Finite Group Databasebeta

← #16822849

Subgroups — #16822849

Order 3712.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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×D116C211yesmaximal, characteristic, complemented
2C2×#11745257C212yesmaximal, complemented
3C2×#11745244C212yesmaximal, complemented
4C2×C4×C22⋊D29C211yesmaximal, characteristic, complemented
5#14271081C212yesmaximal, complemented
6#14271073C212yesmaximal, complemented
7#14271064C212yesmaximal, complemented
8#14271018C212yesmaximal, complemented
9C29×#380C211yesmaximal, characteristic, complemented
10C24×D29C2212yes—
11C23×C4×C29C2211yescharacteristic
12C22×C22⋊D29C2211yescharacteristic
13C22×D116—24no—
14C22×D116—28no—
15C22×D116C2212yes—
16C22×D116C2212yes—
17C22×D116C2214yescomplemented
18C22×C4×D29C2211yescharacteristic, complemented
19C2×C4⋊C4×C29C2212yescomplemented
20C2×C22⋊C4×C29C2212yes—
21C2×C22⋊Dic29C2211yescharacteristic
22C4×C22⋊D29—24no—
23C2×#90781C2214yescomplemented
24C2×#90781C2214yescomplemented
25C2×#90781C2211yescharacteristic, complemented
26C2×#90781C2214yescomplemented
27C2×#90779C2211yescharacteristic, complemented
28C2×C4×Dic29C2211yescharacteristic, complemented
29#11745257—28no—
30#11745244—28no—
31C29×#95C2212yescomplemented
32C24×C29C2311yescharacteristic
33C23×D29—24no—
34C23×D29—28no—
35C23×D29—24no—
36C23×D29—24no—
37C23×D29—44no—
38C23×D29C2314yes—
39C23×D29C2311yescharacteristic
40C22×C4×C29—24no—
41C22×C4×C29D414yescomplemented
42C22×C4×C29C2314yes—
43C22×C4×C29C2311yescharacteristic
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—22no—
46C2×C22⋊D29—22no—
47C2×C22⋊D29D414yescomplemented
48C22×Dic29C2311yescharacteristic
49C22×Dic29C2311yescharacteristic
50C22×Dic29C2311yescharacteristic
51C2×D116—232no—
52C2×D116—28no—
53C2×D116—24no—
54C2×D116—24no—
55C2×D116—24no—
56C2×D116—28no—
57C2×D116—28no—
58C2×D116—44no—
59C2×D116D418yescomplemented
60C2×C4×D29—22no—
61C2×C4×D29—24no—
62C4⋊C4×C29—24no—
63C22⋊C4×C29—24no—
64C22⋊Dic29—22no—
65#90781—28no—
66#90781—22no—
67#90781—28no—
68#90781—44no—
69#90779—22no—
70C4×Dic29—22no—
71C23×C29—22no—
72C23×C29—22no—
73C23×C29C2×D412yes—
74C23×C29C2×D412yes—
75C23×C29#4112yes—
76C23×C29C2411yescharacteristic
77C22×D29—216no—
78C22×D29—21no—
79C22×D29—24no—
80C22×D29—22no—
81C22×D29—24no—
82C22×D29—24no—
83C22×D29—21no—
84C22×D29—24no—
85C22×D29—24no—
86C22×D29—22no—
87C22×D29—24no—
88C22×D29—416no—
89C22×D29—416no—
90C22×D29—48no—
91C22×D29—48no—
92C22×D29C2×D418yes—
93C22×D29C2×D412yescomplemented
94C2×C4×C29—22no—
95C2×C4×C29—28no—
96C2×C4×C29—24no—
97C2×C4×C29—24no—
98C2×C4×C29—24no—
99C2×C4×C29C2×D412yes—
100C2×C4×C29C2×D412yes—
101C2×C4×C29C2×D414yes—
102C22⋊D29—28no—
103C2×Dic29—21no—
104C2×Dic29—21no—
105C2×Dic29—21no—
106C2×Dic29—21no—
107C2×Dic29—22no—
108C2×Dic29—22no—
109C2×Dic29C2×D412yescomplemented
110D116—216no—
111D116—216no—
112D116—416no—
113C4×D29—44no—
114#2408—291nomaximal, Sylow
115C22×C29—22no—
116C22×C29—22no—
117C22×C29—24no—
118C22×C29—24no—
119C22×C29#12114yescomplemented
120C22×C29C22×D411yescharacteristic
121C22×C29C22×D412yes—
122C22×C29C22×D411yescharacteristic
123C22×C29C2×#4111yescharacteristic
124C22×C29H(5,𝔽2)12yes—
125C2×D29—28no—
126C2×D29—22no—
127C2×D29—24no—
128C2×D29—28no—
129C2×D29—28no—
130C2×D29—432no—
131C2×D29—41no—
132C2×D29—44no—
133C2×D29—41no—
134C2×D29—44no—
135C2×D29—44no—
136C2×D29—42no—
137C2×D29—44no—
138C4×C29—22no—
139C4×C29—44no—
140C4×C29#12014yescomplemented
141Dic29—22no—
142Dic29—41no—
143Dic29—41no—
144C23×D4—291no—
145C2×#121—292no—
146C2×#120—292no—
147C2×C4×D4—291no—
148#394—292no—
149#392—292no—
150#390—292no—
151#380D2911yescharacteristic, complemented
152#379—292no—
153C2×C29—24no—
154C2×C29C2×#12011yescharacteristic
155C2×C29C2×#12111yescharacteristic
156C2×C29#53511yescharacteristic
157C2×C29D4212yes—
158C2×C29#54612yes—
