
| 32 | C23×Q32 | C2×C4 | 1 | 2 | yes | — |
| 33 | C23×SD32 | — | 2 | 2 | no | — |
| 34 | C23×SD32 | — | 2 | 2 | no | — |
| 35 | C23×D16 | — | 2 | 1 | no | — |
| 36 | C23×D16 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×C16×C23 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C2×C16×C23 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C2×C16×C23 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×C16×C23 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×C16×C23 | D4 | 1 | 1 | yes | characteristic |
| 42 | C2×C16×C23 | C23 | 1 | 1 | yes | characteristic |
| 43 | C23×#108 | — | 2 | 2 | no | — |
| 44 | C23×#108 | — | 4 | 1 | no | — |
| 45 | C23×C32 | — | 2 | 2 | no | — |
| 46 | #41×C23 | — | 4 | 2 | no | — |
| 47 | #41×C23 | — | 4 | 2 | no | — |
| 48 | #41×C23 | — | 8 | 1 | no | — |
| 49 | #41×C23 | — | 8 | 1 | no | — |
| 50 | C2×Q8×C23 | — | 4 | 1 | no | — |
| 51 | C2×D4×C23 | — | 4 | 1 | no | — |
| 52 | C2×D4×C23 | — | 8 | 1 | no | — |
| 53 | C22×C4×C23 | — | 8 | 1 | no | — |
| 54 | C22×C4×C23 | D8 | 1 | 1 | yes | characteristic |
| 55 | Q16×C23 | — | 2 | 2 | no | — |
| 56 | Q16×C23 | — | 4 | 1 | no | — |
| 57 | SD16×C23 | — | 4 | 2 | no | — |
| 58 | SD16×C23 | — | 4 | 2 | no | — |
| 59 | D8×C23 | — | 2 | 2 | no | — |
| 60 | D8×C23 | — | 4 | 1 | no | — |
| 61 | M16×C23 | — | 4 | 2 | no | — |
| 62 | M16×C23 | — | 8 | 1 | no | — |
| 63 | C2×C8×C23 | — | 8 | 1 | no | — |
| 64 | C2×C8×C23 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C23 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C2×C8×C23 | D8 | 1 | 1 | yes | characteristic |
| 67 | C2×C8×C23 | SD16 | 1 | 1 | yes | characteristic |
| 68 | C2×C8×C23 | SD16 | 1 | 1 | yes | characteristic |
| 69 | C2×C8×C23 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C16×C23 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C16×C23 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 72 | C16×C23 | C22×C4 | 1 | 1 | yes | characteristic |
| 73 | C16×C23 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | #13625 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 75 | C23×C23 | — | 8 | 1 | no | — |
| 76 | C23×C23 | SD32 | 1 | 1 | yes | characteristic |
| 77 | Q8×C23 | — | 4 | 2 | no | — |
| 78 | Q8×C23 | — | 8 | 1 | no | — |
| 79 | D4×C23 | — | 4 | 2 | no | — |
| 80 | D4×C23 | — | 8 | 2 | no | — |
| 81 | D4×C23 | — | 8 | 2 | no | — |
| 82 | D4×C23 | — | 8 | 1 | no | — |
| 83 | C2×C4×C23 | — | 8 | 1 | no | — |
| 84 | C2×C4×C23 | — | 8 | 2 | no | — |
| 85 | C2×C4×C23 | — | 8 | 2 | no | — |
| 86 | C2×C4×C23 | #102 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C23 | #102 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C23 | D16 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C23 | D16 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C23 | SD32 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C23 | C2×D8 | 1 | 1 | yes | characteristic |
| 92 | C8×C23 | — | 8 | 2 | no | — |
| 93 | C8×C23 | #102 | 1 | 1 | yes | characteristic |
| 94 | C8×C23 | #102 | 1 | 1 | yes | characteristic |
