Finite Group Databasebeta

← #16822875

Subgroups — #16822875

Order 3712.

786 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 161 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×#11745248C211yesmaximal, characteristic, complemented
2C2×#11745244C211yesmaximal, characteristic
3C2×#11745239C211yesmaximal, characteristic, complemented
4C22×#90781C211yesmaximal, characteristic, complemented
5#14271085C212yesmaximal, complemented
6#14271081C212yesmaximal, complemented
7#14271064C212yesmaximal, complemented
8#14271055C212yesmaximal, complemented
9#14271042C211yesmaximal, characteristic, complemented
10#14271018C211yesmaximal, characteristic, complemented
11C29×#385C211yesmaximal, characteristic, complemented
12C24×D29C2211yescharacteristic
13C23×C4×C29C2211yescharacteristic
14C22×C22⋊D29C2211yescharacteristic
15C23×Dic29C2211yescharacteristic
16C22×D116C2212yes—
17C2×C4⋊C4×C29C2212yescomplemented
18C2×C22⋊C4×C29C2211yescharacteristic
19C2×C22⋊C4×C29C2211yescharacteristic
20C2×C22⋊Dic29C2211yescharacteristic
21C2×#90781—24no—
22C2×#90781C2212yes—
23C2×#90781C2212yes—
24C2×#90781C2212yes—
25C2×#90781C2212yes—
26C2×#90781C2212yes—
27C2×#90781C2211yescharacteristic, complemented
28C2×#90781C2211yescharacteristic, complemented
29C2×#90781C2212yescomplemented
30C2×C4⋊Dic29C2211yescharacteristic, complemented
31C2×C4⋊Dic29C2211yescharacteristic, complemented
32C2×#90779C2212yescomplemented
33C2×C4×Dic29C2212yescomplemented
34#11745248—24no—
35#11745244—24no—
36#11745239—24no—
37C29×#95C2212yescomplemented
38#11745179C2212yescomplemented
39C24×C29C2311yescharacteristic
40C23×D29—22no—
41C23×D29—24no—
42C23×D29—22no—
43C23×D29—22no—
44C23×D29—42no—
45C23×D29C2311yescharacteristic
46C23×D29C2312yes—
47C22×C4×C29—22no—
48C22×C4×C29—42no—
49C22×C4×C29C2311yescharacteristic
50C22×C4×C29C2311yescharacteristic
51C22×C4×C29C2312yes—
52C22×C4×C29C2312yes—
53C2×C22⋊D29—22no—
54C2×C22⋊D29—22no—
55C2×C22⋊D29—22no—
56C22×Dic29—22no—
57C22×Dic29—22no—
58C22×Dic29D414yescomplemented
59C22×Dic29C2312yes—
60C22×Dic29C2311yescharacteristic
61C22×Dic29C2311yescharacteristic
62C22×Dic29C2311yescharacteristic
63C2×D116—24no—
64C2×D116—24no—
65C2×D116—42no—
66C4⋊C4×C29—42no—
67C22⋊C4×C29—22no—
68C22⋊C4×C29D414yes—
69C22⋊Dic29—22no—
70#90781—24no—
71#90781—24no—
72#90781—24no—
73#90781—24no—
74#90781—22no—
75#90781—22no—
76#90781—42no—
77#90781—42no—
78C4⋊Dic29—22no—
79C4⋊Dic29—22no—
80#90779—24no—
81C4×Dic29—24no—
82C23×C29—22no—
83C23×C29—22no—
84C23×C29C2×D412yes—
85C23×C29C2×D412yes—
86C23×C29#4112yes—
87C23×C29C2411yescharacteristic
88C22×D29—28no—
89C22×D29—22no—
90C22×D29—22no—
91C22×D29—22no—
92C22×D29—22no—
93C22×D29—22no—
94C22×D29—22no—
95C22×D29—21no—
96C22×D29—21no—
