Finite Group Databasebeta

← #16829123

Subgroups — #16829123

Order 3712.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#11745271C211yesmaximal, characteristic, complemented
2C2×M16×D29C211yesmaximal, characteristic, complemented
3#14271675C211yesmaximal, characteristic
4C2×#11745259C211yesmaximal, characteristic, complemented
5#14271422C212yesmaximal, complemented
6#14271410C212yesmaximal, complemented
7#14271405C212yesmaximal, complemented
8#14271393C212yesmaximal, complemented
9#14271171C211yesmaximal, characteristic, complemented
10#14271162C211yesmaximal, characteristic, complemented
11C29×#428C211yesmaximal, characteristic, complemented
12C2×#90805C2211yescharacteristic
13C22×C4×D29C2211yescharacteristic
14C22×Dic58C2211yescharacteristic
15C2×M16×C29C2211yescharacteristic
16C29×#117C2211yescharacteristic
17C2×C4⋊C4×C29C2211yescharacteristic
18C2×#90781—21no—
19C2×C4⋊Dic29—21no—
20C2×#90777C2211yescharacteristic
21#11745271—24no—
22M16×D29—24no—
23C2×Dic116C2212yescomplemented
24C2×#90773C2212yescomplemented
25C2×#90772C2212yes—
26C2×C8×D29C2212yes—
27#11745259—21no—
28#11745259—21no—
29#11745259C2212yes—
30#11745259C2211yescharacteristic
31#11745259C2212yes—
32#11745259C2211yescharacteristic
33#11745257C2211yescharacteristic
34#11745257C2211yescharacteristic
35#11745248—21no—
36#11745244—21no—
37C4×D116—21no—
38C4×D116—21no—
39C29×#107C2212yescomplemented
40C29×#106C2212yescomplemented
41#11745158C2212yescomplemented
42#11745158C2212yescomplemented
43#11745158C2212yescomplemented
44#11745157C2212yescomplemented
45C23×D29D411yescharacteristic
46C22×C4×C29—21no—
47C22×C4×C29C2311yescharacteristic
48C2×C22⋊D29—21no—
49C22×Dic29—21no—
50C22×Dic29D411yescharacteristic
51#90805—22no—
52#90805—22no—
53C2×D116—21no—
54C2×D116—21no—
55C2×D116C2311yescharacteristic
56C2×C4×D29—22no—
57C2×C4×D29—22no—
58C2×C4×D29—21no—
59C2×C4×D29D411yescharacteristic
60C2×C4×D29D411yescharacteristic
61C2×C4×D29C2311yescharacteristic
62C2×C4×D29C2311yescharacteristic
63C2×Dic58—22no—
64C2×Dic58—22no—
65C2×Dic58—21no—
66C2×Dic58C2312yes—
67C2×Dic58C2311yescharacteristic
68M16×C29D414yescomplemented
69C2×C8×C29C2312yes—
70C4⋊C4×C29—21no—
71C4⋊C4×C29C2312yes—
72C4⋊C4×C29C2311yescharacteristic
73C4⋊C4×C29C2311yescharacteristic
74C22⋊C4×C29—21no—
75C42×C29—21no—
76#90781—22no—
77#90781—22no—
78#90781—21no—
79#90781—21no—
80#90781—21no—
81#90781—21no—
82C4⋊Dic29—22no—
83C4⋊Dic29—22no—
84C4⋊Dic29—21no—
85C4⋊Dic29—21no—
86#90777—22no—
87C2×#6597C2312yes—
88Dic116—24no—
89#90773—24no—
90#90772—24no—
91C8×D29—24no—
92C23×C29C2×D411yescharacteristic
93C22×D29—22no—
94C22×D29—22no—
95C22×D29—21no—
96C22×D29—41no—
97C22×D29C2×D411yescharacteristic
98C22×D29C2×D411yescharacteristic
99C2×C4×C29—22no—
100C2×C4×C29—21no—
101C2×C4×C29—21no—
102C2×C4×C29—42no—
103C2×C4×C29C2×D412yes—
104C2×C4×C29C2×D411yescharacteristic
105C2×C4×C29#4112yes—
106C2×C4×C29C2411yescharacteristic
107C22⋊D29—42no—
108C2×Dic29—22no—
109C2×Dic29—22no—
110C2×Dic29—21no—
111C2×Dic29—42no—
112C2×Dic29C2×D411yescharacteristic
113C2×Dic29C2×D411yescharacteristic
114D116—21no—
115D116—21no—
116D116—42no—
117D116—42no—
118C4×D29—22no—
119C4×D29—22no—
120C4×D29—21no—
121C4×D29—21no—
122C4×D29—21no—
123C4×D29—21no—
124C4×D29—42no—
125Dic58—22no—
126Dic58—21no—
127Dic58—22no—
128Dic58—21no—
129Dic58—42no—
130C8×C29C2×D414yes—
131#6597—22no—
132#3056—291nomaximal, Sylow
133C22×C29—21no—
134C22×C29—21no—
135C22×C29#12112yes—
136C22×C29C22×D411yescharacteristic
137C2×D29—21no—
138C2×D29—21no—
139C2×D29—21no—
140C2×D29—21no—
141C2×D29—21no—
142C2×D29—44no—
143C2×D29—41no—
144C2×D29—41no—
145C2×D29—41no—
146C2×D29#13712yescomplemented
147C4×C29—41no—
148C4×C29—41no—
149C4×C29—42no—
150C4×C29#12112yes—
151C4×C29C22×D411yescharacteristic
152C4×C29C2×#4111yescharacteristic
153Dic29—21no—
154Dic29—21no—
155Dic29—42no—
156Dic29—41no—
157C2×#137—291no—
158C22×M16—291no—
159#525—291no—
160C2×C22⋊Q8—291no—
161#470—292no—
162#468—292no—
163#467—292no—
164#465—292no—