| 95 | C8×C23 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C8×C23 | C2×SD16 | 1 | 1 | yes | characteristic |
| 97 | C2×#508 | C2×C23 | 1 | 1 | yes | characteristic |
| 98 | C2×M64 | C2×C23 | 1 | 1 | yes | characteristic, complemented |
| 99 | C2×#368 | C2×C23 | 1 | 1 | yes | characteristic, complemented |
| 100 | #1420 | C2×C23 | 1 | 2 | yes | complemented |
| 101 | #1419 | C2×C23 | 1 | 2 | yes | complemented |
| 102 | C22×C23 | — | 4 | 1 | no | — |
| 103 | C22×C23 | — | 8 | 2 | no | — |
| 104 | C22×C23 | — | 16 | 2 | no | — |
| 105 | C22×C23 | #357 | 1 | 1 | yes | characteristic |
| 106 | C22×C23 | #357 | 1 | 1 | yes | characteristic |
| 107 | C22×C23 | C2×SD32 | 1 | 1 | yes | characteristic |
| 108 | C4×C23 | — | 8 | 2 | no | — |
| 109 | C4×C23 | #357 | 1 | 1 | yes | characteristic |
| 110 | C4×C23 | #357 | 1 | 1 | yes | characteristic |
| 111 | C4×C23 | C2×#102 | 1 | 1 | yes | characteristic |
| 112 | C4×C23 | C2×D16 | 1 | 1 | yes | characteristic |
| 113 | C2×#135 | — | 2 | 1 | no | — |
| 114 | #508 | — | 2 | 1 | no | — |
| 115 | #508 | — | 2 | 1 | no | — |
| 116 | #508 | C4×C23 | 1 | 2 | yes | — |
| 117 | #508 | C4×C23 | 1 | 2 | yes | — |
| 118 | C2×Q32 | C22×C23 | 1 | 1 | yes | characteristic |
| 119 | C2×SD32 | — | 2 | 1 | no | — |
| 120 | C2×D16 | C22×C23 | 1 | 1 | yes | characteristic |
| 121 | C22×C16 | C22×C23 | 1 | 1 | yes | characteristic |
| 122 | C2×#108 | — | 2 | 1 | no | — |
| 123 | M64 | — | 2 | 1 | no | — |
| 124 | M64 | C22×C23 | 1 | 2 | yes | — |
| 125 | C2×C32 | — | 2 | 1 | no | — |
| 126 | #368 | — | 2 | 1 | no | — |
| 127 | #368 | C22×C23 | 1 | 2 | yes | — |
| 128 | C2×C23 | — | 2 | 1 | no | — |
| 129 | C2×C23 | — | 2 | 1 | no | — |
| 130 | C2×C23 | — | 2 | 1 | no | — |
| 131 | C2×C23 | — | 16 | 2 | no | — |
| 132 | C2×C23 | C2×#357 | 1 | 1 | yes | characteristic |
| 133 | C2×#41 | — | 4 | 1 | no | — |
| 134 | #135 | — | 2 | 2 | no | — |
| 135 | #135 | — | 2 | 2 | no | — |
| 136 | #135 | — | 4 | 1 | no | — |
| 137 | #135 | — | 4 | 1 | no | — |
| 138 | C2×Q16 | — | 2 | 1 | no | — |
| 139 | C2×SD16 | — | 4 | 1 | no | — |
| 140 | C2×D8 | — | 2 | 1 | no | — |
| 141 | C2×M16 | — | 4 | 1 | no | — |
| 142 | C22×C8 | D4×C23 | 1 | 1 | yes | characteristic |
| 143 | Q32 | — | 2 | 1 | no | — |
| 144 | Q32 | C2×C4×C23 | 1 | 2 | yes | — |
| 145 | SD32 | — | 2 | 2 | no | — |
| 146 | SD32 | — | 2 | 2 | no | — |
| 147 | D16 | — | 2 | 1 | no | — |
| 148 | D16 | C2×C4×C23 | 1 | 2 | yes | — |
| 149 | C2×C16 | C2×C4×C23 | 1 | 1 | yes | characteristic |
| 150 | C2×C16 | C2×C4×C23 | 1 | 1 | yes | characteristic |
| 151 | C2×C16 | D4×C23 | 1 | 1 | yes | characteristic |
| 152 | C2×C16 | D4×C23 | 1 | 1 | yes | characteristic |
| 153 | C2×C16 | D4×C23 | 1 | 1 | yes | characteristic |
| 154 | C2×C16 | C23×C23 | 1 | 1 | yes | characteristic |
| 155 | #108 | — | 2 | 2 | no | — |
| 156 | #108 | — | 4 | 1 | no | — |
| 157 | C32 | — | 2 | 2 | no | — |
| 158 | C23 | #13625 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 159 | #41 | — | 4 | 2 | no | — |
| 160 | #41 | — | 4 | 2 | no | — |
