Finite Group Databasebeta

← C53×#3549 #20272486

Subgroups — C53×#3549 #20272486

Order 6784.

1258 conjugacy classes of subgroups; only the 768 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 768 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×H(3,𝔽4)C214yesmaximal, complemented
2C53×#555C216yesmaximal, complemented
3D42×C53C216yesmaximal, complemented
4C53×#535C2112yesmaximal, complemented
5C53×#532C213yesmaximal, complemented
6C2×#41×C53C2216yescomplemented
7C22×D4×C53C22112yescomplemented
8C4⋊Q8×C53C2213yescomplemented
9#127×C53C2213yes—
10#127×C53C2218yescomplemented
11#124×C53C2216yescomplemented
12#124×C53C22124yescomplemented
13#121×C53C22112yescomplemented
14#121×C53C22124yescomplemented
15#120×C53C2218yescomplemented
16#120×C53C22124yescomplemented
17C4×D4×C53C22112yescomplemented
18#117×C53C22112yescomplemented
19C2×C42×C53C2211yescharacteristic, complemented
20C24×C53C2318yes—
21#41×C53—412no—
22C2×Q8×C53C2316yes—
23C2×D4×C53—424no—
24C2×D4×C53C23112yes—
25C2×D4×C53C2316yes—
26C2×D4×C53C23124yes—
27C2×D4×C53C23112yes—
28C22×C4×C53C2313yes—
29C22×C4×C53C23112yes—
30C4⋊C4×C53C23112yes—
31C22⋊C4×C53C23124yes—
32C22⋊C4×C53C23124yes—
33C42×C53C2313yes—
34C42×C53C2311yescharacteristic
35C42×C53C2318yes—
36C23×C53—424no—
37C23×C53C2411yescharacteristic
38C23×C53C24112yes—
39Q8×C53—46no—
40D4×C53—224no—
41D4×C53—424no—
42D4×C53—412no—
43D4×C53—46no—
44C2×C4×C53—224no—
45C2×C4×C53—26no—
46C2×C4×C53C24112yes—
47C2×C4×C53C2413yes—
48C2×C4×C53C2413yes—
49C22×C53—23no—
50C22×C53—212no—
51C22×C53—224no—
52C22×C53—816no—
53C22×C53C2511yescharacteristic
54C4×C53—212no—
55C4×C53—23no—
56C4×C53H(5,𝔽2)16yes—
57#3549C5311yesmaximal, characteristic, direct product factor, Sylow
58C2×C53—41no—
59C2×C53—412no—
60C2×C53C2×H(5,𝔽2)13yes—
61H(3,𝔽4)C2×C5314yescomplemented
62#555C2×C5316yescomplemented
63D42C2×C5316yescomplemented
64#535C2×C53112yescomplemented
65#532C2×C5313yescomplemented
66C53#354911yescharacteristic, direct product factor, Sylow
67C2×#41C22×C5316yescomplemented
68C22×D4C22×C53112yescomplemented
69C4⋊Q8C22×C5313yescomplemented
70#127C22×C5313yes—
71#127C22×C5318yescomplemented
72#124C22×C5316yescomplemented
73#124C22×C53124yescomplemented
74#121C22×C53112yescomplemented
75#121C22×C53124yescomplemented
76#120C22×C5318yescomplemented
77#120C22×C53124yescomplemented
78C4×D4C22×C53112yescomplemented
79#117C22×C53112yescomplemented
80C2×C42C22×C5311yescharacteristic, complemented
81C24C23×C5318yes—
82#41—412no—
83C2×Q8C23×C5316yes—
84C2×D4—424no—
85C2×D4C23×C53112yes—
86C2×D4C23×C5316yes—
87C2×D4C23×C53124yes—
88C2×D4C23×C53112yes—
89C22×C4C23×C5313yes—
90C22×C4C23×C53112yes—
91C4⋊C4C23×C53112yes—
92C22⋊C4C23×C53124yes—
93C22⋊C4C23×C53124yes—
94C42C23×C5313yes—
95C42C23×C5311yescharacteristic
96C42C23×C5318yes—
97C23—424no—
98C23C24×C53112yes—
99C23C24×C5311yescharacteristic
100Q8—46no—
101D4—224no—
102D4—424no—
103D4—412no—
104D4—46no—
105C2×C4—224no—
106C2×C4—26no—
107C2×C4C24×C5313yes—
108C2×C4C24×C53112yes—
109C2×C4C24×C5313yes—
110C22—23no—
111C22—212no—
112C22—224no—
113C22—816no—
114C22C25×C5311yescharacteristic
115C4—212no—
116C4—23no—
117C4H(5,𝔽2)×C5316yes—
118C2—412no—
119C2—41no—
120C2C2×H(5,𝔽2)×C5313yes—
subgroupG/H
center, socleC22×C53C25
commutator, Frattini, cosocleC22C25×C53
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×H(3,𝔽4)C2semidirect
2. C53×#555C2semidirect
3. D42×C53C2semidirect
4. C53×#535C2semidirect
5. C53×#532C2semidirect
6. C2×#41×C53C22semidirect
7. C22×D4×C53C22semidirect
8. C4⋊Q8×C53C22semidirect
10. #127×C53C22semidirect
11. #124×C53C22semidirect
12. #124×C53C22semidirect
13. #121×C53C22semidirect
14. #121×C53C22semidirect
15. #120×C53C22semidirect
16. #120×C53C22semidirect
17. C4×D4×C53C22semidirect
18. #117×C53C22semidirect
19. C2×C42×C53C22semidirect
57. #3549C53direct (N × H)
61. H(3,𝔽4)C2×C53semidirect
62. #555C2×C53semidirect
63. D42C2×C53semidirect
64. #535C2×C53semidirect
65. #532C2×C53semidirect
66. C53#3549direct (N × H)
67. C2×#41C22×C53semidirect
68. C22×D4C22×C53semidirect
69. C4⋊Q8C22×C53semidirect
71. #127C22×C53semidirect
72. #124C22×C53semidirect
73. #124C22×C53semidirect
74. #121C22×C53semidirect
75. #121C22×C53semidirect
76. #120C22×C53semidirect
77. #120C22×C53semidirect
78. C4×D4C22×C53semidirect
79. #117C22×C53semidirect
80. C2×C42C22×C53semidirect

Maximal subgroups

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

subgroupindex
C53×H(3,𝔽4)2
C53×#5552
D42×C532
C53×#5352
C53×#5322
#354953