165#428D2911yescharacteristic, complemented
166#419—291no—
167#417—291no—
168C2×C29—22no—
169C2×C29C2×#12111yescharacteristic
170C2×C29C2×#13711yescharacteristic
171C2×C29#57511yescharacteristic
172D29—22no—
173D29—41no—
174D29—81no—
175C2×#41—291no—
176C22×Q8—291no—
177C23×C4—291no—
178#137—584no—
179C2×Q16—292no—
180C2×SD16—292no—
181C2×M16—291no—
182C2×M16—292no—
183C2×M16—584no—
184C2×M16C2×D2911yescharacteristic
185C22×C8—292no—
186#123—581no—
187C22⋊Q8—292no—
188C22⋊Q8—292no—
189C22⋊Q8—291no—
190C22⋊Q8—291no—
191C22⋊Q8—581no—
192C22⋊Q8—581no—
193#121—291no—
194#121—291no—
195#120—581no—
196C4×D4—581no—
197C4×D4—581no—
198#117C2×D2911yescharacteristic
199C2×C4⋊C4—581no—
200C2×C4⋊C4C2×D2911yescharacteristic
201C2×C22⋊C4—581no—
202#107C2×D2912yescomplemented
203#106C2×D2912yescomplemented
204#103—292no—
205#103—292no—
206#103—292no—
207#102—292no—
208C29#305611yescharacteristic, complemented, Sylow
209C24—291no—
210#41—582no—
211#41—582no—
212C2×Q8—292no—
213C2×Q8—291no—
214C2×Q8—581no—
215C2×Q8—582no—
216C2×Q8—582no—
217C2×D4—291no—
218C2×D4—581no—
219C2×D4—581no—
220C2×D4—581no—
221C22×C4—21no—
222C22×C4—291no—
223C22×C4—291no—
224C22×C4—291no—
225C22×C4—291no—
226C22×C4—291no—
227C22×C4—582no—
228C22×C4—582no—
229C22×C4—581no—
230C22×C4—581no—
231C22×C4C22×D2911yescharacteristic
232Q16—584no—
233SD16—584no—
234M16—582no—
235M16—584no—
236M16D11614yescomplemented
237C2×C8—292no—
238C2×C8—584no—
239C2×C8C22×D2912yes—
240C4⋊C4—21no—
241C4⋊C4—582no—
242C4⋊C4—582no—
243C4⋊C4—581no—
244C4⋊C4—581no—
245C4⋊C4C22×D2912yes—
246C4⋊C4C22×D2911yescharacteristic
247C4⋊C4C22×D2911yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—581no—
250C22⋊C4—581no—
251C22⋊C4—582no—
252C22⋊C4—582no—
253C22⋊C4—581no—
254C22⋊C4—581no—
255C42—21no—
256C23—291no—
257C23—291no—
258C23—582no—
259C23—582no—
260C23—581no—
261C23—1161no—
262C23D4×D2911yescharacteristic
263Q8—582no—
264Q8—581no—
265Q8—582no—
266Q8—581no—
267Q8—1162no—
268D4—581no—
269D4—581no—
270D4—1162no—
271D4—1162no—
272D4—1162no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—291no—
278C2×C4—291no—
279C2×C4—582no—
280C2×C4—581no—
281C2×C4—581no—
282C2×C4—581no—
283C2×C4—581no—
284C2×C4—582no—
285C2×C4—582no—
286C2×C4—581no—
287C2×C4—582no—
288C2×C4—1162no—
289C2×C4—1162no—
290C2×C4C2×D11612yes—
291C2×C4D4×D2911yescharacteristic
292C2×C4#9080912yes—
293C2×C4C23×D2911yescharacteristic
294C8—582no—
295C8C2×D11614yes—
296C22—21no—
297C22—21no—
298C22—292no—
299C22—581no—
300C22—581no—
301C22—581no—
302C22—581no—
303C22—581no—
304C22—1164no—
305C22—1161no—
306C22—1161no—
307C22—1161no—
308C22#1174525712yes—
309C22C2×D4×D2911yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—581no—
314C4—581no—
315C4—1161no—
316C4—1162no—
317C4#1174525712yes—
318C4C22×D11611yescharacteristic
319C4C2×#9080911yescharacteristic
320C2—22no—
321C2—582no—
322C2—1161no—
323C2—2321no—
324C2C2×#1174525711yescharacteristic
325C2#137×D2911yescharacteristic
326C2#1427195911yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#428C2
socleC22×C29C22×D4
prime radical#14271675C2
hypercenter#428D29
hypocenterC29#3056
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#11745271C2semidirect
2. C2×M16×D29C2semidirect
4. C2×#11745259C2semidirect
5. #14271422C2semidirect
6. #14271410C2semidirect
7. #14271405C2semidirect
8. #14271393C2semidirect
9. #14271171C2semidirect
10. #14271162C2semidirect
11. C29×#428C2semidirect
23. C2×Dic116C22semidirect
24. C2×#90773C22semidirect
39. C29×#107C22semidirect
40. C29×#106C22semidirect
41. #11745158C22semidirect
42. #11745158C22semidirect
43. #11745158C22semidirect
44. #11745157C22semidirect
68. M16×C29D4semidirect
146. C2×D29#137semidirect
165. #428D29semidirect
202. #107C2×D29semidirect
203. #106C2×D29semidirect
208. C29#3056semidirect
236. M16D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117452712
C2×M16×D292
#142716752
C2×#117452592
#142714222
#142714102
#142714052
#142713932
#142711712
#142711622
C29×#4282
#305629