Finite Group Databasebeta

← #16831102

Subgroups — #16831102

Order 3712.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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×#11745273C212yesmaximal, complemented
2#14271808C212yesmaximal, complemented
3#14271547C211yesmaximal, characteristic, complemented
4#14271410C212yesmaximal, complemented
5#14271378C212yesmaximal, complemented
6#14271326C212yesmaximal, complemented
7#14271298C212yesmaximal, complemented
8#14271242C211yesmaximal, characteristic, complemented
9C4×D8×C29C211yesmaximal, characteristic, complemented
10C2×#90807C2212yes—
11C2×D4×D29C2212yescomplemented
12C2×D8×C29C2211yescharacteristic
13C4×D4×C29C2212yescomplemented
14C2×#90783C2212yescomplemented
15C2×#90782C2212yes—
16#11745292—22no—
17C2×#90781—22no—
18#11745273—28no—
19C2×Dic116C2211yescharacteristic, complemented
20C2×D232C2211yescharacteristic
21C2×#90773C2212yescomplemented
22C2×#90772C2212yescomplemented
23#11745259C2212yescomplemented
24#11745257C2212yes—
25#11745248—22no—
26#11745244—22no—
27#11745237C2212yescomplemented
28C4×D116C2211yescharacteristic
29C29×#107C2211yescharacteristic, complemented
30C29×#102C2212yescomplemented
31C4×C8×C29C2211yescharacteristic, complemented
32#11745168C2212yescomplemented
33#11745158C2212yescomplemented
34#11745157C2212yescomplemented
35C4⋊#6597C2211yescharacteristic, complemented
36C23×D29—22no—
37C2×D4×C29C2312yes—
38C22×C4×C29—22no—
39C2×C22⋊D29—22no—
40C2×C22⋊D29—22no—
41C22×Dic29—22no—
42#90807—24no—
43#90807—42no—
44D4×D29—24no—
45D4×D29—42no—
46C2×D116—22no—
47C2×D116D411yescharacteristic
48C2×D116C2312yes—
49C2×C4×D29C2312yes—
50C2×Dic58C2312yes—
51D8×C29D414yescomplemented
52C2×C8×C29C2311yescharacteristic
53C2×C8×C29C2311yescharacteristic
54C4⋊C4×C29C2312yes—
55C22⋊C4×C29—22no—
56C42×C29C2311yescharacteristic
57#90783—24no—
58#90782—24no—
59#90781—22no—
60#90781—22no—
61#90781—22no—
62#90781—22no—
63#90781D412yes—
64C4⋊Dic29—22no—
65C4⋊Dic29D411yescharacteristic
66C2×#6597C2312yes—
67Dic116—41no—
68D232—41no—
69#90773—24no—
70#90772—24no—
71C23×C29—22no—
72C22×D29—22no—
73C22×D29—44no—
74C22×D29—44no—
75C22×D29—82no—
76C22×D29C2×D412yes—
77D4×C29—22no—
78D4×C29C2×D414yes—
79C2×C4×C29—22no—
80C2×C4×C29—82no—
81C2×C4×C29C2×D411yescharacteristic
82C2×C4×C29C2×D411yescharacteristic
83C2×C4×C29C2411yescharacteristic
84C22⋊D29—44no—
85C22⋊D29—44no—
86C2×Dic29—22no—
87C2×Dic29—44no—
88C2×Dic29C2×D412yes—
89D116—22no—
90D116—22no—
91D116—41no—
92D116—82no—
93C4×D29—22no—
94C4×D29—22no—
95Dic58—22no—
96Dic58—22no—
97C8×C29—21no—
98C8×C29C2×D412yes—
99#6597—22no—
100#3273—291nomaximal, Sylow
101C22×C29—24no—
102C22×C29—42no—
103C22×C29—42no—
104C22×C29C22×D411yescharacteristic
105C2×D29—22no—
106C2×D29—22no—
107C2×D29—22no—
108C2×D29—42no—
109C2×D29—42no—
110C2×D29—42no—
111C2×D29—88no—
112C4×C29—21no—
113C4×C29—21no—
114C4×C29—42no—
115C4×C29C22×D411yescharacteristic
116C4×C29H(5,𝔽2)11yescharacteristic
117Dic29—22no—
118Dic29—42no—
119C2×AGL(1,ℤ/8)—292no—
120#546—292no—
121#495—291no—
122#468—292no—
123#463—292no—
124#452—292no—
125#447—292no—
126C4×D8D2911yescharacteristic, complemented
127#436—291no—
128C2×C29—44no—
129C2×C29D4211yescharacteristic
130C2×C29#57511yescharacteristic
131C2×C29#57611yescharacteristic
132D29—42no—
