Finite Group Databasebeta

← #16822475

Subgroups — #16822475

Order 3712.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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×#11745299C211yesmaximal, characteristic
2C2×C4×C22⋊D29C212yesmaximal, complemented
3C2×C22⋊C4×D29C212yesmaximal, complemented
4#14271066C211yesmaximal, characteristic, complemented
5#14271057C212yesmaximal, complemented
6#14271039C212yesmaximal, complemented
7#14271037C212yesmaximal, complemented
8#14271010C212yesmaximal, complemented
9C29×#386C211yesmaximal, characteristic, complemented
10C24×D29C2211yescharacteristic
11C22×D4×C29C2211yescharacteristic
12C23×C4×C29C2211yescharacteristic, complemented
13C22×C22⋊D29C2212yes—
14C22×C4×D29C2212yes—
15C22×C4×D29C2212yes—
16C2×C22⋊C4×C29C2212yes—
17C2×C22⋊C4×C29C2211yescharacteristic, complemented
18C2×C22⋊Dic29C2212yes—
19C2×C22⋊Dic29C2211yescharacteristic
20C2×C22⋊Dic29C2212yescomplemented
21#11745299C418yescomplemented
22C4×C22⋊D29—28no—
23C2×#90781C2212yes—
24C2×#90781C2212yes—
25C2×#90781C2212yes—
26C2×C4⋊Dic29C2211yescharacteristic, complemented
27C2×#90779C2212yescomplemented
28C2×C4×Dic29C2212yescomplemented
29C2×C4×Dic29C2211yescharacteristic, complemented
30C22⋊C4×D29—28no—
31C29×#95C2212yescomplemented
32#11745179C2212yescomplemented
33#11745179C2212yescomplemented
34C24×C29C2311yescharacteristic
35C24×C29C2311yescharacteristic
36C23×D29—24no—
37C23×D29—24no—
38C23×D29—41no—
39C23×D29—41no—
40C23×D29C2×C414yescomplemented
41C23×D29C2312yes—
42C2×D4×C29—41no—
43C2×D4×C29—41no—
44C2×D4×C29C2×C414yescomplemented
45C22×C4×C29—24no—
46C22×C4×C29—41no—
47C22×C4×C29C2312yes—
48C22×C4×C29C2311yescharacteristic
49C22×C4×C29C2312yes—
50C2×C22⋊D29—24no—
51C2×C22⋊D29—24no—
52C2×C22⋊D29C2×C418yescomplemented
53C22×Dic29C2312yes—
54C22×Dic29C2312yes—
55C22×Dic29C2312yes—
56C2×C4×D29—24no—
57C2×C4×D29—24no—
58C2×C4×D29—24no—
59C2×C4×D29—24no—
60C2×C4×D29—42no—
61C2×C4×D29—42no—
62C22⋊C4×C29—24no—
63C22⋊C4×C29—41no—
64C22⋊Dic29—24no—
65C22⋊Dic29—24no—
66C22⋊Dic29C2×C414yescomplemented
67#90781—24no—
68#90781C2×C418yescomplemented
69#90781D418yescomplemented
70C4⋊Dic29—41no—
71#90779—24no—
72C4×Dic29—24no—
73C4×Dic29—41no—
74C23×C29—22no—
75C23×C29—22no—
76C23×C29—22no—
77C23×C29—22no—
78C23×C29—21no—
79C23×C29—21no—
80C23×C29—21no—
81C23×C29—21no—
82C23×C29C22×C412yes—
83C23×C29C22×C412yes—
84C23×C29C2411yescharacteristic
85C22×D29—22no—
86C22×D29—22no—
87C22×D29—22no—
88C22×D29—22no—
89C22×D29—216no—
90C22×D29—44no—
91C22×D29—44no—
92C22×D29—48no—
93C22×D29—44no—
94C22×D29C22×C414yes—
95C22×D29C2×D414yes—
96C22×D29#4114yes—
97D4×C29—44no—
98C2×C4×C29—22no—
99C2×C4×C29—21no—
100C2×C4×C29—21no—
101C2×C4×C29—22no—
102C2×C4×C29—22no—
103C2×C4×C29—22no—
104C2×C4×C29—22no—
105C2×C4×C29—44no—
106C2×C4×C29C22×C412yes—
107C2×C4×C29C2×D414yes—
108C22⋊D29—216no—
109C2×Dic29—22no—
110C2×Dic29—22no—
111C2×Dic29—22no—
112C2×Dic29—22no—
113C2×Dic29—22no—
114C2×Dic29—22no—
115C2×Dic29—22no—
116C2×Dic29C22×C414yes—
117C2×Dic29C2×D414yes—
118C4×D29—48no—
119C4×D29—48no—
120#2358—291nomaximal, Sylow
121C22×C29—24no—
122C22×C29—24no—
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—41no—
126C22×C29—41no—
127C22×C29—42no—
128C22×C29—42no—
129C22×C29—41no—
130C22×C29—41no—
131C22×C29C23×C411yescharacteristic
132C22×C29C22×D412yes—
133C22×C29C2×#4112yes—
134C22×C29H(5,𝔽2)11yescharacteristic
135C22×C29H(5,𝔽2)11yescharacteristic
136C2×D29—24no—
137C2×D29—24no—
138C2×D29—24no—
139C2×D29—24no—
140C2×D29—24no—
141C2×D29—24no—
142C2×D29—416no—
143C2×D29C4×D418yescomplemented
144C4×C29—22no—
145C4×C29—24no—
146C4×C29—42no—
147Dic29—24no—
148Dic29—24no—
149Dic29—42no—
150C2×#120—291no—
151C2×C4×D4—292no—
152C22×C22⋊C4—292no—
153#390—291no—
154#388—292no—
155#386—292no—
156#386D2911yescharacteristic, complemented
157#385—292no—
158C4×C22⋊C4—292no—
159C2×C29—42no—
160C2×C29—42no—