97C22×D29—22no—
98C22×D29—21no—
99C22×D29—22no—
100C22×D29—48no—
101C22×D29—48no—
102C22×D29—44no—
103C22×D29—44no—
104C22×D29#4114yes—
105C2×C4×C29—22no—
106C2×C4×C29—22no—
107C2×C4×C29—21no—
108C2×C4×C29—21no—
109C2×C4×C29—22no—
110C2×C4×C29—22no—
111C2×C4×C29—22no—
112C2×C4×C29—22no—
113C2×C4×C29—22no—
114C2×C4×C29—44no—
115C2×C4×C29C2×D414yes—
116C22⋊D29—44no—
117C2×Dic29—28no—
118C2×Dic29—21no—
119C2×Dic29—21no—
120C2×Dic29—21no—
121C2×Dic29—21no—
122C2×Dic29—22no—
123C2×Dic29—21no—
124C2×Dic29—21no—
125C2×Dic29—22no—
126C2×Dic29—21no—
127C2×Dic29—22no—
128C2×Dic29C2×D412yes—
129C2×Dic29C2×D412yes—
130D116—48no—
131#2411—291nomaximal, Sylow
132C22×C29—22no—
133C22×C29—22no—
134C22×C29—24no—
135C22×C29—24no—
136C22×C29#12114yes—
137C22×C29C22×D411yescharacteristic
138C22×C29C22×D411yescharacteristic
139C22×C29C2×#4111yescharacteristic
140C22×C29C2×#4112yes—
141C22×C29H(5,𝔽2)12yes—
142C2×D29—24no—
143C2×D29—24no—
144C2×D29—24no—
145C2×D29—416no—
146C2×D29—42no—
147C2×D29—42no—
148C2×D29—42no—
149C2×D29—42no—
150C2×D29—41no—
151C2×D29—41no—
152C2×D29—41no—
153C2×D29—42no—
154C2×D29—42no—
155C4×C29—24no—
156C4×C29—41no—
157C4×C29—41no—
158C4×C29—42no—
159Dic29—22no—
160Dic29—22no—
161Dic29—41no—
162Dic29—42no—
163C2×#123—291no—
164C2×#120—291no—
165C2×#117—291no—
166C22×C22⋊C4—291no—
167#396—292no—
168#394—292no—
169#390—292no—
170#388—292no—
171#386—291no—
172#385D2911yescharacteristic, complemented
173#379—291no—
174C2×C29—24no—
175C2×C29C2×#12111yescharacteristic
176C2×C29#52511yescharacteristic
177C2×C29#53511yescharacteristic
178C2×C29#54612yes—
179C2×C29#55112yes—
180D29—44no—
181D29—81no—
182C25—291no—
183C22×D4—291no—
184C22×D4—292no—
185C23×C4—291no—
186C23×C4C2×D2911yescharacteristic
187#123—584no—
188#120—584no—
189#117—584no—
190C2×C4⋊C4—292no—
191C2×C4⋊C4—291no—
192C2×C4⋊C4—291no—
193C2×C4⋊C4C2×D2912yescomplemented
194C2×C22⋊C4—291no—
195C2×C22⋊C4—292no—
196C2×C22⋊C4—292no—
197C2×C22⋊C4—292no—
198C2×C22⋊C4—292no—
199C2×C22⋊C4—291no—
200C2×C22⋊C4—291no—
201C2×C22⋊C4—292no—
202C2×C22⋊C4—292no—
203C2×C22⋊C4—584no—
204C2×C22⋊C4C2×D2911yescharacteristic
205C2×C22⋊C4C2×D2911yescharacteristic
206C2×C42—292no—
207#95—292no—
208#95C2×D2912yescomplemented
209C29#241111yescharacteristic, complemented, Sylow
210C24—292no—
211C24—291no—
212C24—584no—
213C24—582no—
214C24—582no—
215C24—582no—
216C24—1162no—
217C24C22×D2911yescharacteristic
218C2×D4—584no—
219C2×D4—582no—
220C2×D4—582no—
221C2×D4—582no—
222C2×D4—584no—
223C2×D4—1162no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—294no—
227C22×C4—291no—
228C22×C4—291no—
229C22×C4—291no—
230C22×C4—292no—
231C22×C4—582no—
232C22×C4—582no—
233C22×C4C22×D2912yes—
234C22×C4C22×D2911yescharacteristic
