Finite Group Databasebeta

← C22×C29×#543 #20357570

Subgroups — C22×C29×#543 #20357570

Order 7424.

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×C29×#543C2148yesmaximal, direct product factor
2C22×C29×C22⋊Q8C2112yesmaximal, complemented
3C23×C22⋊C4×C29C213yesmaximal
4C26×C29C2211yescharacteristic
5C23×Q8×C29C2214yescomplemented
6C24×C4×C29C2216yes—
7C29×#543C221256yesdirect product factor
8C2×C29×C22⋊Q8C221288yescomplemented
9C22×C4⋊C4×C29C22112yescomplemented
10C22×C22⋊C4×C29C22172yes—
11C22×C22⋊C4×C29C22112yes—
12C25×C29—48no—
13C25×C29Q8116yescomplemented
14C25×C29C2313yes—
15C25×C29C23112yes—
16C22×Q8×C29—416no—
17C22×Q8×C29C23148yescomplemented
18C23×C4×C29—424no—
19C23×C4×C29C23112yes—
20C23×C4×C29C23172yes—
21C29×C22⋊Q8C231768yescomplemented
22C2×C4⋊C4×C29C231144yescomplemented
23C2×C22⋊C4×C29C231144yes—
24C2×C22⋊C4×C29C231192yes—
25C24×C29—224no—
26C24×C29—464no—
27C24×C29—448no—
28C24×C29C2×Q8124yes—
29C24×C29C2×Q8196yescomplemented
30C24×C29C24118yes—
31C24×C29C24116yes—
32C24×C29C2411yescharacteristic
33C2×Q8×C29—496no—
34C2×Q8×C29C24164yescomplemented
35C22×C4×C29—248no—
36C22×C4×C29—4144no—
37C22×C4×C29C24172yes—
38C22×C4×C29C24196yes—
39C4⋊C4×C29C241192yescomplemented
40C22⋊C4×C29C241192yes—
41C22×#543C2911yesmaximal, characteristic, direct product factor, Sylow
42C23×C29—296no—
43C23×C29—272no—
44C23×C29—4192no—
45C23×C29—432no—
46C23×C29C22×Q8172yes—
47C23×C29C22×Q814yes—
48C23×C29C22×Q8164yescomplemented
49C23×C29H(5,𝔽2)18yes—
50C23×C29C25112yes—
51C23×C29C2513yes—
52Q8×C29—464no—
53C2×C4×C29—2144no—
54C2×C4×C29—496no—
55C2×C4×C29C25148yes—
56C2×#543C2×C29148yesdirect product factor
57C22×C22⋊Q8C2×C29112yescomplemented
58C23×C22⋊C4C2×C2913yes—
59C22×C29—224no—
60C22×C29—2144no—
61C22×C29—464no—
62C22×C29#543116yesdirect product factor
63C22×C29C23×Q8124yes—
64C22×C29C23×Q816yes—
65C22×C29C2×H(5,𝔽2)112yes—
66C22×C29C2611yescharacteristic
67C4×C29—248no—
68C26C22×C2911yescharacteristic
69C23×Q8C22×C2914yescomplemented
70C24×C4C22×C2916yes—
71#543C22×C291256yesdirect product factor
72C2×C22⋊Q8C22×C291288yescomplemented
73C22×C4⋊C4C22×C29112yescomplemented
74C22×C22⋊C4C22×C29112yes—
75C22×C22⋊C4C22×C29172yes—
76C2×C29—224no—
77C2×C29C2×#543112yesdirect product factor
78C2×C29C24×Q811yescharacteristic
79C2×C29C22×H(5,𝔽2)12yes—
80C25—48no—
81C25Q8×C29116yescomplemented
82C25C23×C2913yes—
83C25C23×C29112yes—
84C22×Q8—416no—
85C22×Q8C23×C29148yescomplemented
86C23×C4—424no—
87C23×C4C23×C29112yes—
88C23×C4C23×C29172yes—
89C22⋊Q8C23×C291768yescomplemented
90C2×C4⋊C4C23×C291144yescomplemented
91C2×C22⋊C4C23×C291144yes—
92C2×C22⋊C4C23×C291192yes—
93C29C22×#54311yescharacteristic, direct product factor, Sylow
94C24—224no—
95C24—464no—
96C24—448no—
97C24C2×Q8×C29124yes—
98C24C2×Q8×C29196yescomplemented
99C24C24×C29118yes—
100C24C24×C29116yes—
101C24C24×C2911yescharacteristic
102C2×Q8—496no—
103C2×Q8C24×C29164yescomplemented
104C22×C4—248no—
105C22×C4—4144no—
106C22×C4C24×C29172yes—
107C22×C4C24×C29196yes—
108C4⋊C4C24×C291192yescomplemented
109C22⋊C4C24×C291192yes—
110C23—296no—
111C23—272no—
112C23—4192no—
113C23—432no—
114C23C22×Q8×C29172yes—
115C23C22×Q8×C2914yes—
116C23C22×Q8×C29164yescomplemented
117C23C29×H(5,𝔽2)18yes—
118C23C25×C29112yes—
119C23C25×C2913yes—
120Q8—464no—
121C2×C4—2144no—
122C2×C4—496no—
123C2×C4C25×C29148yes—
124C22—2144no—
125C22—224no—
126C22—464no—
127C22C29×#543116yesdirect product factor
128C22C23×Q8×C29124yes—
129C22C23×Q8×C2916yes—
130C22C2×C29×H(5,𝔽2)112yes—
131C22C26×C2911yescharacteristic
132C4—248no—
133C2—224no—
134C2C2×C29×#543112yesdirect product factor
135C2C24×Q8×C2911yescharacteristic
136C2C22×C29×H(5,𝔽2)12yes—
subgroupG/H
center, socleC24×C29C24
commutator, Frattini, cosocleC22C26×C29
prime radicalC26×C29C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C29 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C29×#543C2direct (N × H)
2. C22×C29×C22⋊Q8C2semidirect
5. C23×Q8×C29C22semidirect
7. C29×#543C22direct (N × H)
8. C2×C29×C22⋊Q8C22semidirect
9. C22×C4⋊C4×C29C22semidirect
13. C25×C29Q8semidirect
17. C22×Q8×C29C23semidirect
21. C29×C22⋊Q8C23semidirect
22. C2×C4⋊C4×C29C23semidirect
29. C24×C29C2×Q8semidirect
34. C2×Q8×C29C24semidirect
39. C4⋊C4×C29C24semidirect
41. C22×#543C29direct (N × H)
48. C23×C29C22×Q8semidirect
56. C2×#543C2×C29direct (N × H)
57. C22×C22⋊Q8C2×C29semidirect
62. C22×C29#543direct (N × H)
69. C23×Q8C22×C29semidirect
71. #543C22×C29direct (N × H)
72. C2×C22⋊Q8C22×C29semidirect
73. C22×C4⋊C4C22×C29semidirect
77. C2×C29C2×#543direct (N × H)
81. C25Q8×C29semidirect
85. C22×Q8C23×C29semidirect
89. C22⋊Q8C23×C29semidirect
90. C2×C4⋊C4C23×C29semidirect
93. C29C22×#543direct (N × H)
98. C24C2×Q8×C29semidirect
103. C2×Q8C24×C29semidirect
108. C4⋊C4C24×C29semidirect
116. C23C22×Q8×C29semidirect
127. C22C29×#543direct (N × H)
134. C2C2×C29×#543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C29×#5432
C22×C29×C22⋊Q82
C23×C22⋊C4×C292
C22×#54329