Finite Group Databasebeta

← #16822819

Subgroups — #16822819

Order 3712.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D29C211yesmaximal, characteristic
2C2×#11745291C212yesmaximal, complemented
3#14271078C214yesmaximal, complemented
4#14271042C212yesmaximal, complemented
5#14271035C212yesmaximal, complemented
6#14271019C211yesmaximal, characteristic, complemented
7#14271018C212yesmaximal, complemented
8C22×C22⋊C4×C29C211yesmaximal, characteristic, complemented
9C25×C29C2211yescharacteristic
10C24×D29C2211yescharacteristic
11C23×C4×C29C2211yescharacteristic, complemented
12C23×C4×C29C2211yescharacteristic, complemented
13C22×C22⋊D29—24no—
14C22×C22⋊D29—28no—
15C22×C22⋊D29C2212yes—
16C22×C22⋊D29C2212yes—
17C23×Dic29C2211yescharacteristic
18C2×C22⋊C4×C29—24no—
19C2×C22⋊C4×C29C2212yes—
20C2×C22⋊C4×C29C2212yescomplemented
21C2×C22⋊Dic29C2212yes—
22C2×C22⋊Dic29C2212yescomplemented
23C2×C22⋊Dic29C2212yescomplemented
24#11745291—28no—
25C2×#90781C2214yescomplemented
26C2×#90781C2214yescomplemented
27C2×#90779C2214yescomplemented
28#11745179C2214yescomplemented
29C24×C29—24no—
30C24×C29—24no—
31C24×C29—42no—
32C24×C29D414yes—
33C24×C29C2311yescharacteristic
34C24×C29C2311yescharacteristic
35C24×C29C2311yescharacteristic
36C23×D29—22no—
37C23×D29—24no—
38C23×D29—24no—
39C23×D29—42no—
40C23×D29C2312yes—
41C22×C4×C29—24no—
42C22×C4×C29—22no—
43C22×C4×C29—22no—
44C22×C4×C29—42no—
45C22×C4×C29C2312yes—
46C22×C4×C29C2312yes—
47C2×C22⋊D29—232no—
48C2×C22⋊D29—24no—
49C2×C22⋊D29—24no—
50C2×C22⋊D29—24no—
51C2×C22⋊D29—24no—
52C2×C22⋊D29—24no—
53C2×C22⋊D29D418yes—
54C22×Dic29—22no—
55C22×Dic29—24no—
56C22×Dic29C2314yes—
57C22×Dic29C2312yes—
58C22⋊C4×C29—24no—
59C22⋊C4×C29—24no—
60C22⋊Dic29—24no—
61C22⋊Dic29—24no—
62C22⋊Dic29—42no—
63#90781—28no—
64#90781—44no—
65#90779—28no—
66C23×C29—22no—
67C23×C29—22no—
68C23×C29—22no—
69C23×C29—22no—
70C23×C29—28no—
71C23×C29—28no—
72C23×C29—22no—
73C23×C29—22no—
74C23×C29—22no—
75C23×C29—44no—
76C23×C29—416no—
77C23×C29C2×D414yes—
78C23×C29C2×D412yes—
79C23×C29C2×D412yes—
80C23×C29C2×D412yes—
81C23×C29#4114yes—
82C23×C29C2411yescharacteristic
83C22×D29—28no—
84C22×D29—22no—
85C22×D29—22no—
86C22×D29—24no—
87C22×D29—24no—
88C22×D29—416no—
89C22×D29—44no—
90C22×D29—44no—
91C22×D29C2×D414yes—
92C2×C4×C29—24no—
93C2×C4×C29—22no—
94C2×C4×C29—22no—
95C2×C4×C29—24no—
96C2×C4×C29—44no—
97C2×C4×C29—44no—
98C22⋊D29—216no—
99C22⋊D29—216no—
100C2×Dic29—28no—
101C2×Dic29—22no—
102C2×Dic29—22no—
103C2×Dic29—24no—
104C2×Dic29—24no—
105C2×Dic29—24no—
106C2×Dic29C2×D414yes—
107#2405—291nomaximal, Sylow
108C22×C29—22no—
109C22×C29—22no—
110C22×C29—22no—
111C22×C29—22no—
112C22×C29—28no—
113C22×C29—28no—
114C22×C29—22no—
115C22×C29—22no—
116C22×C29—22no—
117C22×C29—416no—
118C22×C29—44no—
119C22×C29#12014yes—
120C22×C29#12314yescomplemented
121C22×C29C22×D412yes—
122C22×C29C22×D411yescharacteristic
123C22×C29C2×#4112yes—
124C22×C29H(5,𝔽2)12yes—
125C2×D29—24no—
126C2×D29—28no—
127C2×D29—416no—
128C2×D29—42no—
129C2×D29—42no—
130C2×D29—44no—
131C4×C29—42no—
132C4×C29—42no—
133Dic29—24no—
134Dic29—44no—
135C23×D4—291no—
136C2×#123—292no—
137C22×C22⋊C4D2911yescharacteristic, complemented
138#394—294no—
139#386—292no—
140#385—292no—
141#379—291no—
142#379—292no—
143C2×C29—24no—
144C2×C29—24no—
145C2×C29—42no—
146C2×C29C2×#12011yescharacteristic
