Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
362 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 362 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 | #22621724 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22612009 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22607972 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22607908 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C73×#1722 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #17989393 | — | 2 | 2 | no | — |
| 7 | #17989028 | — | 2 | 2 | no | — |
| 8 | #17988847 | — | 2 | 2 | no | — |
| 9 | #17988769 | — | 2 | 1 | no | — |
| 10 | #17988754 | — | 2 | 1 | no | — |
| 11 | #17988705 | C22 | 1 | 2 | yes | — |
| 12 | #17988408 | C22 | 1 | 2 | yes | complemented |
| 13 | #17988403 | — | 2 | 2 | no | — |
| 14 | #500×C73 | — | 2 | 1 | no | — |
| 15 | #494×C73 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | #492×C73 | C22 | 1 | 1 | yes | characteristic |
| 17 | #486×C73 | — | 2 | 1 | no | — |
| 18 | #322×C73 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | #14556639 | — | 2 | 1 | no | — |
| 20 | #14556636 | — | 2 | 1 | no | — |
| 21 | #14556622 | — | 2 | 2 | no | — |
| 22 | #14556620 | — | 4 | 2 | no | — |
| 23 | #14556618 | — | 4 | 2 | no | — |
| 24 | #14556617 | — | 2 | 2 | no | — |
| 25 | #14556613 | D4 | 1 | 2 | yes | — |
| 26 | #14556612 | — | 4 | 2 | no | — |
| 27 | SD16×D73 | — | 4 | 2 | no | — |
| 28 | #14556597 | — | 2 | 2 | no | — |
| 29 | #14556537 | — | 4 | 2 | no | — |
| 30 | #14556534 | — | 4 | 2 | no | — |
| 31 | #14556531 | — | 2 | 2 | no | — |
| 32 | #14556527 | D4 | 1 | 2 | yes | — |
| 33 | #14556519 | — | 2 | 2 | no | — |
| 34 | #14556510 | — | 2 | 1 | no | — |
| 35 | #14556508 | — | 2 | 1 | no | — |
| 36 | C2×Q16×C73 | — | 2 | 2 | no | — |
| 37 | C2×SD16×C73 | — | 2 | 2 | no | — |
| 38 | C4⋊Q8×C73 | — | 2 | 1 | no | — |
| 39 | C4⋊Q8×C73 | D4 | 1 | 1 | yes | characteristic |
| 40 | #127×C73 | D4 | 1 | 1 | yes | characteristic |
| 41 | #107×C73 | — | 4 | 2 | no | — |
| 42 | #102×C73 | — | 4 | 2 | no | — |
| 43 | C4×C8×C73 | — | 2 | 1 | no | — |
| 44 | C4×C8×C73 | C23 | 1 | 1 | yes | characteristic |
| 45 | #11924029 | — | 4 | 2 | no | — |
| 46 | #11924028 | — | 2 | 2 | no | — |
| 47 | #11924028 | — | 8 | 1 | no | — |
| 48 | #11924027 | — | 4 | 2 | no | — |
| 49 | Q8×D73 | — | 4 | 2 | no | — |
| 50 | C2×C22⋊D73 | — | 8 | 1 | no | — |
| 51 | D4×D73 | — | 4 | 2 | no | — |
| 52 | Dic292 | D8 | 1 | 2 | yes | — |
| 53 | #11924018 | — | 4 | 2 | no | — |
