Finite Group Databasebeta

← C2×C29×#560 #16817239

Subgroups — C2×C29×#560 #16817239

Order 3712.

1344 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 776 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
1C29×#560C2116yesmaximal, direct product factor
2C2×C29×#127C211yesmaximal, characteristic, complemented
3C2×C29×#126C212yesmaximal, complemented
4C2×C29×#123C213yesmaximal, complemented
5C2×C29×#121C216yesmaximal, complemented
6C2×C29×#120C213yesmaximal, complemented
7C25×C29C2211yescharacteristic
8C22×D4×C29C2216yes—
9C22×D4×C29C2213yes—
10C22×D4×C29C2213yescomplemented
11C23×C4×C29C2213yescomplemented
12C29×#127C2218yescomplemented
13C29×#126C22116yescomplemented
14C29×#123C22124yescomplemented
15C29×#121C22148yescomplemented
16C29×#120C22124yescomplemented
17C2×C4⋊C4×C29C2216yescomplemented
18C2×C22⋊C4×C29C2216yes—
19C2×C22⋊C4×C29C2213yes—
20C2×C22⋊C4×C29C2213yescomplemented
21C2×C42×C29C2211yescharacteristic, complemented
22C24×C29—46no—
23C24×C29C2312yes—
24C24×C29C2311yescharacteristic
25C24×C29C2313yes—
26C24×C29C2314yes—
27C2×D4×C29—412no—
28C2×D4×C29—46no—
29C2×D4×C29—46no—
30C2×D4×C29C23124yes—
31C2×D4×C29C23112yes—
32C2×D4×C29C23112yescomplemented
33C22×C4×C29—46no—
34C22×C4×C29C2313yes—
35C22×C4×C29C2316yes—
36C22×C4×C29C23112yescomplemented
37C4⋊C4×C29C23124yescomplemented
38C22⋊C4×C29C23124yes—
39C22⋊C4×C29C23112yes—
40C22⋊C4×C29C23112yescomplemented
41C42×C29C2314yescomplemented
42C23×C29—26no—
43C23×C29—212no—
44C23×C29—212no—
45C23×C29—26no—
46C23×C29—412no—
47C23×C29—416no—
48C23×C29C2414yes—
49C23×C29C2412yes—
50C23×C29C2416yes—
51C23×C29C2411yescharacteristic
52D4×C29—424no—
53D4×C29—412no—
54D4×C29—412no—
55C2×C4×C29—212no—
56C2×C4×C29—26no—
57C2×C4×C29—412no—
58C2×C4×C29C2416yes—
59C2×C4×C29C24112yes—
60C2×#560C2911yesmaximal, characteristic, direct product factor, Sylow
61C22×C29—26no—
62C22×C29—212no—
63C22×C29—212no—
64C22×C29—26no—
65C22×C29—416no—
66C22×C29—412no—
67C22×C29—48no—
68C22×C29—44no—
69C22×C29H(5,𝔽2)16yes—
70C22×C29C2511yescharacteristic
71C4×C29—212no—
72C4×C29—26no—
73#560C2×C29116yesdirect product factor
74C2×#127C2×C2911yescharacteristic, complemented
75C2×#126C2×C2912yescomplemented
76C2×#123C2×C2913yescomplemented
77C2×#121C2×C2916yescomplemented
78C2×#120C2×C2913yescomplemented
79C2×C29—46no—
80C2×C29—44no—
81C2×C29—42no—
82C2×C29#56014yesdirect product factor
83C2×C29C2×H(5,𝔽2)13yes—
84C25C22×C2911yescharacteristic
85C22×D4C22×C2916yes—
86C22×D4C22×C2913yes—
87C22×D4C22×C2913yescomplemented
88C23×C4C22×C2913yescomplemented
89#127C22×C2918yescomplemented
90#126C22×C29116yescomplemented
91#123C22×C29124yescomplemented
92#121C22×C29148yescomplemented
93#120C22×C29124yescomplemented
94C2×C4⋊C4C22×C2916yescomplemented
95C2×C22⋊C4C22×C2916yes—
96C2×C22⋊C4C22×C2913yes—
97C2×C22⋊C4C22×C2913yescomplemented
98C2×C42C22×C2911yescharacteristic, complemented
99C29C2×#56011yescharacteristic, direct product factor, Sylow
100C24—46no—
101C24C23×C2912yes—
102C24C23×C2911yescharacteristic
103C24C23×C2913yes—
104C24C23×C2914yes—
105C2×D4—412no—
106C2×D4—46no—
107C2×D4—46no—
108C2×D4C23×C29124yes—
109C2×D4C23×C29112yes—
110C2×D4C23×C29112yescomplemented
111C22×C4—46no—
112C22×C4C23×C2913yes—
113C22×C4C23×C2916yes—
114C22×C4C23×C29112yescomplemented
115C4⋊C4C23×C29124yescomplemented
116C22⋊C4C23×C29124yes—
117C22⋊C4C23×C29112yes—
118C22⋊C4C23×C29112yescomplemented
119C42C23×C2914yescomplemented
120C23—26no—
121C23—212no—
122C23—212no—
123C23—26no—
124C23—412no—
125C23—416no—
126C23C24×C2914yes—
127C23C24×C2912yes—
128C23C24×C2916yes—
129C23C24×C2911yescharacteristic
130D4—424no—
131D4—412no—
132D4—412no—
133C2×C4—212no—
134C2×C4—26no—
135C2×C4—412no—
136C2×C4C24×C2916yes—
137C2×C4C24×C29112yes—
138C22—26no—
139C22—212no—
140C22—212no—
141C22—26no—
142C22—416no—
143C22—48no—
144C22—44no—
145C22—412no—
146C22C29×H(5,𝔽2)16yes—
147C22C25×C2911yescharacteristic
148C4—212no—
149C4—26no—
150C2—44no—
151C2—42no—
152C2—46no—
153C2C29×#56014yesdirect product factor
154C2C2×C29×H(5,𝔽2)13yes—
subgroupG/H
center, socleC23×C29C24
commutator, Frattini, cosocleC22C25×C29
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C29 ◁ G

