Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
122 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 15 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 | C5×#338⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10597974 | — | 3 | 1 | no | maximal, Hall |
| 3 | C4⋊(C5⋊S4) | — | 4 | 1 | no | maximal |
| 4 | #67994 | — | 5 | 1 | no | maximal, Hall |
| 5 | #65424 | — | 3 | 1 | no | — |
| 6 | C5×#338 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #64773 | — | 3 | 1 | no | — |
| 8 | C2×C5⋊S4 | — | 4 | 2 | no | — |
| 9 | C4×C5×A4 | — | 4 | 1 | no | — |
| 10 | #338⋊C3 | D5 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C5×M16 | — | 3 | 1 | no | — |
| 12 | #4076 | — | 6 | 1 | no | — |
| 13 | C2×D40 | — | 3 | 1 | no | — |
| 14 | #3953 | — | 6 | 1 | no | — |
| 15 | #3949 | — | 3 | 1 | no | — |
| 16 | #3944 | — | 6 | 1 | no | — |
| 17 | #1341 | — | 15 | 1 | no | Sylow |
| 18 | C2×C5×A4 | — | 4 | 1 | no | — |
| 19 | C5⋊S4 | — | 8 | 2 | no | — |
| 20 | D60 | — | 16 | 1 | no | — |
| 21 | C4⋊S4 | — | 20 | 1 | no | — |
| 22 | C22×C4×C5 | S4 | 1 | 1 | yes | characteristic |
| 23 | C2×C22⋊D5 | — | 12 | 1 | no | — |
| 24 | C2×D20 | — | 6 | 1 | no | — |
| 25 | C5×M16 | — | 12 | 1 | no | — |
| 26 | C2×C5×C8 | — | 3 | 1 | no | — |
| 27 | C2×C5×C8 | — | 3 | 1 | no | — |
| 28 | #685 | — | 12 | 1 | no | — |
| 29 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 30 | D40 | — | 12 | 1 | no | — |
| 31 | #672 | — | 12 | 1 | no | — |
| 32 | #469 | — | 15 | 1 | no | — |
| 33 | #349 | — | 15 | 1 | no | — |
| 34 | #338 | D15 | 1 | 1 | yes | characteristic, complemented |
| 35 | C2×D15 | — | 16 | 2 | no | — |
| 36 | C5×A4 | — | 4 | 1 | no | — |
| 37 | C3×C4×C5 | — | 16 | 1 | no | — |
| 38 | C2×S4 | — | 20 | 2 | no | — |
| 39 | C4×A4 | — | 4 | 1 | no | — |
| 40 | C23×C5 | GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 41 | C22×D5 | — | 12 | 1 | no | — |
| 42 | C2×C4×C5 | — | 3 | 1 | no | — |
| 43 | C2×C4×C5 | — | 3 | 1 | no | — |
| 44 | C22⋊D5 | — | 24 | 2 | no | — |
| 45 | C2×Dic5 | — | 12 | 1 | no | — |
| 46 | D20 | — | 12 | 1 | no | — |
| 47 | D20 | — | 12 | 1 | no | — |
| 48 | C5×C8 | — | 6 | 1 | no | — |
| 49 | C5×C8 | — | 6 | 1 | no | — |
| 50 | C2×D8 | — | 15 | 1 | no | — |
| 51 | C2×M16 | — | 3 | 1 | no | — |
| 52 | #121 | — | 30 | 1 | no | — |
| 53 | M32 | — | 30 | 1 | no | — |
| 54 | #106 | — | 15 | 1 | no | — |
| 55 | #102 | — | 30 | 1 | no | — |
| 56 | C2×C3×C5 | — | 16 | 1 | no | — |
| 57 | D15 | — | 32 | 2 | no | — |
| 58 | C2×A4 | — | 4 | 1 | no | — |
| 59 | S4 | — | 40 | 2 | no | — |