| 54 | #11924017 | — | 2 | 2 | no | — |
| 55 | #11924016 | — | 4 | 2 | no | — |
| 56 | #11924015 | — | 4 | 2 | no | — |
| 57 | D584 | D8 | 1 | 2 | yes | — |
| 58 | #11924013 | — | 4 | 2 | no | — |
| 59 | #11924012 | — | 2 | 2 | no | — |
| 60 | #11924011 | — | 4 | 2 | no | — |
| 61 | C8×D73 | — | 4 | 2 | no | — |
| 62 | #11924003 | — | 4 | 2 | no | — |
| 63 | C2×Q8×C73 | — | 2 | 1 | no | — |
| 64 | C2×Q8×C73 | — | 4 | 1 | no | — |
| 65 | C2×D4×C73 | — | 2 | 1 | no | — |
| 66 | C2×D4×C73 | — | 4 | 1 | no | — |
| 67 | Q16×C73 | — | 4 | 2 | no | — |
| 68 | Q16×C73 | — | 4 | 2 | no | — |
| 69 | SD16×C73 | — | 4 | 2 | no | — |
| 70 | SD16×C73 | — | 4 | 2 | no | — |
| 71 | C2×C8×C73 | — | 2 | 2 | no | — |
| 72 | C2×C8×C73 | C2×D4 | 1 | 2 | yes | — |
| 73 | C4⋊C4×C73 | — | 4 | 1 | no | — |
| 74 | C4⋊C4×C73 | — | 4 | 1 | no | — |
| 75 | C4⋊C4×C73 | — | 4 | 1 | no | — |
| 76 | C42×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C23×C73 | — | 4 | 1 | no | — |
| 78 | C22×D73 | — | 8 | 1 | no | — |
| 79 | Q8×C73 | — | 4 | 2 | no | — |
| 80 | Q8×C73 | — | 8 | 2 | no | — |
| 81 | D4×C73 | — | 4 | 2 | no | — |
| 82 | D4×C73 | — | 8 | 2 | no | — |
| 83 | C2×C4×C73 | — | 2 | 1 | no | — |
| 84 | C2×C4×C73 | — | 4 | 1 | no | — |
| 85 | C2×C4×C73 | — | 8 | 1 | no | — |
| 86 | C2×C4×C73 | #120 | 1 | 1 | yes | characteristic |
| 87 | C22⋊D73 | — | 4 | 2 | no | — |
| 88 | C22⋊D73 | — | 8 | 2 | no | — |
| 89 | C2×Dic73 | — | 8 | 1 | no | — |
| 90 | D292 | — | 2 | 2 | no | — |
| 91 | D292 | — | 8 | 1 | no | — |
| 92 | C4×D73 | — | 4 | 2 | no | — |
| 93 | C4×D73 | — | 8 | 2 | no | — |
| 94 | Dic146 | — | 2 | 2 | no | — |
| 95 | Dic146 | — | 8 | 1 | no | — |
| 96 | C8×C73 | — | 4 | 2 | no | — |
| 97 | C8×C73 | C2×D8 | 1 | 2 | yes | — |
| 98 | C8×C73 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 99 | #10596831 | — | 4 | 2 | no | — |
| 100 | C22×C73 | — | 8 | 1 | no | — |
| 101 | C22×C73 | — | 16 | 1 | no | — |
| 102 | C22×C73 | #453 | 1 | 1 | yes | characteristic |
| 103 | C2×D73 | — | 4 | 2 | no | — |
| 104 | C2×D73 | — | 32 | 1 | no | — |
| 105 | C4×C73 | — | 2 | 2 | no | — |
| 106 | C4×C73 | — | 8 | 1 | no | — |
| 107 | C4×C73 | — | 16 | 1 | no | — |
| 108 | C4×C73 | #447 | 1 | 2 | yes | — |
| 109 | Dic73 | — | 4 | 2 | no | — |
| 110 | #12315 | — | 73 | 1 | no | maximal, Sylow |
| 111 | C2×C73 | — | 2 | 1 | no | — |
| 112 | C2×C73 | — | 16 | 1 | no | — |
| 113 | C2×C73 | #1619 | 1 | 1 | yes | characteristic |
| 114 | D73 | — | 8 | 2 | no | — |
| 115 | #3291 | — | 73 | 1 | no | — |