133D29—82no—
134C2×#41—292no—
135C22×D4—292no—
136AGL(1,ℤ/8)—588no—
137C2×Q16—291no—
138C2×SD16—292no—
139C2×SD16—292no—
140C2×D8—292no—
141C2×D8—291no—
142C2×D8C2×D2911yescharacteristic
143C2×M16—292no—
144#124—292no—
145#123—582no—
146C22⋊Q8—292no—
147#121—292no—
148#121—582no—
149#120—582no—
150C4×D4—291no—
151C4×D4C2×D2912yescomplemented
152C2×C22⋊C4—582no—
153#107C2×D2911yescharacteristic, complemented
154C4⋊C8—291no—
155#103—292no—
156#102—292no—
157#102C2×D2912yescomplemented
158C22⋊C8—292no—
159C4×C8C2×D2911yescharacteristic, complemented
160C29#327311yescharacteristic, complemented, Sylow
161C24—582no—
162#41—584no—
163#41—1162no—
164C2×Q8—292no—
165C2×D4—291no—
166C2×D4—292no—
167C2×D4—582no—
168C2×D4—582no—
169C2×D4—584no—
170C2×D4—582no—
171C2×D4—1162no—
172C2×D4C22×D2912yes—
173C22×C4—22no—
174C22×C4—292no—
175C22×C4—582no—
176Q16—1161no—
177SD16—584no—
178SD16—584no—
179D8—584no—
180D8—1161no—
181D8D11614yescomplemented
182M16—584no—
183C2×C8—292no—
184C2×C8C22×D2911yescharacteristic
185C2×C8C22×D2911yescharacteristic
186C4⋊C4—291no—
187C4⋊C4—582no—
188C4⋊C4C22×D2912yes—
189C22⋊C4—22no—
190C22⋊C4—292no—
191C22⋊C4—582no—
192C22⋊C4—582no—
193C22⋊C4—582no—
194C22⋊C4—582no—
195C42C22×D2911yescharacteristic
196C23—22no—
197C23—292no—
198C23—582no—
199C23—1164no—
200C23—1164no—
201C23—2322no—
202Q8—582no—
203Q8—582no—
204D4—22no—
205D4—582no—
206D4—582no—
207D4—1161no—
208D4—1164no—
209D4—1164no—
210D4—2322no—
211D4C2×D11614yes—
212C2×C4—22no—
213C2×C4—82no—
214C2×C4—292no—
215C2×C4—582no—
216C2×C4—582no—
217C2×C4—582no—
218C2×C4—1164no—
219C2×C4D4×D2911yescharacteristic
220C2×C4D4×D2911yescharacteristic
221C2×C4C23×D2911yescharacteristic
222C8—21no—
223C8—582no—
224C8C2×D11612yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—582no—
229C22—582no—
230C22—582no—
231C22—1162no—
232C22—1162no—
233C22—1162no—
234C22—2328no—
235C22C2×D4×D2911yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—582no—
240C4—1162no—
241C4C22×D11611yescharacteristic
242C4#1174536411yescharacteristic
243C2—44no—
244C2—1162no—
245C2—2322no—
246C2D4×D11611yescharacteristic
247C2#1427196011yescharacteristic
248C2#1427196411yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC4×D8×C29C2
socleC22×C29C22×D4
hypercenterC4×D8D29
hypocenterC29#3273
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. #14271808C2semidirect
3. #14271547C2semidirect
4. #14271410C2semidirect
5. #14271378C2semidirect
6. #14271326C2semidirect
7. #14271298C2semidirect
8. #14271242C2semidirect
9. C4×D8×C29C2semidirect
11. C2×D4×D29C22semidirect
13. C4×D4×C29C22semidirect
14. C2×#90783C22semidirect
19. C2×Dic116C22semidirect
21. C2×#90773C22semidirect
22. C2×#90772C22semidirect
23. #11745259C22semidirect
27. #11745237C22semidirect
29. C29×#107C22semidirect
30. C29×#102C22semidirect
31. C4×C8×C29C22semidirect
32. #11745168C22semidirect
33. #11745158C22semidirect
34. #11745157C22semidirect
35. C4⋊#6597C22semidirect
51. D8×C29D4semidirect
126. C4×D8D29semidirect
151. C4×D4C2×D29semidirect
153. #107C2×D29semidirect
157. #102C2×D29semidirect
159. C4×C8C2×D29semidirect
160. C29#3273semidirect
181. D8D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
#142718082
#142715472
#142714102
#142713782
#142713262
#142712982
#142712422
C4×D8×C292
#327329