Complements

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

NH ≅ G/Ntype
1. C29×#560C2direct (N × H)
2. C2×C29×#127C2semidirect
3. C2×C29×#126C2semidirect
4. C2×C29×#123C2semidirect
5. C2×C29×#121C2semidirect
6. C2×C29×#120C2semidirect
10. C22×D4×C29C22semidirect
11. C23×C4×C29C22semidirect
12. C29×#127C22semidirect
13. C29×#126C22semidirect
14. C29×#123C22semidirect
15. C29×#121C22semidirect
16. C29×#120C22semidirect
17. C2×C4⋊C4×C29C22semidirect
20. C2×C22⋊C4×C29C22semidirect
21. C2×C42×C29C22semidirect
32. C2×D4×C29C23semidirect
36. C22×C4×C29C23semidirect
37. C4⋊C4×C29C23semidirect
40. C22⋊C4×C29C23semidirect
41. C42×C29C23semidirect
60. C2×#560C29direct (N × H)
73. #560C2×C29direct (N × H)
74. C2×#127C2×C29semidirect
75. C2×#126C2×C29semidirect
76. C2×#123C2×C29semidirect
77. C2×#121C2×C29semidirect
78. C2×#120C2×C29semidirect
82. C2×C29#560direct (N × H)
87. C22×D4C22×C29semidirect
88. C23×C4C22×C29semidirect
89. #127C22×C29semidirect
90. #126C22×C29semidirect
91. #123C22×C29semidirect
92. #121C22×C29semidirect
93. #120C22×C29semidirect
94. C2×C4⋊C4C22×C29semidirect
97. C2×C22⋊C4C22×C29semidirect
98. C2×C42C22×C29semidirect
99. C29C2×#560direct (N × H)
110. C2×D4C23×C29semidirect
114. C22×C4C23×C29semidirect
115. C4⋊C4C23×C29semidirect
118. C22⋊C4C23×C29semidirect
119. C42C23×C29semidirect
153. C2C29×#560direct (N × H)

Maximal subgroups

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

subgroupindex
C29×#5602
C2×C29×#1272
C2×C29×#1262
C2×C29×#1232
C2×C29×#1212
C2×C29×#1202
C2×#56029