| 161 | #41 | — | 8 | 1 | no | — |
| 162 | #41 | — | 8 | 1 | no | — |
| 163 | C2×Q8 | — | 4 | 1 | no | — |
| 164 | C2×D4 | — | 4 | 1 | no | — |
| 165 | C2×D4 | — | 8 | 1 | no | — |
| 166 | C22×C4 | — | 8 | 1 | no | — |
| 167 | C22×C4 | D8×C23 | 1 | 1 | yes | characteristic |
| 168 | Q16 | — | 2 | 2 | no | — |
| 169 | Q16 | — | 4 | 1 | no | — |
| 170 | SD16 | — | 4 | 2 | no | — |
| 171 | SD16 | — | 4 | 2 | no | — |
| 172 | D8 | — | 2 | 2 | no | — |
| 173 | D8 | — | 4 | 1 | no | — |
| 174 | M16 | — | 4 | 2 | no | — |
| 175 | M16 | — | 8 | 1 | no | — |
| 176 | C2×C8 | — | 8 | 1 | no | — |
| 177 | C2×C8 | C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 178 | C2×C8 | C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 179 | C2×C8 | D8×C23 | 1 | 1 | yes | characteristic |
| 180 | C2×C8 | SD16×C23 | 1 | 1 | yes | characteristic |
| 181 | C2×C8 | SD16×C23 | 1 | 1 | yes | characteristic |
| 182 | C2×C8 | C2×D4×C23 | 1 | 1 | yes | characteristic |
| 183 | C16 | C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 184 | C16 | C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 185 | C16 | C22×C4×C23 | 1 | 1 | yes | characteristic |
| 186 | C16 | C2×D4×C23 | 1 | 1 | yes | characteristic |
| 187 | C23 | — | 8 | 1 | no | — |
| 188 | C23 | C23×SD32 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 4 | 2 | no | — |
| 190 | Q8 | — | 8 | 1 | no | — |
| 191 | D4 | — | 4 | 2 | no | — |
| 192 | D4 | — | 8 | 2 | no | — |
| 193 | D4 | — | 8 | 2 | no | — |
| 194 | D4 | — | 8 | 1 | no | — |
| 195 | C2×C4 | — | 8 | 1 | no | — |
| 196 | C2×C4 | — | 8 | 2 | no | — |
| 197 | C2×C4 | — | 8 | 2 | no | — |
| 198 | C2×C4 | C23×#102 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | C23×#102 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | C23×D16 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | C23×D16 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | C23×SD32 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | C2×D8×C23 | 1 | 1 | yes | characteristic |
| 204 | C8 | — | 8 | 2 | no | — |
| 205 | C8 | C23×#102 | 1 | 1 | yes | characteristic |
| 206 | C8 | C23×#102 | 1 | 1 | yes | characteristic |
| 207 | C8 | C2×C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 208 | C8 | C2×SD16×C23 | 1 | 1 | yes | characteristic |
| 209 | C22 | — | 4 | 1 | no | — |
| 210 | C22 | — | 8 | 2 | no | — |
| 211 | C22 | — | 16 | 2 | no | — |
| 212 | C22 | C23×#357 | 1 | 1 | yes | characteristic |
| 213 | C22 | C23×#357 | 1 | 1 | yes | characteristic |
| 214 | C22 | C2×C23×SD32 | 1 | 1 | yes | characteristic |
| 215 | C4 | — | 8 | 2 | no | — |
| 216 | C4 | C23×#357 | 1 | 1 | yes | characteristic |
| 217 | C4 | C23×#357 | 1 | 1 | yes | characteristic |
| 218 | C4 | C2×C23×#102 | 1 | 1 | yes | characteristic |
| 219 | C4 | C2×C23×D16 | 1 | 1 | yes | characteristic |
| 220 | C2 | — | 2 | 1 | no | — |
| 221 | C2 | — | 2 | 1 | no | — |
| 222 | C2 | — | 2 | 1 | no | — |
| 223 | C2 | — | 16 | 2 | no | — |
| 224 | C2 | C2×C23×#357 | 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