Finite Group Databasebeta

← #16827836

Subgroups — #16827836

Order 3712.

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

normal subgroups only

This group has 924 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
1C2×#11745302C211yesmaximal, characteristic, complemented
2C2×#11745257C211yesmaximal, characteristic, complemented
3C2×#11745246C214yesmaximal, complemented
4C2×#11745300C211yesmaximal, characteristic, complemented
5C2×C42×D29C211yesmaximal, characteristic, complemented
6#14271065C211yesmaximal, characteristic, complemented
7#14271063C214yesmaximal, complemented
8#14271032C211yesmaximal, characteristic, complemented
9C29×#390C211yesmaximal, characteristic, complemented
10C22×D4×C29C2211yescharacteristic
11C22×C22⋊D29C2214yes—
12C22×C22⋊D29C2212yes—
13C22×D116C2211yescharacteristic
14C22×D116C2212yes—
15C22×C4×D29C2211yescharacteristic
16C22×C4×D29C2212yes—
17C2×C4⋊C4×C29C2211yescharacteristic
18C2×C22⋊C4×C29C2214yes—
19C2×C42×C29C2211yescharacteristic
20C2×C22⋊Dic29C2212yes—
21#11745302—24no—
22#11745300—24no—
23C2×#90781C2212yes—
24C2×#90781C2214yes—
25C2×C4⋊Dic29C2211yescharacteristic
26C2×#90779C2214yes—
27C2×C4×Dic29C2212yes—
28C2×C4×Dic29C2211yescharacteristic
29#11745257—24no—
30#11745246—216no—
31C42×D29—24no—
32C24×C29C2312yes—
33C23×D29C2312yes—
34C23×D29C2311yescharacteristic
35C2×D4×C29—22no—
36C2×D4×C29—22no—
37C2×D4×C29—22no—
38C22×C4×C29C2312yes—
39C22×C4×C29C2312yes—
40C22×C4×C29C2311yescharacteristic
41C2×C22⋊D29—24no—
42C2×C22⋊D29—28no—
43C2×C22⋊D29—28no—
44C2×C22⋊D29—28no—
45C2×C22⋊D29—28no—
46C22×Dic29C2312yes—
47C22×Dic29C2311yescharacteristic
48C22×Dic29C2312yes—
49C2×D116—28no—
50C2×D116—24no—
51C2×D116—22no—
52C2×D116—22no—
53C2×D116—22no—
54C2×C4×D29—22no—
55C2×C4×D29—22no—
56C2×C4×D29—24no—
57C2×C4×D29—24no—
58C2×C4×D29D414yescomplemented
59C2×C4×D29D418yescomplemented
60C4⋊C4×C29—22no—
61C22⋊C4×C29—28no—
62C42×C29—22no—
63C22⋊Dic29—24no—
64#90781—28no—
65#90781—24no—
66C4⋊Dic29—22no—
67#90779—28no—
68C4×Dic29—24no—
69C4×Dic29D414yescomplemented
70C23×C29—22no—
71C23×C29—24no—
72C23×C29—22no—
73C23×C29—24no—
74C23×C29—22no—
75C23×C29C2411yescharacteristic
76C22×D29—21no—
77C22×D29—21no—
78C22×D29—22no—
79C22×D29—22no—
80C22×D29—24no—
81C22×D29—22no—
82C22×D29—22no—
83C22×D29—24no—
84C22×D29C2×D412yes—
85C22×D29C2×D414yes—
86D4×C29—44no—
87C2×C4×C29—22no—
88C2×C4×C29—22no—
89C2×C4×C29—24no—
90C2×C4×C29—22no—
91C2×C4×C29C2×D414yes—
92C2×C4×C29C2×D412yes—
93C2×C4×C29C2×D412yes—
94C2×C4×C29#4112yes—
95C22⋊D29—48no—
96C22⋊D29—416no—
97C2×Dic29—21no—
98C2×Dic29—21no—
99C2×Dic29—24no—
100C2×Dic29—22no—
101C2×Dic29C2×D412yes—
102C2×Dic29C2×D414yes—
103C2×Dic29C2×D414yes—
104C2×Dic29#4114yes—
105D116—48no—
106D116—44no—
107C4×D29—28no—
108C4×D29—216no—
109#2856—291nomaximal, Sylow
110C22×C29—44no—
111C22×C29—42no—
112C22×C29—42no—
113C22×C29—44no—
114C22×C29—42no—
115C22×C29C22×D411yescharacteristic
116C22×C29C22×D412yes—
117C22×C29C22×D411yescharacteristic
118C22×C29C2×#4111yescharacteristic
119C22×C29C2×#4112yes—
120C2×D29—24no—
121C2×D29—22no—
122C2×D29—24no—
123C2×D29—22no—
124C2×D29—42no—
125C2×D29—44no—
126C2×D29—42no—
127C2×D29—44no—
128C2×D29—42no—
129C2×D29#12714yescomplemented
130C4×C29—24no—
131C4×C29—42no—
132C4×C29#12114yescomplemented
133Dic29—22no—
134Dic29—42no—
135Dic29#12118yescomplemented
136C2×#127—291no—
137C2×#121—291no—
138C2×#121—291no—
139C2×#121—294no—
140C22×C42—291no—
141#390—294no—
142#390—291no—
