Finite Group Databasebeta

← C22×C53×#543 #27270182

Subgroups — C22×C53×#543 #27270182

Order 13568.

9310 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.

normal subgroups only

This group has 9310 conjugacy classes of subgroups — too many to draw. Only its 6366 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
1C2×C53×#543C2148yesmaximal, direct product factor
2C22×C22⋊Q8×C53C2112yesmaximal, complemented
3C23×C22⋊C4×C53C213yesmaximal
4C26×C53C2211yescharacteristic
5C23×Q8×C53C2214yescomplemented
6C24×C4×C53C2216yes—
7C53×#543C221256yesdirect product factor
8C2×C22⋊Q8×C53C221288yescomplemented
9C22×C4⋊C4×C53C22112yescomplemented
10C22×C22⋊C4×C53C22172yes—
11C22×C22⋊C4×C53C22112yes—
12C25×C53—48no—
13C25×C53Q8116yescomplemented
14C25×C53C2313yes—
15C25×C53C23112yes—
16C22×Q8×C53—416no—
17C22×Q8×C53C23148yescomplemented
18C23×C4×C53—424no—
19C23×C4×C53C23112yes—
20C23×C4×C53C23172yes—
21C22⋊Q8×C53C231768yescomplemented
22C2×C4⋊C4×C53C231144yescomplemented
23C2×C22⋊C4×C53C231144yes—
24C2×C22⋊C4×C53C231192yes—
25C24×C53—224no—
26C24×C53—464no—
27C24×C53—448no—
28C24×C53C2×Q8124yes—
29C24×C53C2×Q8196yescomplemented
30C24×C53C24118yes—
31C24×C53C24116yes—
32C24×C53C2411yescharacteristic
33C2×Q8×C53—496no—
34C2×Q8×C53C24164yescomplemented
35C22×C4×C53—248no—
36C22×C4×C53—4144no—
37C22×C4×C53C24172yes—
38C22×C4×C53C24196yes—
39C4⋊C4×C53C241192yescomplemented
40C22⋊C4×C53C241192yes—
41C23×C53—296no—
42C23×C53—272no—
43C23×C53—4192no—
44C23×C53—432no—
45C23×C53C22×Q8172yes—
46C23×C53C22×Q814yes—
47C23×C53C22×Q8164yescomplemented
48C23×C53H(5,𝔽2)18yes—
49C23×C53C25112yes—
50C23×C53C2513yes—
51Q8×C53—464no—
52C2×C4×C53—2144no—
53C2×C4×C53—496no—
54C2×C4×C53C25148yes—
55C22×#543C5311yesmaximal, characteristic, direct product factor, Sylow
56C22×C53—224no—
57C22×C53—2144no—
58C22×C53—464no—
59C22×C53#543116yesdirect product factor
60C22×C53C23×Q8124yes—
61C22×C53C23×Q816yes—
62C22×C53C2×H(5,𝔽2)112yes—
63C22×C53C2611yescharacteristic
64C4×C53—248no—
65C2×#543C2×C53148yesdirect product factor
66C22×C22⋊Q8C2×C53112yescomplemented
67C23×C22⋊C4C2×C5313yes—
68C2×C53—224no—
69C2×C53C2×#543112yesdirect product factor
70C2×C53C24×Q811yescharacteristic
71C2×C53C22×H(5,𝔽2)12yes—
72C26C22×C5311yescharacteristic
73C23×Q8C22×C5314yescomplemented
74C24×C4C22×C5316yes—
75#543C22×C531256yesdirect product factor
76C2×C22⋊Q8C22×C531288yescomplemented
77C22×C4⋊C4C22×C53112yescomplemented
78C22×C22⋊C4C22×C53112yes—
79C22×C22⋊C4C22×C53172yes—
80C53C22×#54311yescharacteristic, direct product factor, Sylow
81C25—48no—
82C25Q8×C53116yescomplemented
83C25C23×C5313yes—
84C25C23×C53112yes—
85C22×Q8—416no—
86C22×Q8C23×C53148yescomplemented
87C23×C4—424no—
88C23×C4C23×C53112yes—
89C23×C4C23×C53172yes—
90C22⋊Q8C23×C531768yescomplemented
91C2×C4⋊C4C23×C531144yescomplemented
92C2×C22⋊C4C23×C531144yes—
93C2×C22⋊C4C23×C531192yes—
94C24—224no—
95C24—464no—
96C24—448no—
97C24C2×Q8×C53124yes—
98C24C2×Q8×C53196yescomplemented
99C24C24×C53118yes—
100C24C24×C53116yes—
101C24C24×C5311yescharacteristic
102C2×Q8—496no—
103C2×Q8C24×C53164yescomplemented
104C22×C4—248no—
105C22×C4—4144no—
106C22×C4C24×C53172yes—
107C22×C4C24×C53196yes—
108C4⋊C4C24×C531192yescomplemented
109C22⋊C4C24×C531192yes—
110C23—296no—
111C23—272no—
112C23—4192no—
113C23—432no—
114C23C22×Q8×C53172yes—
115C23C22×Q8×C5314yes—
116C23C22×Q8×C53164yescomplemented
117C23H(5,𝔽2)×C5318yes—
118C23C25×C53112yes—
119C23C25×C5313yes—
120Q8—464no—
121C2×C4—2144no—
122C2×C4—496no—
123C2×C4C25×C53148yes—
124C22—2144no—
125C22—224no—
126C22—464no—
127C22C53×#543116yesdirect product factor
128C22C23×Q8×C53124yes—
129C22C23×Q8×C5316yes—
130C22C2×H(5,𝔽2)×C53112yes—
131C22C26×C5311yescharacteristic
132C4—248no—
133C2—224no—
134C2C2×C53×#543112yesdirect product factor
135C2C24×Q8×C5311yescharacteristic
136C2C22×H(5,𝔽2)×C5312yes—
subgroupG/H
center, socleC24×C53C24
commutator, Frattini, cosocleC22C26×C53
prime radicalC26×C53C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#543C2direct (N × H)
2. C22×C22⋊Q8×C53C2semidirect
5. C23×Q8×C53C22semidirect
7. C53×#543C22direct (N × H)
8. C2×C22⋊Q8×C53C22semidirect
9. C22×C4⋊C4×C53C22semidirect
13. C25×C53Q8semidirect
17. C22×Q8×C53C23semidirect
21. C22⋊Q8×C53C23semidirect
22. C2×C4⋊C4×C53C23semidirect
29. C24×C53C2×Q8semidirect
34. C2×Q8×C53C24semidirect
39. C4⋊C4×C53C24semidirect
47. C23×C53C22×Q8semidirect
55. C22×#543C53direct (N × H)
59. C22×C53#543direct (N × H)
65. C2×#543C2×C53direct (N × H)
66. C22×C22⋊Q8C2×C53semidirect
69. C2×C53C2×#543direct (N × H)
73. C23×Q8C22×C53semidirect
75. #543C22×C53direct (N × H)
76. C2×C22⋊Q8C22×C53semidirect
77. C22×C4⋊C4C22×C53semidirect
80. C53C22×#543direct (N × H)
82. C25Q8×C53semidirect
86. C22×Q8C23×C53semidirect
90. C22⋊Q8C23×C53semidirect
91. C2×C4⋊C4C23×C53semidirect
98. C24C2×Q8×C53semidirect
103. C2×Q8C24×C53semidirect
108. C4⋊C4C24×C53semidirect
116. C23C22×Q8×C53semidirect
127. C22C53×#543direct (N × H)
134. C2C2×C53×#543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5432
C22×C22⋊Q8×C532
C23×C22⋊C4×C532
C22×#54353