159D29—42no—
160D29—48no—
161C25—292no—
162C22×D4—291no—
163C22×D4—292no—
164C22×D4—292no—
165C22×D4—294no—
166C22×D4—588no—
167C22×D4—584no—
168C23×C4—291no—
169C23×C4C2×D2911yescharacteristic
170#121—588no—
171#120—588no—
172C4×D4—584no—
173C2×C4⋊C4—291no—
174C2×C4⋊C4C2×D2912yescomplemented
175C2×C22⋊C4—291no—
176C2×C22⋊C4—291no—
177C2×C22⋊C4—294no—
178C2×C22⋊C4—294no—
179C2×C22⋊C4—294no—
180C2×C22⋊C4C2×D2912yes—
181C2×C42—291no—
182#95C2×D2912yescomplemented
183C29#240811yescharacteristic, complemented, Sylow
184C24—294no—
185C24—291no—
186C24—588no—
187C24—584no—
188C24—584no—
189C24—584no—
190C24—1164no—
191C24C22×D2911yescharacteristic
192C2×D4—294no—
193C2×D4—298no—
194C2×D4—588no—
195C2×D4—5832no—
196C2×D4—588no—
197C2×D4—588no—
198C2×D4—582no—
199C2×D4—582no—
200C2×D4—584no—
201C2×D4—584no—
202C2×D4—584no—
203C2×D4—1164no—
204C22×C4—24no—
205C22×C4—291no—
206C22×C4—291no—
207C22×C4—291no—
208C22×C4—582no—
209C22×C4—584no—
210C22×C4D11614yescomplemented
211C22×C4C22×D2914yes—
212C22×C4C22×D2911yescharacteristic
213C22×C4C22×D2911yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—582no—
216C22⋊C4—24no—
217C22⋊C4—588no—
218C22⋊C4—588no—
219C22⋊C4—582no—
220C22⋊C4—582no—
221C22⋊C4—1164no—
222C42—582no—
223C23—22no—
224C23—22no—
225C23—298no—
226C23—292no—
227C23—5816no—
228C23—581no—
229C23—584no—
230C23—584no—
231C23—582no—
232C23—584no—
233C23—581no—
234C23—584no—
235C23—584no—
236C23—582no—
237C23—584no—
238C23—11616no—
239C23—11616no—
240C23—1168no—
241C23—1168no—
242C23C2×D11612yes—
243C23D4×D2912yes—
244C23#9080912yes—
245C23C23×D2911yescharacteristic
246D4—5816no—
247D4—588no—
248D4—5816no—
249D4—11616no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—292no—
256C2×C4—581no—
257C2×C4—581no—
258C2×C4—581no—
259C2×C4—581no—
260C2×C4—582no—
261C2×C4—582no—
262C2×C4—1164no—
263C2×C4C2×D11612yes—
264C2×C4C2×D11612yes—
265C2×C4D4×D2914yes—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22—584no—
271C22—588no—
272C22—588no—
273C22—588no—
274C22—582no—
275C22—11632no—
276C22—1164no—
277C22—1164no—
278C22—1162no—
279C22—1161no—
280C22—1164no—
281C22—1164no—
282C22—1161no—
283C22#1174525714yescomplemented
284C22C22×D11611yescharacteristic
285C22C2×D4×D2911yescharacteristic
286C22C2×D4×D2912yes—
287C22C2×#9080911yescharacteristic
288C22#1174536412yes—
289C4—22no—
290C4—44no—
291C4—582no—
292C4—1161no—
293C4—1161no—
294C4#1174524414yescomplemented
295C2—24no—
296C2—1168no—
297C2—1162no—
298C2C2×#1174524411yescharacteristic
299C2C2×#1174525711yescharacteristic
300C2#1427173211yescharacteristic
301C2#1427179712yes—
302C2#1427180412yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#380C2
hypercenter#380D29
hypocenterC29#2408
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D116C2semidirect
2. C2×#11745257C2semidirect
3. C2×#11745244C2semidirect
4. C2×C4×C22⋊D29C2semidirect
5. #14271081C2semidirect
6. #14271073C2semidirect
7. #14271064C2semidirect
8. #14271018C2semidirect
9. C29×#380C2semidirect
17. C22×D116C22semidirect
18. C22×C4×D29C22semidirect
19. C2×C4⋊C4×C29C22semidirect
23. C2×#90781C22semidirect
24. C2×#90781C22semidirect
25. C2×#90781C22semidirect
26. C2×#90781C22semidirect
27. C2×#90779C22semidirect
28. C2×C4×Dic29C22semidirect
31. C29×#95C22semidirect
41. C22×C4×C29D4semidirect
47. C2×C22⋊D29D4semidirect
59. C2×D116D4semidirect
93. C22×D29C2×D4semidirect
109. C2×Dic29C2×D4semidirect
119. C22×C29#121semidirect
140. C4×C29#120semidirect
151. #380D29semidirect
174. C2×C4⋊C4C2×D29semidirect
182. #95C2×D29semidirect
183. C29#2408semidirect
210. C22×C4D116semidirect
283. C22#11745257semidirect
294. C4#11745244semidirect

Maximal subgroups

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

subgroupindex
C23×D1162
C2×#117452572
C2×#117452442
C2×C4×C22⋊D292
#142710812
#142710732
#142710642
#142710182
C29×#3802
#240829