143#390D2911yescharacteristic, complemented
144#384—291no—
145C2×C29—82no—
146C2×C29C2×#12112yes—
147C2×C29C2×#12111yescharacteristic
148C2×C29C2×#12711yescharacteristic
149C2×C29#53212yes—
150C2×C29#53211yescharacteristic
151D29—24no—
152D29—82no—
153C22×D4—292no—
154C22×D4—294no—
155C22×D4—292no—
156C22×D4—291no—
157C22×D4C2×D2911yescharacteristic
158C23×C4—292no—
159C23×C4—291no—
160#127—584no—
161#121—5816no—
162#121—584no—
163#121—584no—
164C2×C4⋊C4—294no—
165C2×C4⋊C4—291no—
166C2×C4⋊C4C2×D2911yescharacteristic
167C2×C22⋊C4—292no—
168C2×C22⋊C4—292no—
169C2×C22⋊C4—294no—
170C2×C22⋊C4C2×D2914yes—
171C2×C42—291no—
172C2×C42—292no—
173C2×C42—584no—
174C2×C42C2×D2911yescharacteristic
175C29#285611yescharacteristic, complemented, Sylow
176C24—291no—
177C24—292no—
178C24C22×D2912yes—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—588no—
183C2×D4—584no—
184C2×D4—588no—
185C2×D4—588no—
186C2×D4—588no—
187C2×D4—588no—
188C2×D4—584no—
189C2×D4—582no—
190C2×D4—582no—
191C2×D4—582no—
192C22×C4—292no—
193C22×C4—291no—
194C22×C4—298no—
195C22×C4—294no—
196C22×C4—292no—
197C22×C4—584no—
198C22×C4—584no—
199C22×C4—582no—
200C22×C4—582no—
201C22×C4C22×D2912yes—
202C22×C4C22×D2912yes—
203C22×C4C22×D2911yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—588no—
206C4⋊C4—582no—
207C22⋊C4—28no—
208C22⋊C4—584no—
209C22⋊C4—584no—
210C22⋊C4—588no—
211C42—22no—
212C42—294no—
213C42—584no—
214C23—22no—
215C23—24no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—294no—
220C23—292no—
221C23—581no—
222C23—581no—
223C23—582no—
224C23—582no—
225C23—584no—
226C23—582no—
227C23—582no—
228C23—584no—
229C23C23×D2911yescharacteristic
230D4—44no—
231D4—1168no—
232D4—11616no—
233D4—1168no—
234D4—1164no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—294no—
240C2×C4—294no—
241C2×C4—294no—
242C2×C4—292no—
243C2×C4—5816no—
244C2×C4—581no—
245C2×C4—581no—
246C2×C4—588no—
247C2×C4—584no—
248C2×C4—582no—
249C2×C4#9080512yes—
250C2×C4D4×D2914yes—
251C2×C4D4×D2912yes—
252C2×C4D4×D2912yes—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—294no—
259C22—584no—
260C22—582no—
261C22—584no—
262C22—582no—
263C22—1164no—
264C22—1162no—
265C22—1162no—
266C22—1164no—
267C22—1162no—
268C22C2×#9080511yescharacteristic
269C22C2×D4×D2911yescharacteristic
270C22C2×D4×D2912yes—
271C22C2×D4×D2911yescharacteristic
272C22#41×D2912yes—
273C4—24no—
274C4—42no—
275C4—298no—
276C4—582no—
277C4—1162no—
278C4#1174524614yescomplemented
279C2—82no—
280C2—584no—
281C2—2322no—
282C2#121×D2912yes—
283C2C2×#1174524611yescharacteristic
284C2#127×D2911yescharacteristic
285C2#1427171012yes—
286C2#1427171211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#390C2
hypercenter#390D29
hypocenterC29#2856
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#11745302C2semidirect
2. C2×#11745257C2semidirect
3. C2×#11745246C2semidirect
4. C2×#11745300C2semidirect
5. C2×C42×D29C2semidirect
6. #14271065C2semidirect
7. #14271063C2semidirect
8. #14271032C2semidirect
9. C29×#390C2semidirect
58. C2×C4×D29D4semidirect
59. C2×C4×D29D4semidirect
69. C4×Dic29D4semidirect
129. C2×D29#127semidirect
132. C4×C29#121semidirect
135. Dic29#121semidirect
143. #390D29semidirect
175. C29#2856semidirect
278. C4#11745246semidirect

Maximal subgroups

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

subgroupindex
C2×#117453022
C2×#117452572
C2×#117452462
C2×#117453002
C2×C42×D292
#142710652
#142710632
#142710322
C29×#3902
#285629