Finite Group Databasebeta

← C53×#63040 #27270552

Subgroups — C53×#63040 #27270552

Order 13568.

8508 conjugacy classes of subgroups; only the 5718 normal subgroups are shown.

normal subgroups only

This group has 8508 conjugacy classes of subgroups — too many to draw. Only its 5718 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3549C2127yesmaximal, complemented
2C53×#3498C2127yesmaximal, complemented
3C53×#3483C219yesmaximal, complemented
4C2×H(5,𝔽2)×C53C22118yescomplemented
5C53×H(3,𝔽4)C22136yescomplemented
6Q82×C53C2213yescomplemented
7C53×#555C22181yescomplemented
8C53×#550C22154yescomplemented
9D42×C53C221162yescomplemented
10C53×#535C221162yescomplemented
11C53×#532C22181yescomplemented
12C2×#127×C53C22127yescomplemented
13C4×#41×C53C22127yescomplemented
14H(5,𝔽2)×C53—472no—
15C2×#41×C53C23127yes—
16C2×#41×C53C23154yes—
17C22×D4×C53C231162yes—
18C4⋊Q8×C53C23127yes—
19#127×C53C23181yes—
20#127×C53C23154yes—
21#124×C53C231162yes—
22#121×C53C231324yes—
23#120×C53C231216yes—
24C4×Q8×C53C23118yes—
25C4×D4×C53C231162yes—
26#117×C53C23181yes—
27C2×C42×C53C23127yes—
28C24×C53C24136yes—
29#41×C53—2108no—
30#41×C53—4108no—
31C2×Q8×C53C24118yes—
32C2×Q8×C53C2413yes—
33C2×D4×C53—4324no—
34C2×D4×C53C24181yes—
35C2×D4×C53C241162yes—
36C22×C4×C53C24181yes—
37C4⋊C4×C53C24154yes—
38C22⋊C4×C53C241162yes—
39C42×C53C24127yes—
40C42×C53C24127yes—
41C23×C53—4108no—
42C23×C53C25127yes—
43Q8×C53—418no—
44Q8×C53H(5,𝔽2)112yescomplemented
45D4×C53—2162no—
46D4×C53—4162no—
47C2×C4×C53—2162no—
48C2×C4×C53C25127yes—
49C2×C4×C53C2519yes—
50#63040C5311yesmaximal, characteristic, direct product factor, Sylow
51C22×C53—281no—
52C22×C53—1636no—
53C22×C53C2611yescharacteristic
54C4×C53—227no—
55C4×C53C2×H(5,𝔽2)118yes—
56#3549C2×C53127yescomplemented
57#3498C2×C53127yescomplemented
58#3483C2×C5319yescomplemented
59C2×C53—427no—
60C2×C53C22×H(5,𝔽2)13yes—
61C2×H(5,𝔽2)C22×C53118yescomplemented
62H(3,𝔽4)C22×C53136yescomplemented
63Q82C22×C5313yescomplemented
64#555C22×C53181yescomplemented
65#550C22×C53154yescomplemented
66D42C22×C531162yescomplemented
67#535C22×C531162yescomplemented
68#532C22×C53181yescomplemented
69C2×#127C22×C53127yescomplemented
70C4×#41C22×C53127yescomplemented
71C53#6304011yescharacteristic, direct product factor, Sylow
72H(5,𝔽2)—472no—
73C2×#41C23×C53127yes—
74C2×#41C23×C53154yes—
75C22×D4C23×C531162yes—
76C4⋊Q8C23×C53127yes—
77#127C23×C53181yes—
78#127C23×C53154yes—
79#124C23×C531162yes—
80#121C23×C531324yes—
81#120C23×C531216yes—
82C4×Q8C23×C53118yes—
83C4×D4C23×C531162yes—
84#117C23×C53181yes—
85C2×C42C23×C53127yes—
86C24C24×C53136yes—
87#41—2108no—
88#41—4108no—
89C2×Q8C24×C53118yes—
90C2×Q8C24×C5313yes—
91C2×D4—4324no—
92C2×D4C24×C53181yes—
93C2×D4C24×C531162yes—
94C22×C4C24×C53181yes—
95C4⋊C4C24×C53154yes—
96C22⋊C4C24×C531162yes—
97C42C24×C53127yes—
98C42C24×C53127yes—
99C23—4108no—
100C23C25×C53127yes—
101Q8—418no—
102Q8H(5,𝔽2)×C53112yescomplemented
103D4—2162no—
104D4—4162no—
105C2×C4—2162no—
106C2×C4C25×C53127yes—
107C2×C4C25×C5319yes—
108C22—281no—
109C22—1636no—
110C22C26×C5311yescharacteristic
111C4—227no—
112C4C2×H(5,𝔽2)×C53118yes—
113C2—427no—
114C2C22×H(5,𝔽2)×C5313yes—
subgroupG/H
center, socleC22×C53C26
commutator, Frattini, cosocleC22C26×C53
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C53×#3549C2semidirect
2. C53×#3498C2semidirect
3. C53×#3483C2semidirect
4. C2×H(5,𝔽2)×C53C22semidirect
5. C53×H(3,𝔽4)C22semidirect
6. Q82×C53C22semidirect
7. C53×#555C22semidirect
8. C53×#550C22semidirect
9. D42×C53C22semidirect
10. C53×#535C22semidirect
11. C53×#532C22semidirect
12. C2×#127×C53C22semidirect
13. C4×#41×C53C22semidirect
44. Q8×C53H(5,𝔽2)semidirect
50. #63040C53direct (N × H)
56. #3549C2×C53semidirect
57. #3498C2×C53semidirect
58. #3483C2×C53semidirect
61. C2×H(5,𝔽2)C22×C53semidirect
62. H(3,𝔽4)C22×C53semidirect
63. Q82C22×C53semidirect
64. #555C22×C53semidirect
65. #550C22×C53semidirect
66. D42C22×C53semidirect
67. #535C22×C53semidirect
68. #532C22×C53semidirect
69. C2×#127C22×C53semidirect
70. C4×#41C22×C53semidirect
71. C53#63040direct (N × H)
102. Q8H(5,𝔽2)×C53semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C53×#35492
C53×#34982
C53×#34832
#6304053