Finite Group Databasebeta

← #16822731

Subgroups — #16822731

Order 3712.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×Dic29C211yesmaximal, characteristic, complemented
2C2×#11745243C212yesmaximal, complemented
3C2×#11745254C211yesmaximal, characteristic, complemented
4#14271093C211yesmaximal, characteristic, complemented
5#14271090C211yesmaximal, characteristic, complemented
6#14271086C212yesmaximal, complemented
7#14271078C212yesmaximal, complemented
8#14271071C211yesmaximal, characteristic
9#14271014C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic29C212yesmaximal, complemented
11C29×#394C211yesmaximal, characteristic, complemented
12C22×D4×C29C2211yescharacteristic
13C22×C22⋊D29C2212yes—
14C23×Dic29C2212yes—
15C22×C4×D29C2211yescharacteristic
16C2×C4⋊C4×C29C2211yescharacteristic
17C2×C22⋊C4×C29C2212yes—
18C2×C22⋊C4×C29C2212yes—
19C2×C22⋊Dic29C2212yes—
20C2×C22⋊Dic29C2212yes—
21D4×Dic29—24no—
22C2×#90781C2211yescharacteristic
23C2×#90781C2211yescharacteristic
24C2×#90781C2212yes—
25C2×C4⋊Dic29C2211yescharacteristic
26C2×C4⋊Dic29C2211yescharacteristic
27C2×#90779C2212yes—
28C2×#90779C2212yes—
29C2×C4×Dic29C2211yescharacteristic
30C2×C4×Dic29C2212yes—
31C2×C4×Dic29C2211yescharacteristic
32#11745254—24no—
33#11745243—28no—
34C29×#95C2211yescharacteristic
35#11745179C2212yes—
36#11745179C2212yes—
37#11745179C2211yescharacteristic
38C24×C29C2312yes—
39C23×D29C2311yescharacteristic
40C2×D4×C29—24no—
41C2×D4×C29—22no—
42C22×C4×C29C2312yes—
43C22×C4×C29C2311yescharacteristic
44C22×C4×C29C2311yescharacteristic
45C22×C4×C29C2311yescharacteristic
46C2×C22⋊D29—24no—
47C2×C22⋊D29—24no—
48C2×C22⋊D29—24no—
49C22×Dic29—24no—
50C22×Dic29—24no—
51C22×Dic29—24no—
52C22×Dic29C2311yescharacteristic
53C22×Dic29C2312yes—
54C22×Dic29C2311yescharacteristic
55C22×Dic29C2312yes—
56C22×Dic29C2311yescharacteristic
57C2×C4×D29—22no—
58C2×C4×D29—22no—
59C2×C4×D29D414yescomplemented
60C4⋊C4×C29—22no—
61C22⋊C4×C29—24no—
62C22⋊C4×C29—24no—
63C22⋊Dic29—24no—
64C22⋊Dic29—24no—
65#90781—24no—
66#90781—22no—
67#90781—22no—
68C4⋊Dic29—22no—
69C4⋊Dic29—22no—
70#90779—24no—
71#90779—24no—
72C4×Dic29—24no—
73C4×Dic29—22no—
74C4×Dic29D414yescomplemented
75C23×C29—22no—
76C23×C29—22no—
77C23×C29—22no—
78C23×C29—22no—
79C23×C29—22no—
80C23×C29—22no—
81C23×C29#4114yes—
82C23×C29C2411yescharacteristic
83C22×D29—21no—
84C22×D29—21no—
85C22×D29—22no—
86C22×D29—22no—
87C22×D29C2×D412yes—
88D4×C29—28no—
89C2×C4×C29—21no—
90C2×C4×C29—21no—
91C2×C4×C29—22no—
92C2×C4×C29—22no—
93C2×C4×C29—21no—
94C2×C4×C29—22no—
95C2×C4×C29—22no—
96C2×C4×C29—22no—
97C2×C4×C29C2×D412yes—
98C2×C4×C29C2×D412yes—
99C22⋊D29—216no—
100C2×Dic29—22no—
101C2×Dic29—22no—
102C2×Dic29—22no—
103C2×Dic29—21no—
104C2×Dic29—22no—
105C2×Dic29—22no—
106C2×Dic29—22no—
107C2×Dic29—22no—
108C2×Dic29—48no—
109C2×Dic29C2×D412yes—
110C2×Dic29C2×D412yes—
111C2×Dic29C2×D412yes—
112C2×Dic29#4114yes—
113C2×Dic29#4112yes—
114C4×D29—28no—
115#2392—291nomaximal, Sylow
116C22×C29—24no—
117C22×C29—24no—
118C22×C29—24no—
119C22×C29—42no—
120C22×C29—42no—
121C22×C29—42no—
122C22×C29—42no—
123C22×C29C22×D411yescharacteristic
124C22×C29C22×D411yescharacteristic
125C22×C29C2×#4112yes—
126C22×C29C2×#4111yescharacteristic
127C22×C29C2×#4112yes—
128C2×D29—22no—
129C2×D29—24no—
130C2×D29—41no—
131C2×D29—42no—
132C2×D29—41no—
133C4×C29—22no—
134C4×C29—22no—
135C4×C29—41no—
136C4×C29—42no—
137Dic29—24no—
138Dic29—22no—
139Dic29—42no—
140Dic29—41no—