161C2×C29C2×C4×D412yes—
162C2×C29#51811yescharacteristic
163C2×C29D4211yescharacteristic
164C2×C29#54612yes—
165C2×C29#55111yescharacteristic
166D29—28no—
167C25—291no—
168C22×D4—292no—
169C22×D4C2×D2911yescharacteristic
170C23×C4—292no—
171C23×C4—292no—
172C23×C4C2×D2911yescharacteristic, complemented
173#120—298no—
174C4×D4—588no—
175C2×C4⋊C4—292no—
176C2×C4⋊C4—291no—
177C2×C22⋊C4—291no—
178C2×C22⋊C4—292no—
179C2×C22⋊C4—292no—
180C2×C22⋊C4—292no—
181C2×C22⋊C4—292no—
182C2×C22⋊C4—292no—
183C2×C22⋊C4—588no—
184C2×C22⋊C4C2×D2912yes—
185C2×C22⋊C4C2×D2911yescharacteristic, complemented
186C2×C42—292no—
187C2×C42—291no—
188#95—292no—
189#95—292no—
190#95C2×D2912yescomplemented
191C29#235811yescharacteristic, complemented, Sylow
192C24—292no—
193C24—294no—
194C24—584no—
195C24—584no—
196C24—1161no—
197C24—1161no—
198C24C22×D2911yescharacteristic
199C24C22×D2911yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—298no—
203C2×D4—584no—
204C2×D4—584no—
205C2×D4C4×D2914yescomplemented
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—292no—
209C22×C4—292no—
210C22×C4—292no—
211C22×C4—584no—
212C22×C4—584no—
213C22×C4—584no—
214C22×C4—584no—
215C22×C4—1162no—
216C22×C4—1162no—
217C22×C4C22×D2912yes—
218C22×C4C22×D2911yescharacteristic
219C22×C4C22×D2912yes—
220C4⋊C4—584no—
221C4⋊C4—1161no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—294no—
225C22⋊C4—298no—
226C22⋊C4—298no—
227C22⋊C4—584no—
228C22⋊C4—584no—
229C22⋊C4—584no—
230C42—584no—
231C42—1161no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—294no—
241C23—294no—
242C23—294no—
243C23—5816no—
244C23—582no—
245C23—582no—
246C23—582no—
247C23—582no—
248C23—1164no—
249C23—1164no—
250C23—1168no—
251C23—1164no—
252C23C2×C4×D2912yes—
253C23C2×C4×D2912yes—
254C23C23×D2911yescharacteristic
255D4—44no—
256D4—5816no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—44no—
265C2×C4—294no—
266C2×C4—294no—
267C2×C4—582no—
268C2×C4—582no—
269C2×C4—582no—
270C2×C4—582no—
271C2×C4—582no—
272C2×C4—582no—
273C2×C4—582no—
274C2×C4—1168no—
275C2×C4—1168no—
276C2×C4C2×C4×D2912yes—
277C2×C4D4×D2914yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—298no—
289C22—584no—
290C22—584no—
291C22—584no—
292C22—584no—
293C22—584no—
294C22—584no—
295C22—11616no—
296C22C22×C4×D2911yescharacteristic
297C22C2×D4×D2912yes—
298C22#1174535911yescharacteristic
299C22#1174535911yescharacteristic
300C22#41×D2912yes—
301C4—24no—
302C4—22no—
303C4—42no—
304C4—584no—
305C4—584no—
306C4—1162no—
307C2—42no—
308C2—42no—
309C2—588no—
310C2C4×D4×D2912yes—
311C2#1427163611yescharacteristic
312C2#1427179511yescharacteristic
313C2#1427181012yes—
314C2#1427184111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutatorC22×C29C23×C4
FittingC29×#386C2
socle, cosocleC23×C29C24
prime radicalC2×#11745299C2
hypercenter#386D29
hypocenterC29#2358
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D29C2semidirect
3. C2×C22⋊C4×D29C2semidirect
4. #14271066C2semidirect
5. #14271057C2semidirect
6. #14271039C2semidirect
7. #14271037C2semidirect
8. #14271010C2semidirect
9. C29×#386C2semidirect
12. C23×C4×C29C22semidirect
17. C2×C22⋊C4×C29C22semidirect
20. C2×C22⋊Dic29C22semidirect
21. #11745299C4semidirect
26. C2×C4⋊Dic29C22semidirect
27. C2×#90779C22semidirect
28. C2×C4×Dic29C22semidirect
29. C2×C4×Dic29C22semidirect
31. C29×#95C22semidirect
32. #11745179C22semidirect
33. #11745179C22semidirect
40. C23×D29C2×C4semidirect
44. C2×D4×C29C2×C4semidirect
52. C2×C22⋊D29C2×C4semidirect
66. C22⋊Dic29C2×C4semidirect
68. #90781C2×C4semidirect
69. #90781D4semidirect
143. C2×D29C4×D4semidirect
156. #386D29semidirect
172. C23×C4C2×D29semidirect
185. C2×C22⋊C4C2×D29semidirect
190. #95C2×D29semidirect
191. C29#2358semidirect
205. C2×D4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452992
C2×C4×C22⋊D292
C2×C22⋊C4×D292
#142710662
#142710572
#142710392
#142710372
#142710102
C29×#3862
#235829