Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
177 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 28 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 | #11717397 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#87838 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11717394 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11717373 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11716806 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11716782 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11716781 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#6039 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6037 | C22 | 1 | 1 | yes | characteristic |
| 10 | #87880 | C22 | 1 | 1 | yes | characteristic |
| 11 | #87867 | C22 | 1 | 1 | yes | characteristic |
| 12 | #87867 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#6034 | C22 | 1 | 1 | yes | characteristic |
| 14 | #87860 | C22 | 1 | 1 | yes | characteristic |
| 15 | #87839 | — | 2 | 1 | no | — |
| 16 | #87838 | — | 2 | 1 | no | — |
| 17 | #87838 | — | 2 | 1 | no | — |
| 18 | #87838 | — | 2 | 1 | no | — |
| 19 | #87838 | — | 2 | 1 | no | — |
| 20 | #87837 | — | 2 | 1 | no | — |
| 21 | #87834 | — | 2 | 1 | no | — |
| 22 | D5⋊#170 | — | 2 | 1 | no | — |
| 23 | C22×C5×D5 | — | 2 | 1 | no | — |
| 24 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 25 | C5×C22⋊D5 | — | 2 | 1 | no | — |
| 26 | C2×C5×Dic5 | — | 2 | 1 | no | — |
| 27 | #6039 | — | 2 | 1 | no | — |
| 28 | #6039 | — | 2 | 1 | no | — |
| 29 | #6039 | D4 | 1 | 1 | yes | characteristic |
| 30 | #6037 | — | 2 | 1 | no | — |
| 31 | #6037 | — | 2 | 1 | no | — |
| 32 | #6037 | D4 | 1 | 1 | yes | characteristic |
| 33 | D5×Dic5 | — | 2 | 1 | no | — |
| 34 | #6034 | — | 2 | 1 | no | — |
| 35 | #6034 | — | 2 | 1 | no | — |
| 36 | #6034 | D4 | 1 | 1 | yes | characteristic |
| 37 | #6034 | D4 | 1 | 1 | yes | characteristic |
| 38 | #6033 | — | 2 | 1 | no | — |
| 39 | #4131 | — | 10 | 1 | no | — |
| 40 | #4013 | — | 10 | 1 | no | — |
| 41 | #4005 | — | 10 | 1 | no | — |
| 42 | C22×C52 | #41 | 1 | 1 | yes | characteristic |
| 43 | C2×C5×D5 | — | 2 | 1 | no | — |
| 44 | C2×C5×D5 | — | 4 | 1 | no | — |
| 45 | C2×C5×D5 | — | 8 | 1 | no | — |
| 46 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 47 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C5×Dic5 | — | 2 | 1 | no | — |
| 49 | C5×Dic5 | — | 4 | 1 | no | — |
| 50 | C2×C22⋊D5 | — | 10 | 1 | no | — |
| 51 | #711 | — | 10 | 1 | no | — |
| 52 | C2×Dic10 | — | 10 | 1 | no | — |
| 53 | #704 | — | 10 | 1 | no | — |
| 54 | #704 | — | 10 | 1 | no | — |
| 55 | #704 | — | 10 | 1 | no | — |
| 56 | #704 | — | 10 | 1 | no | — |
| 57 | #704 | — | 10 | 1 | no | — |
| 58 | #704 | — | 10 | 1 | no | — |
| 59 | C2×#170 | — | 10 | 1 | no | — |
| 60 | C2×#170 | — | 10 | 1 | no | — |
| 61 | #700 | — | 10 | 1 | no | — |
| 62 | D5⋊C8 | — | 10 | 1 | no | — |
| 63 | #471 | — | 25 | 1 | no | maximal, Sylow |
| 64 | C2×C52 | — | 2 | 1 | no | — |
| 65 | C2×C52 | #121 | 1 | 1 | yes | characteristic |
| 66 | C5×D5 | — | 4 | 1 | no | — |
| 67 | C5×D5 | — | 8 | 1 | no | — |
| 68 | C23×C5 | — | 10 | 1 | no | — |
| 69 | C22×D5 | — | 2 | 1 | no | — |
| 70 | C5×D4 | — | 10 | 1 | no | — |
| 71 | C2×C4×C5 | — | 10 | 1 | no | — |
| 72 | C22⋊D5 | — | 2 | 1 | no | — |
| 73 | C22⋊D5 | — | 10 | 1 | no | — |
| 74 | C22⋊D5 | — | 20 | 1 | no | — |
| 75 | C22⋊D5 | — | 20 | 1 | no | — |
| 76 | C2×Dic5 | — | 2 | 1 | no | — |
| 77 | C2×Dic5 | — | 10 | 1 | no | — |
| 78 | C2×Dic5 | — | 10 | 1 | no | — |
| 79 | C2×Dic5 | — | 10 | 1 | no | — |
| 80 | C2×Dic5 | — | 10 | 1 | no | — |
| 81 | C4×D5 | — | 10 | 1 | no | — |
| 82 | Dic10 | — | 10 | 1 | no | — |
| 83 | Dic10 | — | 20 | 1 | no | — |
| 84 | Dic10 | — | 20 | 1 | no | — |
| 85 | #170 | — | 10 | 1 | no | — |
