Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
220 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 33 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 | #11717386 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11717383 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11717379 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11716914 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C8×#1044 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #87877 | C22 | 1 | 2 | yes | — |
| 7 | #87872 | C22 | 1 | 2 | yes | — |
| 8 | C4×#1044 | C22 | 1 | 1 | yes | characteristic |
| 9 | #87862 | C22 | 1 | 1 | yes | characteristic |
| 10 | #87839 | — | 2 | 2 | no | — |
| 11 | #87837 | — | 2 | 2 | no | — |
| 12 | #87835 | — | 2 | 4 | no | — |
| 13 | C8×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 14 | C5×#677 | — | 2 | 2 | no | — |
| 15 | #87772 | — | 2 | 2 | no | — |
| 16 | C2×D52 | — | 2 | 2 | no | — |
| 17 | C2×#1044 | D4 | 1 | 2 | yes | — |
| 18 | C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 19 | C5×D20 | — | 2 | 2 | no | — |
| 20 | C5×Dic10 | — | 2 | 2 | no | — |
| 21 | #6039 | — | 2 | 2 | no | — |
| 22 | #6037 | — | 2 | 2 | no | — |
| 23 | #6036 | — | 2 | 2 | no | — |
| 24 | #6034 | D4 | 1 | 2 | yes | — |
| 25 | C52×C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | #6029 | D4 | 1 | 1 | yes | characteristic |
| 27 | D5×SD16 | — | 10 | 2 | no | — |
| 28 | C8×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 29 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 30 | C2×C5×D5 | — | 4 | 2 | no | — |
| 31 | D52 | — | 4 | 4 | no | — |
| 32 | #1044 | — | 2 | 2 | no | — |
| 33 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 34 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 35 | C5×Dic5 | — | 4 | 2 | no | — |
| 36 | Q8×D5 | — | 10 | 2 | no | — |
| 37 | D4×D5 | — | 10 | 2 | no | — |
| 38 | C4×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 39 | D5⋊C8 | — | 10 | 1 | no | — |
| 40 | C5×SD16 | — | 10 | 2 | no | — |
| 41 | #688 | — | 10 | 2 | no | — |
| 42 | #687 | — | 10 | 2 | no | — |
| 43 | #677 | — | 2 | 2 | no | — |
| 44 | C8×D5 | — | 10 | 1 | no | — |
| 45 | C8×D5 | — | 10 | 2 | no | — |
| 46 | #495 | — | 25 | 1 | no | maximal, Sylow |
| 47 | C2×C52 | #127 | 1 | 1 | yes | characteristic |
| 48 | C5⋊D5 | #135 | 1 | 2 | yes | — |
| 49 | C5×D5 | — | 8 | 2 | no | — |
| 50 | C22×D5 | — | 20 | 2 | no | — |
| 51 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 52 | C5×Q8 | — | 10 | 2 | no | — |
| 53 | C5×D4 | — | 10 | 2 | no | — |
| 54 | C22⋊D5 | — | 20 | 2 | no | — |
| 55 | D20 | — | 2 | 2 | no | — |
| 56 | C4×D5 | — | 10 | 1 | no | — |
| 57 | C4×D5 | — | 10 | 2 | no | — |
| 58 | C4×D5 | — | 20 | 2 | no | — |
| 59 | Dic10 | — | 2 | 2 | no | — |
| 60 | Dic10 | — | 20 | 2 | no | — |
| 61 | #170 | — | 10 | 2 | no | — |
| 62 | C5×C8 | — | 2 | 1 | no | — |
| 63 | C5×C8 | — | 2 | 2 | no | — |
| 64 | #168 | — | 10 | 1 | no | — |
| 65 | #168 | — | 10 | 2 | no | — |
| 66 | C2×Q16 | — | 25 | 1 | no | — |
| 67 | C2×SD16 | — | 25 | 2 | no | — |
| 68 | C2×D8 | — | 25 | 1 | no | — |
| 69 | #124 | — | 25 | 2 | no | — |
| 70 | C4×C8 | — | 25 | 1 | no | — |
| 71 | C52 | #495 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 72 | C22×C5 | — | 20 | 2 | no | — |
| 73 | C2×D5 | — | 4 | 2 | no | — |
| 74 | C2×D5 | — | 10 | 2 | no | — |
| 75 | C2×D5 | — | 10 | 1 | no | — |
| 76 | C2×D5 | — | 40 | 4 | no | — |
| 77 | AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 78 | C4×C5 | — | 2 | 2 | no | — |
| 79 | C4×C5 | — | 2 | 1 | no | — |
| 80 | C4×C5 | — | 20 | 2 | no | — |
| 81 | Dic5 | — | 4 | 2 | no | — |
| 82 | Dic5 | — | 10 | 1 | no | — |
| 83 | Dic5 | — | 10 | 2 | no | — |
| 84 | C2×Q8 | — | 25 | 2 | no | — |
| 85 | C2×D4 | — | 25 | 2 | no | — |
| 86 | Q16 | — | 50 | 2 | no | — |
| 87 | SD16 | — | 10 | 2 | no | — |
| 88 | SD16 | — | 50 | 2 | no | — |
| 89 | D8 | — | 50 | 2 | no | — |
| 90 | C2×C8 | — | 25 | 1 | no | — |
| 91 | C2×C8 | — | 25 | 1 | no | — |
| 92 | C22⋊C4 | — | 50 | 4 | no | — |
| 93 | C42 | — | 25 | 1 | no | — |
| 94 | C2×C5 | — | 2 | 2 | no | — |
| 95 | C2×C5 | — | 2 | 1 | no | — |
| 96 | C2×C5 | — | 40 | 2 | no | — |
| 97 | D5 | — | 8 | 2 | no | — |
| 98 | D5 | — | 10 | 2 | no | — |
| 99 | D5 | — | 10 | 4 | no | — |
| 100 | C23 | — | 50 | 2 | no | — |
| 101 | Q8 | — | 10 | 2 | no | — |
| 102 | Q8 | — | 50 | 2 | no | — |
| 103 | D4 | — | 10 | 2 | no | — |
| 104 | D4 | — | 50 | 2 | no | — |
| 105 | C2×C4 | — | 25 | 1 | no | — |
| 106 | C2×C4 | — | 25 | 2 | no | — |
| 107 | C2×C4 | — | 50 | 2 | no | — |
| 108 | C8 | — | 25 | 2 | no | — |
| 109 | C8 | — | 25 | 1 | no | — |
| 110 | C8 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 111 | C5 | — | 2 | 2 | no | — |
| 112 | C5 | — | 2 | 1 | no | — |
| 113 | C22 | — | 20 | 2 | no | — |
| 114 | C22 | — | 25 | 1 | no | — |
| 115 | C22 | — | 100 | 4 | no | — |
| 116 | C4 | — | 20 | 2 | no | — |
| 117 | C4 | — | 25 | 1 | no | — |
| 118 | C4 | — | 50 | 2 | no | — |
| 119 | C4 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 120 | C2 | — | 25 | 2 | no | — |
| 121 | C2 | — | 40 | 2 | no | — |
| 122 | C2 | #11717389 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717386 | C2 | semidirect |
| 3. #11717379 | C2 | semidirect |
| 4. #11716914 | C2 | semidirect |
| 5. C8×#1044 | C2 | semidirect |
| 71. C52 | #495 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717386 | 2 |
| #11717383 | 2 |
| #11717379 | 2 |
| #11716914 | 2 |
| C8×#1044 | 2 |
| #495 | 25 |