| 60 | D12 | — | 80 | 1 | no | — |
| 61 | C22×C5 | — | 3 | 1 | no | — |
| 62 | C22×C5 | — | 4 | 1 | no | — |
| 63 | C2×D5 | — | 24 | 1 | no | — |
| 64 | C2×D5 | — | 24 | 1 | no | — |
| 65 | C2×D5 | — | 24 | 1 | no | — |
| 66 | C4×C5 | — | 3 | 1 | no | — |
| 67 | C4×C5 | #857 | 1 | 1 | yes | characteristic |
| 68 | Dic5 | — | 24 | 1 | no | — |
| 69 | C2×D4 | — | 30 | 1 | no | — |
| 70 | C2×D4 | — | 60 | 1 | no | — |
| 71 | C22×C4 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 72 | D8 | — | 60 | 1 | no | — |
| 73 | M16 | — | 12 | 1 | no | — |
| 74 | C2×C8 | — | 3 | 1 | no | — |
| 75 | C2×C8 | — | 3 | 1 | no | — |
| 76 | C4⋊C4 | — | 30 | 1 | no | — |
| 77 | C22⋊C4 | — | 60 | 1 | no | — |
| 78 | C16 | — | 60 | 1 | no | — |
| 79 | C3×C5 | — | 16 | 1 | no | Hall |
| 80 | C2×D3 | — | 80 | 2 | no | — |
| 81 | A4 | — | 4 | 1 | no | — |
| 82 | C3×C4 | — | 16 | 1 | no | — |
| 83 | C2×C5 | — | 6 | 1 | no | — |
| 84 | C2×C5 | #4622 | 1 | 1 | yes | characteristic |
| 85 | D5 | — | 48 | 1 | no | — |
| 86 | C23 | — | 60 | 1 | no | — |
| 87 | C23 | C5⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 88 | D4 | — | 60 | 1 | no | — |
| 89 | D4 | — | 60 | 1 | no | — |
| 90 | D4 | — | 120 | 2 | no | — |
| 91 | C2×C4 | — | 3 | 1 | no | — |
| 92 | C2×C4 | — | 3 | 1 | no | — |
| 93 | C2×C4 | — | 60 | 1 | no | — |
| 94 | C8 | — | 6 | 1 | no | — |
| 95 | C8 | — | 6 | 1 | no | — |
| 96 | C2×C3 | — | 16 | 1 | no | — |
| 97 | D3 | — | 160 | 2 | no | — |
| 98 | C5 | #67994 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 99 | C22 | — | 3 | 1 | no | — |
| 100 | C22 | — | 4 | 1 | no | — |
| 101 | C22 | — | 120 | 1 | no | — |
| 102 | C22 | — | 120 | 1 | no | — |
| 103 | C22 | — | 120 | 1 | no | — |
| 104 | C4 | — | 3 | 1 | no | — |
| 105 | C4 | — | 120 | 1 | no | — |
| 106 | C4 | C5⋊#857 | 1 | 1 | yes | characteristic |
| 107 | C3 | — | 16 | 1 | no | Sylow |
| 108 | C2 | — | 6 | 1 | no | — |
| 109 | C2 | — | 240 | 1 | no | — |
| 110 | C2 | C5⋊#4622 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C5⋊#4622 |
| commutator, cosocle, hypocenter | C5×#338⋊C3 | C2 |
| Frattini | C22×C4 | C5⋊S4 |
| Fitting | C5×#338 | D3 |
| socle | C2×C5 | #4622 |
| hypercenter | C4 | C5⋊#857 |
| LCS | G ▷ C5×#338⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#338⋊C3 | C2 | semidirect |
| 6. C5×#338 | D3 | semidirect |
| 10. #338⋊C3 | D5 | semidirect |
| 34. #338 | D15 | semidirect |
| 98. C5 | #67994 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#338⋊C3 | 2 |
| #10597974 | 3 |
| C4⋊(C5⋊S4) | 4 |
| #67994 | 5 |