Finite Group Databasebeta

← #16822807

Subgroups — #16822807

Order 3712.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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×#11745299C212yesmaximal, complemented
2C22×#90781C211yesmaximal, characteristic, complemented
3#14271074C212yesmaximal, complemented
4#14271071C212yesmaximal, complemented
5#14271043C212yesmaximal, complemented
6#14271042C214yesmaximal, complemented
7#14271026C211yesmaximal, characteristic, complemented
8C22×#41×C29C211yesmaximal, characteristic, complemented
9C24×D29C2211yescharacteristic
10C2×#41×C29—28no—
11C2×#41×C29—24no—
12C22×Q8×C29C2211yescharacteristic, complemented
13C22×D4×C29C2212yes—
14C22×D4×C29C2211yescharacteristic, complemented
15C23×C4×C29C2211yescharacteristic
16C23×C4×C29C2212yescomplemented
17C22×C22⋊D29C2214yes—
18C23×Dic29C2211yescharacteristic
19C2×C22⋊Dic29C2212yes—
20C2×C22⋊Dic29C2214yescomplemented
21#11745299—28no—
22C2×#90781—24no—
23C2×#90781C2212yes—
24C2×#90781C2214yes—
25C2×#90781C2212yes—
26#11745179C2212yescomplemented
27#11745179C2212yescomplemented
28#11745179C2214yescomplemented
29C24×C29C2312yes—
30C24×C29C2311yescharacteristic
31C23×D29—22no—
32C23×D29—24no—
33C23×D29—24no—
34C23×D29—42no—
35C23×D29C2312yes—
36#41×C29—232no—
37C2×Q8×C29—24no—
38C2×Q8×C29—22no—
39C2×D4×C29—24no—
40C2×D4×C29—24no—
41C2×D4×C29—24no—
42C2×D4×C29—22no—
43C2×D4×C29D418yescomplemented
44C22×C4×C29—24no—
45C22×C4×C29—24no—
46C22×C4×C29—24no—
47C22×C4×C29—24no—
48C22×C4×C29D414yescomplemented
49C22×C4×C29C2311yescharacteristic
50C22×C4×C29C2311yescharacteristic
51C22×C4×C29C2312yes—
52C2×C22⋊D29—28no—
53C2×C22⋊D29—28no—
54C2×C22⋊D29—44no—
55C22×Dic29—22no—
56C22×Dic29—42no—
57C22×Dic29C2312yes—
58C22×Dic29C2314yes—
59C22⋊Dic29—24no—
60C22⋊Dic29—44no—
61#90781—24no—
62#90781—24no—
63#90781—28no—
64C23×C29—22no—
65C23×C29—22no—
66C23×C29—22no—
67C23×C29—22no—
68C23×C29—24no—
69C23×C29—24no—
70C23×C29C2×D414yes—
71C23×C29C2×D414yes—
72C23×C29C2×D412yes—
73C23×C29C2411yescharacteristic
74C22×D29—28no—
75C22×D29—22no—
76C22×D29—24no—
77C22×D29—22no—
78C22×D29—24no—
79C22×D29—416no—
80C22×D29—44no—
81C22×D29—44no—
82C22×D29#4114yescomplemented
83Q8×C29—28no—
84D4×C29—28no—
85D4×C29—216no—
86C2×C4×C29—216no—
87C2×C4×C29—28no—
88C2×C4×C29—22no—
89C2×C4×C29—22no—
90C2×C4×C29—22no—
91C2×C4×C29—22no—
92C2×C4×C29C2×D412yes—
93C2×C4×C29C2×D414yes—
94C2×C4×C29C2×D412yes—
95C22⋊D29—416no—
96C2×Dic29—22no—
97C2×Dic29—24no—
98C2×Dic29—24no—
99C2×Dic29—24no—
100C2×Dic29—24no—
101C2×Dic29—44no—
102#2404—291nomaximal, Sylow
103C22×C29—22no—
104C22×C29—22no—
105C22×C29—28no—
106C22×C29—24no—
107C22×C29—28no—
108C22×C29—42no—
109C22×C29—42no—
110C22×C29—44no—
111C22×C29#12014yes—
112C22×C29C22×D411yescharacteristic
113C22×C29C22×D412yes—
114C22×C29C2×#4112yes—
115C22×C29H(5,𝔽2)12yes—
116C2×D29—24no—
117C2×D29—28no—
118C2×D29—416no—
119C2×D29—42no—
120C2×D29—42no—
121C2×D29—44no—
122C4×C29—22no—
123C4×C29—24no—
124C4×C29—22no—
125Dic29—42no—
126Dic29—44no—
127C22×#41D2911yescharacteristic, complemented
128C2×#120—292no—
129C22×C22⋊C4—291no—
130#393—292no—
131#392—292no—
132#386—292no—
133#386—294no—
134#381—291no—
135C2×C29—24no—
136C2×C29—44no—
