Finite Group Databasebeta

← #16822852

Subgroups — #16822852

Order 3712.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D29C211yesmaximal, characteristic, complemented
2C2×#11745258C212yesmaximal, complemented
3C2×#11745255C211yesmaximal, characteristic
4#14271083C211yesmaximal, characteristic, complemented
5#14271076C211yesmaximal, characteristic, complemented
6#14271074C211yesmaximal, characteristic, complemented
7#14271067C212yesmaximal, complemented
8#14271041C212yesmaximal
9#14271034C212yesmaximal, complemented
10#14271022C211yesmaximal, characteristic, complemented
11C29×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C29C2211yescharacteristic
13C2×Q8×D29—24no—
14C2×Q8×D29—28no—
15C22×D116C2211yescharacteristic
16C22×C4×D29C2212yes—
17C22×C4×D29C2211yescharacteristic
18C22×C4×D29C2211yescharacteristic
19C22×Dic58C2211yescharacteristic
20C22×Dic58C2212yes—
21C2×C4⋊C4×C29C2212yes—
22C2×C4⋊C4×C29C2211yescharacteristic
23C2×#90781C2212yes—
24C2×#90781C2211yescharacteristic
25C2×#90781C2212yes—
26C2×#90781C2212yes—
27C2×#90781C2211yescharacteristic
28C2×#90779C2212yes—
29C2×#90779C2212yes—
30C2×C4×Dic29C2211yescharacteristic
31#11745258—28no—
32#11745255—24no—
33C29×#95C2211yescharacteristic
34C29×#95C2212yes—
35#11745179C2211yescharacteristic
36#11745179C2212yes—
37#11745179C2212yes—
38#11745179C2212yes—
39C23×D29C2311yescharacteristic
40C23×D29C2311yescharacteristic
41C2×Q8×C29—22no—
42C2×Q8×C29—24no—
43C22×C4×C29C2311yescharacteristic
44C22×C4×C29C2312yes—
45C22×C4×C29C2311yescharacteristic
46C22×C4×C29C2312yes—
47C22×C4×C29C2311yescharacteristic
48C22×Dic29C2311yescharacteristic
49C22×Dic29C2312yes—
50C22×Dic29C2312yes—
51C22×Dic29C2311yescharacteristic
52Q8×D29—232no—
53C2×D116—22no—
54C2×D116—22no—
55C2×D116Q814yescomplemented
56C2×C4×D29—24no—
57C2×C4×D29—24no—
58C2×C4×D29—24no—
59C2×C4×D29—24no—
60C2×C4×D29—22no—
61C2×C4×D29—24no—
62C2×C4×D29D414yes—
63C2×Dic58—22no—
64C2×Dic58—24no—
65C2×Dic58—24no—
66C2×Dic58—24no—
67C2×Dic58D418yescomplemented
68C4⋊C4×C29—22no—
69C4⋊C4×C29—24no—
70#90781—22no—
71#90781—24no—
72#90781—22no—
73#90781—24no—
74#90781—24no—
75#90779—24no—
76#90779—24no—
77C4×Dic29—22no—
78C23×C29C2411yescharacteristic
79C22×D29—22no—
80C22×D29—22no—
81C22×D29—22no—
82C22×D29—21no—
83C22×D29—21no—
84C22×D29—22no—
85C22×D29C2×D412yes—
86C22×D29C2×Q812yes—
87C22×D29C2×Q812yescomplemented
88C22×D29#4112yes—
89Q8×C29—28no—
90C2×C4×C29—21no—
91C2×C4×C29—22no—
92C2×C4×C29—22no—
93C2×C4×C29—22no—
94C2×C4×C29—21no—
95C2×C4×C29—22no—
96C2×C4×C29—22no—
97C2×C4×C29—21no—
98C2×C4×C29—22no—
99C2×C4×C29—22no—
100C2×C4×C29C2×D412yes—
101C2×C4×C29C2×D414yes—
102C2×C4×C29C2×Q812yescomplemented
103C2×Dic29—22no—
104C2×Dic29—22no—
105C2×Dic29—21no—
106C2×Dic29—22no—
107C2×Dic29—22no—
108C2×Dic29—22no—
109C2×Dic29—22no—
110C2×Dic29C2×D414yes—
111C2×Dic29C2×D412yes—
112C2×Dic29C2×D414yes—
113D116—28no—
114C4×D29—216no—
115C4×D29—28no—
116C4×D29—44no—
117Dic58—28no—
118Dic58—216no—
119#2409—291nomaximal, Sylow
120C22×C29C22×D411yescharacteristic
121C22×C29C22×D412yes—
122C22×C29C22×Q811yescharacteristic
123C22×C29C2×#4111yescharacteristic
124C22×C292−1+412yes—
125C2×D29—22no—
126C2×D29—22no—
127C2×D29—24no—
128C2×D29—24no—
129C2×D29—22no—
130C2×D29—24no—
131C2×D29—41no—
132C2×D29—41no—
133C2×D29—42no—
134C2×D29C22⋊Q814yescomplemented
135C4×C29—22no—
136C4×C29—24no—
137C4×C29—22no—
138C4×C29—41no—
139C4×C29—42no—
140Dic29—24no—
141Dic29—41no—
142Dic29—42no—
143Dic29#12014yes—
144C23×Q8—291no—
145C2×C22⋊Q8—292no—
146C2×C4×D4—291no—