| 86 | #170 | — | 10 | 1 | no | — |
| 87 | #170 | — | 10 | 1 | no | — |
| 88 | #170 | — | 10 | 1 | no | — |
| 89 | #170 | — | 20 | 1 | no | — |
| 90 | #170 | — | 20 | 1 | no | — |
| 91 | #170 | — | 20 | 1 | no | — |
| 92 | #170 | — | 20 | 1 | no | — |
| 93 | #137 | — | 25 | 1 | no | — |
| 94 | AGL(1,ℤ/8) | — | 25 | 1 | no | — |
| 95 | C2×SD16 | — | 25 | 1 | no | — |
| 96 | #131 | — | 25 | 1 | no | — |
| 97 | #108 | — | 25 | 1 | no | — |
| 98 | #101 | — | 25 | 1 | no | — |
| 99 | #100 | — | 25 | 1 | no | — |
| 100 | C52 | #471 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 101 | C22×C5 | — | 2 | 1 | no | — |
| 102 | C22×C5 | — | 2 | 1 | no | — |
| 103 | C22×C5 | — | 2 | 1 | no | — |
| 104 | C22×C5 | — | 10 | 1 | no | — |
| 105 | C22×C5 | — | 20 | 1 | no | — |
| 106 | C22×C5 | — | 40 | 1 | no | — |
| 107 | C2×D5 | — | 2 | 1 | no | — |
| 108 | C2×D5 | — | 4 | 1 | no | — |
| 109 | C2×D5 | — | 8 | 1 | no | — |
| 110 | C4×C5 | — | 10 | 1 | no | — |
| 111 | C4×C5 | — | 20 | 1 | no | — |
| 112 | Dic5 | — | 2 | 1 | no | — |
| 113 | Dic5 | — | 4 | 1 | no | — |
| 114 | Dic5 | — | 10 | 1 | no | — |
| 115 | Dic5 | — | 10 | 1 | no | — |
| 116 | Dic5 | — | 10 | 1 | no | — |
| 117 | Dic5 | — | 10 | 1 | no | — |
| 118 | Dic5 | — | 10 | 1 | no | — |
| 119 | Dic5 | — | 10 | 1 | no | — |
| 120 | #41 | — | 25 | 1 | no | — |
| 121 | C2×Q8 | — | 25 | 1 | no | — |
| 122 | C2×D4 | — | 25 | 1 | no | — |
| 123 | Q16 | — | 50 | 1 | no | — |
| 124 | SD16 | — | 50 | 1 | no | — |
| 125 | SD16 | — | 50 | 1 | no | — |
| 126 | SD16 | — | 50 | 1 | no | — |
| 127 | SD16 | — | 50 | 1 | no | — |
| 128 | D8 | — | 50 | 1 | no | — |
| 129 | M16 | — | 25 | 1 | no | — |
| 130 | M16 | — | 25 | 1 | no | — |
| 131 | M16 | — | 25 | 1 | no | — |
| 132 | M16 | — | 50 | 1 | no | — |
| 133 | C2×C8 | — | 25 | 1 | no | — |
| 134 | C2×C8 | — | 50 | 1 | no | — |
| 135 | C2×C5 | — | 2 | 1 | no | — |
| 136 | C2×C5 | — | 2 | 1 | no | — |
| 137 | C2×C5 | — | 2 | 1 | no | — |
| 138 | C2×C5 | — | 4 | 1 | no | — |
| 139 | C2×C5 | — | 4 | 1 | no | — |
| 140 | C2×C5 | — | 4 | 1 | no | — |
| 141 | C2×C5 | — | 20 | 1 | no | — |
| 142 | C2×C5 | — | 40 | 1 | no | — |
| 143 | D5 | — | 4 | 1 | no | — |
| 144 | D5 | — | 8 | 1 | no | — |
| 145 | C23 | — | 10 | 1 | no | — |
| 146 | Q8 | — | 25 | 1 | no | — |
| 147 | Q8 | — | 50 | 1 | no | — |
| 148 | Q8 | — | 50 | 1 | no | — |
| 149 | D4 | — | 10 | 1 | no | — |
| 150 | D4 | — | 25 | 1 | no | — |
| 151 | D4 | — | 50 | 1 | no | — |
| 152 | D4 | — | 50 | 1 | no | — |
| 153 | C2×C4 | — | 10 | 1 | no | — |
| 154 | C2×C4 | — | 25 | 1 | no | — |
| 155 | C2×C4 | — | 50 | 1 | no | — |
| 156 | C8 | — | 25 | 1 | no | — |
| 157 | C8 | — | 25 | 1 | no | — |
| 158 | C8 | — | 50 | 1 | no | — |
| 159 | C8 | — | 50 | 1 | no | — |
| 160 | C8 | — | 50 | 1 | no | — |
| 161 | C5 | — | 2 | 1 | no | — |
| 162 | C5 | — | 2 | 1 | no | — |
| 163 | C5 | — | 2 | 1 | no | — |
| 164 | C22 | — | 10 | 1 | no | — |
| 165 | C22 | — | 20 | 1 | no | — |
| 166 | C22 | — | 40 | 1 | no | — |
| 167 | C22 | (D5⋊AGL(1,𝔽5))⋊C2 | 1 | 1 | yes | characteristic |
| 168 | C4 | — | 10 | 1 | no | — |
| 169 | C4 | — | 20 | 1 | no | — |
| 170 | C4 | — | 25 | 1 | no | — |
| 171 | C4 | — | 25 | 1 | no | — |
| 172 | C2 | — | 2 | 1 | no | — |
| 173 | C2 | — | 20 | 1 | no | — |
| 174 | C2 | — | 40 | 1 | no | — |
| 175 | C2 | #11717309 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717397 | C2 | semidirect |
| 2. C2×#87838 | C2 | semidirect |
| 4. #11717373 | C2 | semidirect |
| 5. #11716806 | C2 | semidirect |
| 6. #11716782 | C2 | semidirect |
| 7. #11716781 | C2 | semidirect |
| 100. C52 | #471 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717397 | 2 |
| C2×#87838 | 2 |
| #11717394 | 2 |
| #11717373 | 2 |
| #11716806 | 2 |
| #11716782 | 2 |
| #11716781 | 2 |
| #471 | 25 |