137C2×C29C2×#12011yescharacteristic
138C2×C29#52511yescharacteristic
139C2×C29#53511yescharacteristic
140C2×C29#54614yes—
141D29—44no—
142C25—291no—
143C2×#41—28no—
144C2×#41—24no—
145C22×Q8C2×D2911yescharacteristic, complemented
146C22×D4—294no—
147C22×D4C2×D2912yes—
148C22×D4C2×D2911yescharacteristic, complemented
149C23×C4—291no—
150C23×C4C2×D2911yescharacteristic
151C23×C4C2×D2912yescomplemented
152#120—588no—
153C2×C22⋊C4—292no—
154C2×C22⋊C4—292no—
155C2×C22⋊C4—292no—
156C2×C22⋊C4—294no—
157C2×C22⋊C4—294no—
158C2×C22⋊C4—584no—
159#95—292no—
160#95—292no—
161#95—294no—
162C29#240411yescharacteristic, complemented, Sylow
163C24—292no—
164C24—584no—
165C24—582no—
166C24—584no—
167C24—1162no—
168C24C22×D2912yes—
169C24C22×D2911yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—588no—
178C2×D4—588no—
179C2×D4—1164no—
180C2×D4C22⋊D2918yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—294no—
186C22×C4—292no—
187C22×C4—582no—
188C22×C4—1162no—
189C22×C4C22⋊D2914yescomplemented
190C22×C4C22×D2911yescharacteristic
191C22×C4C22×D2911yescharacteristic
192C22×C4C22×D2912yes—
193C22⋊C4—584no—
194C22⋊C4—584no—
195C22⋊C4—584no—
196C22⋊C4—588no—
197C22⋊C4—1164no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—294no—
205C23—588no—
206C23—582no—
207C23—584no—
208C23—582no—
209C23—584no—
210C23—11616no—
211C23—1164no—
212C23—1164no—
213C23C2×C22⋊D2914yes—
214C23C2×C22⋊D2914yes—
215C23C2×C22⋊D2912yes—
216C23C23×D2911yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—11616no—
221C2×C4—216no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—584no—
228C2×C4—584no—
229C2×C4—584no—
230C2×C4—582no—
231C2×C4—584no—
232C2×C4—1164no—
233C2×C4C2×C22⋊D2912yes—
234C2×C4C2×C22⋊D2914yes—
235C2×C4C2×C22⋊D2912yes—
236C22—28no—
237C22—22no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—588no—
245C22—584no—
246C22—11616no—
247C22—1164no—
248C22—1162no—
249C22—1162no—
250C22#1174531514yes—
251C22#41×D2912yes—
252C22#1174536412yes—
253C22C22×C22⋊D2912yes—
254C22C22×C22⋊D2911yescharacteristic
255C4—22no—
256C4—24no—
257C4—22no—
258C4—1164no—
259C4—1162no—
260C2—24no—
261C2—44no—
262C2—1164no—
263C2C2×#1174531511yescharacteristic
264C2#1427167011yescharacteristic
265C2#1427172711yescharacteristic
266C2#1427180114yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC22×#41×C29C2
hypercenterC22×#41D29
hypocenterC29#2404
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11745299C2semidirect
2. C22×#90781C2semidirect
3. #14271074C2semidirect
4. #14271071C2semidirect
5. #14271043C2semidirect
6. #14271042C2semidirect
7. #14271026C2semidirect
8. C22×#41×C29C2semidirect
12. C22×Q8×C29C22semidirect
14. C22×D4×C29C22semidirect
16. C23×C4×C29C22semidirect
20. C2×C22⋊Dic29C22semidirect
26. #11745179C22semidirect
27. #11745179C22semidirect
28. #11745179C22semidirect
43. C2×D4×C29D4semidirect
48. C22×C4×C29D4semidirect
82. C22×D29#41semidirect
127. C22×#41D29semidirect
145. C22×Q8C2×D29semidirect
148. C22×D4C2×D29semidirect
151. C23×C4C2×D29semidirect
162. C29#2404semidirect
180. C2×D4C22⋊D29semidirect
189. C22×C4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452992
C22×#907812
#142710742
#142710712
#142710432
#142710422
#142710262
C22×#41×C292
#240429