147#395—291no—
148#395D2911yescharacteristic, complemented
149#393—291no—
150#393—291no—
151#391—292no—
152#386—292no—
153#385—292no—
154#380—291no—
155C2×C29C2×#12011yescharacteristic
156C2×C29C2×C22⋊Q811yescharacteristic
157C2×C29#53611yescharacteristic
158C2×C29#54812yes—
159C2×C29D4×Q812yes—
160D29—24no—
161D29—42no—
162C22×Q8—292no—
163C22×Q8—291no—
164C22×Q8—584no—
165C22×Q8—588no—
166C22×Q8C2×D2911yescharacteristic
167C22×D4—291no—
168C23×C4—291no—
169C23×C4—291no—
170C23×C4—292no—
171C22⋊Q8—588no—
172C4×D4—584no—
173C2×C4⋊C4—292no—
174C2×C4⋊C4—292no—
175C2×C4⋊C4C2×D2911yescharacteristic
176C2×C4⋊C4C2×D2912yes—
177C2×C22⋊C4—291no—
178C2×C22⋊C4—291no—
179C2×C22⋊C4—292no—
180C2×C22⋊C4—292no—
181C2×C22⋊C4—292no—
182C2×C42—291no—
183#95—291no—
184#95—292no—
185#95—292no—
186#95—292no—
187#95C2×D2911yescharacteristic
188#95C2×D2912yes—
189C29#240911yescharacteristic, complemented, Sylow
190C24—291no—
191C24—291no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—298no—
195C2×Q8—584no—
196C2×Q8—584no—
197C2×Q8—584no—
198C2×Q8—5832no—
199C2×Q8—582no—
200C2×D4—294no—
201C2×D4—582no—
202C2×D4—582no—
203C22×C4—291no—
204C22×C4—292no—
205C22×C4—294no—
206C22×C4—291no—
207C22×C4—292no—
208C22×C4—584no—
209C22×C4—584no—
210C22×C4—582no—
211C22×C4—584no—
212C22×C4—584no—
213C22×C4—584no—
214C22×C4C22×D2912yes—
215C22×C4C22×D2911yescharacteristic
216C22×C4C22×D2911yescharacteristic
217C22×C4C22×D2912yes—
218C22×C4C22×D2911yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—584no—
222C4⋊C4—584no—
223C22⋊C4—584no—
224C22⋊C4—582no—
225C22⋊C4—584no—
226C22⋊C4—582no—
227C22⋊C4—584no—
228C42—582no—
229C23—292no—
230C23—292no—
231C23—292no—
232C23—292no—
233C23—582no—
234C23—582no—
235C23—582no—
236C23—581no—
237C23—582no—
238C23—581no—
239C23C23×D2911yescharacteristic
240Q8—28no—
241Q8—5816no—
242Q8—588no—
243D4—588no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—292no—
255C2×C4—294no—
256C2×C4—294no—
257C2×C4—588no—
258C2×C4—5816no—
259C2×C4—582no—
260C2×C4—582no—
261C2×C4—581no—
262C2×C4—582no—
263C2×C4—582no—
264C2×C4—582no—
265C2×C4—582no—
266C2×C4—1164no—
267C2×C4D4×D2912yes—
268C2×C4D4×D2914yes—
269C2×C4Q8×D2912yescomplemented
270C22—294no—
271C22—584no—
272C22—584no—
273C22—582no—
274C22—582no—
275C22—584no—
276C22—582no—
277C22—1162no—
278C22—1161no—
279C22—1161no—
280C22C2×D4×D2912yes—
281C22C2×D4×D2911yescharacteristic
282C22C2×Q8×D2911yescharacteristic
283C22#1174536212yes—
284C22#41×D2911yescharacteristic
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—294no—
291C4—584no—
292C4—1162no—
293C4—1161no—
294C2—584no—
295C2—1162no—
296C2#120×D2911yescharacteristic
297C2C22⋊Q8×D2911yescharacteristic
298C2#1427173811yescharacteristic
299C2#1427182312yes—
300C2#1427182812yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#395C2
prime radicalC22×D116C22
hypercenter#395D29
hypocenterC29#2409
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D29C2semidirect
2. C2×#11745258C2semidirect
4. #14271083C2semidirect
5. #14271076C2semidirect
6. #14271074C2semidirect
7. #14271067C2semidirect
9. #14271034C2semidirect
10. #14271022C2semidirect
11. C29×#395C2semidirect
55. C2×D116Q8semidirect
67. C2×Dic58D4semidirect
87. C22×D29C2×Q8semidirect
102. C2×C4×C29C2×Q8semidirect
134. C2×D29C22⋊Q8semidirect
148. #395D29semidirect
189. C29#2409semidirect
269. C2×C4Q8×D29semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D292
C2×#117452582
C2×#117452552
#142710832
#142710762
#142710742
#142710672
#142710412
#142710342
#142710222
C29×#3952
#240929