235C22×C4C22×D2911yescharacteristic
236C22×C4C22×D2912yes—
237C4⋊C4—42no—
238C4⋊C4—582no—
239C4⋊C4—582no—
240C4⋊C4—584no—
241C22⋊C4—22no—
242C22⋊C4—584no—
243C22⋊C4—584no—
244C22⋊C4—582no—
245C22⋊C4—582no—
246C22⋊C4—584no—
247C22⋊C4—582no—
248C22⋊C4—584no—
249C22⋊C4—1162no—
250C22⋊C4—1162no—
251C22⋊C4D11614yes—
252C42—584no—
253C23—22no—
254C23—22no—
255C23—294no—
256C23—588no—
257C23—582no—
258C23—582no—
259C23—582no—
260C23—582no—
261C23—582no—
262C23—582no—
263C23—581no—
264C23—581no—
265C23—582no—
266C23—581no—
267C23—582no—
268C23—1168no—
269C23—1168no—
270C23—1164no—
271C23—1164no—
272C23C2×D11612yes—
273C23D4×D2912yes—
274C23#9080912yes—
275C23C23×D2911yescharacteristic
276D4—1168no—
277D4—1164no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—292no—
289C2×C4—292no—
290C2×C4—581no—
291C2×C4—581no—
292C2×C4—581no—
293C2×C4—581no—
294C2×C4—588no—
295C2×C4—582no—
296C2×C4—581no—
297C2×C4—581no—
298C2×C4—582no—
299C2×C4—581no—
300C2×C4—582no—
301C2×C4C2×D11614yes—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—584no—
307C22—584no—
308C22—584no—
309C22—11616no—
310C22—1162no—
311C22—1162no—
312C22—1162no—
313C22—1162no—
314C22—1162no—
315C22—1162no—
316C22—1161no—
317C22—1161no—
318C22—1161no—
319C22#1174525714yes—
320C22C22×D11611yescharacteristic
321C22C2×D4×D2911yescharacteristic
322C22#1174535912yes—
323C22C2×#9080911yescharacteristic
324C22#41×D2912yes—
325C4—24no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—582no—
330C4—582no—
331C4—1162no—
332C4—1161no—
333C2—24no—
334C2—1164no—
335C2—2321no—
336C2C2×#1174525711yescharacteristic
337C2#1427167111yescharacteristic
338C2#1427172811yescharacteristic
339C2#1427180812yes—
340C2#1427184012yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#385C2
prime radicalC2×#11745244C2
hypercenter#385D29
hypocenterC29#2411
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#11745248C2semidirect
3. C2×#11745239C2semidirect
4. C22×#90781C2semidirect
5. #14271085C2semidirect
6. #14271081C2semidirect
7. #14271064C2semidirect
8. #14271055C2semidirect
9. #14271042C2semidirect
10. #14271018C2semidirect
11. C29×#385C2semidirect
17. C2×C4⋊C4×C29C22semidirect
27. C2×#90781C22semidirect
28. C2×#90781C22semidirect
29. C2×#90781C22semidirect
30. C2×C4⋊Dic29C22semidirect
31. C2×C4⋊Dic29C22semidirect
32. C2×#90779C22semidirect
33. C2×C4×Dic29C22semidirect
37. C29×#95C22semidirect
38. #11745179C22semidirect
58. C22×Dic29D4semidirect
172. #385D29semidirect
193. C2×C4⋊C4C2×D29semidirect
208. #95C2×D29semidirect
209. C29#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#117452482
C2×#117452442
C2×#117452392
C22×#907812
#142710852
#142710812
#142710642
#142710552
#142710422
#142710182
C29×#3852
#241129