147C2×C29C2×#12311yescharacteristic
148C2×C29#53411yescharacteristic
149C2×C29#54614yes—
150D29—44no—
151C25—291no—
152C25C2×D2911yescharacteristic
153C22×D4—292no—
154C22×D4—292no—
155C22×D4—588no—
156C22×D4—584no—
157C23×C4—291no—
158C23×C4C2×D2911yescharacteristic, complemented
159C23×C4C2×D2911yescharacteristic, complemented
160#123—588no—
161C2×C4⋊C4—294no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—292no—
164C2×C22⋊C4—292no—
165C2×C22⋊C4—294no—
166C2×C22⋊C4—292no—
167C2×C22⋊C4—294no—
168C2×C22⋊C4C2×D2912yes—
169C2×C22⋊C4C2×D2912yescomplemented
170#95—294no—
171C29#240511yescharacteristic, complemented, Sylow
172C24—24no—
173C24—24no—
174C24—42no—
175C24—292no—
176C24—584no—
177C24—582no—
178C24—584no—
179C24—1162no—
180C24C22⋊D2914yes—
181C24C22×D2911yescharacteristic
182C24C22×D2911yescharacteristic
183C24C22×D2911yescharacteristic
184C2×D4—298no—
185C2×D4—584no—
186C2×D4—584no—
187C2×D4—5832no—
188C2×D4—584no—
189C2×D4—584no—
190C2×D4—584no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—292no—
196C22×C4—294no—
197C22×C4—584no—
198C22×C4—582no—
199C22×C4C22×D2912yes—
200C22×C4C22×D2912yes—
201C4⋊C4—588no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—588no—
205C22⋊C4—584no—
206C22⋊C4—584no—
207C22⋊C4—1164no—
208C22⋊C4—1162no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—294no—
221C23—588no—
222C23—584no—
223C23—582no—
224C23—584no—
225C23—582no—
226C23—11616no—
227C23—1164no—
228C23—1164no—
229C23#9080514yes—
230C23D4×D2914yes—
231C23C2×C22⋊D2912yes—
232C23C2×C22⋊D2912yes—
233C23C2×C22⋊D2912yes—
234C23C23×D2911yescharacteristic
235D4—5816no—
236D4—5816no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—294no—
244C2×C4—588no—
245C2×C4—582no—
246C2×C4—582no—
247C2×C4—584no—
248C2×C4—584no—
249C2×C4—584no—
250C22—28no—
251C22—28no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—588no—
262C22—584no—
263C22—11616no—
264C22—1164no—
265C22—1162no—
266C22—1162no—
267C22#1174529114yescomplemented
268C22#1174529914yes—
269C22C2×#9080512yes—
270C22C2×D4×D2912yes—
271C22#1174535912yes—
272C22C22×C22⋊D2911yescharacteristic
273C4—42no—
274C4—42no—
275C4—584no—
276C4—1164no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—1164no—
281C2C2×#1174529911yescharacteristic
282C2C2×#1174529111yescharacteristic
283C2#1427172211yescharacteristic
284C2#1427180314yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC22×C22⋊C4×C29C2
prime radicalC23×C22⋊D29C2
hypercenterC22×C22⋊C4D29
hypocenterC29#2405
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11745291C2semidirect
3. #14271078C2semidirect
4. #14271042C2semidirect
5. #14271035C2semidirect
6. #14271019C2semidirect
7. #14271018C2semidirect
8. C22×C22⋊C4×C29C2semidirect
11. C23×C4×C29C22semidirect
12. C23×C4×C29C22semidirect
20. C2×C22⋊C4×C29C22semidirect
22. C2×C22⋊Dic29C22semidirect
23. C2×C22⋊Dic29C22semidirect
25. C2×#90781C22semidirect
26. C2×#90781C22semidirect
27. C2×#90779C22semidirect
28. #11745179C22semidirect
120. C22×C29#123semidirect
137. C22×C22⋊C4D29semidirect
158. C23×C4C2×D29semidirect
159. C23×C4C2×D29semidirect
169. C2×C22⋊C4C2×D29semidirect
171. C29#2405semidirect
267. C22#11745291semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D292
C2×#117452912
#142710782
#142710422
#142710352
#142710192
#142710182
C22×C22⋊C4×C292
#240529