| 116 | #2214 | — | 73 | 2 | no | — |
| 117 | #1722 | D73 | 1 | 1 | yes | characteristic, complemented |
| 118 | #1359 | — | 73 | 1 | no | — |
| 119 | #1336 | — | 73 | 2 | no | — |
| 120 | C73 | #12315 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 121 | #577 | — | 146 | 2 | no | — |
| 122 | #510 | — | 146 | 2 | no | — |
| 123 | #500 | — | 2 | 1 | no | — |
| 124 | #494 | C2×D73 | 1 | 1 | yes | characteristic, complemented |
| 125 | #492 | C2×D73 | 1 | 1 | yes | characteristic |
| 126 | #486 | — | 2 | 1 | no | — |
| 127 | #471 | — | 146 | 2 | no | — |
| 128 | #456 | — | 146 | 1 | no | — |
| 129 | #453 | — | 146 | 1 | no | — |
| 130 | #443 | — | 73 | 2 | no | — |
| 131 | #364 | — | 73 | 2 | no | — |
| 132 | #362 | — | 146 | 2 | no | — |
| 133 | #322 | C2×D73 | 1 | 1 | yes | characteristic, complemented |
| 134 | 2−1+4 | — | 146 | 1 | no | — |
| 135 | H(5,𝔽2) | — | 146 | 1 | no | — |
| 136 | #137 | — | 292 | 2 | no | — |
| 137 | #137 | — | 146 | 2 | no | — |
| 138 | AGL(1,ℤ/8) | — | 292 | 2 | no | — |
| 139 | AGL(1,ℤ/8) | — | 146 | 2 | no | — |
| 140 | #135 | — | 292 | 2 | no | — |
| 141 | #135 | — | 73 | 2 | no | — |
| 142 | C2×Q16 | — | 2 | 2 | no | — |
| 143 | C2×SD16 | — | 2 | 2 | no | — |
| 144 | C2×SD16 | — | 292 | 2 | no | — |
| 145 | #131 | — | 146 | 2 | no | — |
| 146 | C4⋊Q8 | — | 2 | 1 | no | — |
| 147 | C4⋊Q8 | C22⋊D73 | 1 | 1 | yes | characteristic |
| 148 | #127 | C22⋊D73 | 1 | 1 | yes | characteristic |
| 149 | Q32 | — | 292 | 2 | no | — |
| 150 | SD32 | — | 292 | 2 | no | — |
| 151 | M32 | — | 146 | 2 | no | — |
| 152 | #108 | — | 73 | 2 | no | — |
| 153 | #107 | — | 4 | 2 | no | — |
| 154 | #104 | — | 146 | 2 | no | — |
| 155 | #102 | — | 4 | 2 | no | — |
| 156 | #101 | — | 146 | 1 | no | — |
| 157 | #100 | — | 146 | 1 | no | — |
| 158 | C4×C8 | — | 2 | 1 | no | — |
| 159 | C4×C8 | C22×D73 | 1 | 1 | yes | characteristic |
| 160 | #41 | — | 292 | 2 | no | — |
| 161 | #41 | — | 292 | 2 | no | — |
| 162 | #41 | — | 584 | 1 | no | — |
| 163 | #41 | — | 146 | 2 | no | — |
| 164 | C2×Q8 | — | 2 | 1 | no | — |
| 165 | C2×Q8 | — | 4 | 1 | no | — |
| 166 | C2×Q8 | — | 292 | 2 | no | — |
| 167 | C2×D4 | — | 2 | 1 | no | — |
| 168 | C2×D4 | — | 4 | 1 | no | — |
| 169 | C2×D4 | — | 292 | 2 | no | — |
| 170 | C2×D4 | — | 584 | 1 | no | — |
| 171 | Q16 | — | 4 | 2 | no | — |
| 172 | Q16 | — | 4 | 2 | no | — |
| 173 | Q16 | — | 292 | 2 | no | — |
| 174 | Q16 | — | 73 | 2 | no | — |
| 175 | SD16 | — | 4 | 2 | no | — |
| 176 | SD16 | — | 4 | 2 | no | — |
| 177 | SD16 | — | 292 | 2 | no | — |