141Dic29#12114yes—
142C2×C4×D4—291no—
143C2×C4×D4—292no—
144C2×#117—291no—
145#398—291no—
146#397—291no—
147#396—292no—
148#394—292no—
149#394D2911yescharacteristic, complemented
150#392—291no—
151C4×C4⋊C4—291no—
152C4×C22⋊C4—292no—
153C2×C29—44no—
154C2×C29C2×#12111yescharacteristic
155C2×C29#52511yescharacteristic
156C2×C29#52512yes—
157C2×C29#52912yes—
158C2×C29#53211yescharacteristic
159D29—42no—
160C22×D4—292no—
161C22×D4C2×D2911yescharacteristic
162C23×C4—291no—
163C23×C4—292no—
164C4×D4—584no—
165C4×D4—588no—
166#117—584no—
167C2×C4⋊C4—291no—
168C2×C4⋊C4—292no—
169C2×C4⋊C4—291no—
170C2×C4⋊C4—292no—
171C2×C4⋊C4C2×D2911yescharacteristic
172C2×C22⋊C4—292no—
173C2×C22⋊C4—292no—
174C2×C22⋊C4—291no—
175C2×C22⋊C4—291no—
176C2×C22⋊C4—292no—
177C2×C22⋊C4C2×D2912yes—
178C2×C22⋊C4C2×D2912yes—
179C2×C42—291no—
180C2×C42—291no—
181C2×C42—292no—
182#95—292no—
183#95—292no—
184#95—291no—
185#95C2×D2911yescharacteristic
186C29#239211yescharacteristic, complemented, Sylow
187C24—291no—
188C24C22×D2912yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—584no—
192C2×D4—584no—
193C2×D4—584no—
194C22×C4—291no—
195C22×C4—292no—
196C22×C4—291no—
197C22×C4—294no—
198C22×C4—291no—
199C22×C4—292no—
200C22×C4—582no—
201C22×C4—582no—
202C22×C4—584no—
203C22×C4—584no—
204C22×C4—584no—
205C22×C4C22×D2911yescharacteristic
206C22×C4C22×D2912yes—
207C22×C4C22×D2911yescharacteristic
208C22×C4C22×D2911yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—582no—
211C4⋊C4—584no—
212C4⋊C4—584no—
213C4⋊C4—582no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—582no—
217C22⋊C4—584no—
218C22⋊C4—584no—
219C22⋊C4—584no—
220C22⋊C4—582no—
221C42—294no—
222C42—582no—
223C42—584no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—292no—
231C23—581no—
232C23—582no—
233C23—581no—
234C23—582no—
235C23#9080714yes—
236C23C23×D2911yescharacteristic
237D4—28no—
238D4—5816no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—292no—
248C2×C4—292no—
249C2×C4—292no—
250C2×C4—292no—
251C2×C4—294no—
252C2×C4—588no—
253C2×C4—582no—
254C2×C4—582no—
255C2×C4—582no—
256C2×C4—581no—
257C2×C4—582no—
258C2×C4—582no—
259C2×C4—582no—
260C2×C4—582no—
261C2×C4—1168no—
262C2×C4D4×D2912yes—
263C2×C4D4×D2912yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—582no—
272C22—584no—
273C22—1161no—
274C22—1162no—
275C22—1161no—
276C22C2×D4×D2911yescharacteristic
277C22C2×D4×D2911yescharacteristic
278C22C2×#9080712yes—
279C22#41×D2911yescharacteristic
280C22#41×D2912yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—294no—
286C4—584no—
287C4—582no—
288C4—1162no—
289C4—1161no—
290C2—44no—
291C2—1162no—
292C2#121×D2911yescharacteristic
293C2#1427166911yescharacteristic
294C2#1427167312yes—
295C2#1427169012yes—
296C2#1427171211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#394C2
prime radical#14271071C2
hypercenter#394D29
hypocenterC29#2392
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic29C2semidirect
2. C2×#11745243C2semidirect
3. C2×#11745254C2semidirect
4. #14271093C2semidirect
5. #14271090C2semidirect
6. #14271086C2semidirect
7. #14271078C2semidirect
9. #14271014C2semidirect
10. C22⋊C4×Dic29C2semidirect
11. C29×#394C2semidirect
59. C2×C4×D29D4semidirect
74. C4×Dic29D4semidirect
149. #394D29semidirect
186. C29#2392semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic292
C2×#117452432
C2×#117452542
#142710932
#142710902
#142710862
#142710782
#142710712
#142710142
C22⋊C4×Dic292
C29×#3942
#239229