| 178 | SD16 | — | 292 | 2 | no | — |
| 179 | SD16 | — | 146 | 2 | no | — |
| 180 | D8 | — | 292 | 2 | no | — |
| 181 | D8 | — | 73 | 2 | no | — |
| 182 | M16 | — | 292 | 2 | no | — |
| 183 | M16 | — | 146 | 2 | no | — |
| 184 | C2×C8 | — | 2 | 2 | no | — |
| 185 | C2×C8 | — | 292 | 2 | no | — |
| 186 | C2×C8 | D4×D73 | 1 | 2 | yes | — |
| 187 | C4⋊C4 | — | 4 | 1 | no | — |
| 188 | C4⋊C4 | — | 4 | 1 | no | — |
| 189 | C4⋊C4 | — | 4 | 1 | no | — |
| 190 | C42 | C2×C22⋊D73 | 1 | 1 | yes | characteristic |
| 191 | C16 | — | 292 | 2 | no | — |
| 192 | C23 | — | 4 | 1 | no | — |
| 193 | C23 | — | 584 | 1 | no | — |
| 194 | Q8 | — | 4 | 2 | no | — |
| 195 | Q8 | — | 8 | 2 | no | — |
| 196 | Q8 | — | 584 | 1 | no | — |
| 197 | Q8 | — | 146 | 2 | no | — |
| 198 | D4 | — | 4 | 2 | no | — |
| 199 | D4 | — | 8 | 2 | no | — |
| 200 | D4 | — | 292 | 2 | no | — |
| 201 | D4 | — | 584 | 1 | no | — |
| 202 | D4 | — | 584 | 2 | no | — |
| 203 | D4 | — | 146 | 2 | no | — |
| 204 | C2×C4 | — | 2 | 1 | no | — |
| 205 | C2×C4 | — | 4 | 1 | no | — |
| 206 | C2×C4 | — | 8 | 1 | no | — |
| 207 | C2×C4 | — | 292 | 2 | no | — |
| 208 | C2×C4 | — | 584 | 2 | no | — |
| 209 | C2×C4 | — | 584 | 1 | no | — |
| 210 | C2×C4 | #14556559 | 1 | 1 | yes | characteristic |
| 211 | C8 | — | 4 | 2 | no | — |
| 212 | C8 | — | 292 | 2 | no | — |
| 213 | C8 | D8×D73 | 1 | 2 | yes | — |
| 214 | C8 | #14556615 | 1 | 2 | yes | — |
| 215 | C22 | — | 8 | 1 | no | — |
| 216 | C22 | — | 16 | 1 | no | — |
| 217 | C22 | — | 2336 | 1 | no | — |
| 218 | C22 | — | 292 | 2 | no | — |
| 219 | C22 | #17988754 | 1 | 1 | yes | characteristic |
| 220 | C4 | — | 2 | 2 | no | — |
| 221 | C4 | — | 8 | 1 | no | — |
| 222 | C4 | — | 16 | 1 | no | — |
| 223 | C4 | — | 292 | 2 | no | — |
| 224 | C4 | #17988719 | 1 | 2 | yes | — |
| 225 | C2 | — | 2 | 1 | no | — |
| 226 | C2 | — | 16 | 1 | no | — |
| 227 | C2 | — | 584 | 2 | no | — |
| 228 | C2 | #22608983 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22612009 | C2 | semidirect |
| 3. #22607972 | C2 | semidirect |
| 4. #22607908 | C2 | semidirect |
| 5. C73×#1722 | C2 | semidirect |
| 12. #17988408 | C22 | semidirect |
| 15. #494×C73 | C22 | semidirect |
| 18. #322×C73 | C22 | semidirect |
| 117. #1722 | D73 | semidirect |
| 120. C73 | #12315 | semidirect |
| 124. #494 | C2×D73 | semidirect |
| 133. #322 | C2×D73 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22621724 | 2 |
| #22612009 | 2 |
| #22607972 | 2 |
| #22607908 | 2 |
| C73×#1722 | 2 |
| #12315 | 73 |