Finite Group Databasebeta

← #16822877

Subgroups — #16822877

Order 3712.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11745291C211yesmaximal, characteristic, complemented
2C2×#11745315C211yesmaximal, characteristic
3C2×#11745288C211yesmaximal, characteristic, complemented
4#14271086C212yesmaximal, complemented
5#14271079C212yesmaximal, complemented
6#14271063C212yesmaximal, complemented
7#14271057C212yesmaximal, complemented
8#14271042C211yesmaximal, characteristic, complemented
9#14271036C211yesmaximal, characteristic, complemented
10#14271019C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C29C211yesmaximal, characteristic, complemented
12C25×C29C2211yescharacteristic
13C23×C4×C29C2211yescharacteristic
14C23×C4×C29C2211yescharacteristic
15C22×C22⋊D29C2211yescharacteristic
16C22×C22⋊D29C2212yes—
17C2×C22⋊C4×C29—24no—
18C2×C22⋊C4×C29C2212yes—
19C2×C22⋊C4×C29C2212yes—
20#11745315—24no—
21C2×C22⋊Dic29C2211yescharacteristic
22C2×C22⋊Dic29C2211yescharacteristic
23C2×C22⋊Dic29C2212yes—
24C2×C22⋊Dic29C2212yes—
25C2×C22⋊Dic29C2212yes—
26C2×C22⋊Dic29C2211yescharacteristic
27#11745291—24no—
28C2×#90781C2211yescharacteristic, complemented
29C2×#90781C2211yescharacteristic, complemented
30C2×#90781C2212yescomplemented
31#11745288—24no—
32C2×C4⋊Dic29C2212yescomplemented
33C2×#90779C2211yescharacteristic, complemented
34C2×#90779C2211yescharacteristic, complemented
35C2×#90779C2212yescomplemented
36C2×C4×Dic29C2212yescomplemented
37#11745179C2212yescomplemented
38#11745179C2212yescomplemented
39C24×C29—22no—
40C24×C29—24no—
41C24×C29—22no—
42C24×C29—22no—
43C24×C29—42no—
44C24×C29C2312yes—
45C24×C29C2311yescharacteristic
46C23×D29C2311yescharacteristic
47C22×C4×C29—22no—
48C22×C4×C29—22no—
49C22×C4×C29—22no—
50C22×C4×C29—42no—
51C22×C4×C29D414yescomplemented
52C22×C4×C29C2312yes—
53C22×C4×C29C2311yescharacteristic
54C22×C4×C29C2311yescharacteristic
55C2×C22⋊D29—22no—
56C2×C22⋊D29—22no—
57C2×C22⋊D29—24no—
58C2×C22⋊D29—22no—
59C2×C22⋊D29—24no—
60C2×C22⋊D29—42no—
61C22×Dic29C2312yes—
62C22×Dic29C2311yescharacteristic
63C22×Dic29C2311yescharacteristic
64C22×Dic29C2311yescharacteristic
65C22×Dic29C2312yes—
66C22⋊C4×C29—24no—
67C22⋊C4×C29—24no—
68C22⋊Dic29—22no—
69C22⋊Dic29—22no—
70C22⋊Dic29—24no—
71C22⋊Dic29—24no—
72C22⋊Dic29—42no—
73C22⋊Dic29D414yes—
74#90781—22no—
75#90781—22no—
76#90781—42no—
77C4⋊Dic29—24no—
78#90779—22no—
79#90779—22no—
80#90779—42no—
81C4×Dic29—24no—
82C23×C29—22no—
83C23×C29—22no—
84C23×C29—28no—
85C23×C29—22no—
86C23×C29—22no—
87C23×C29—22no—
88C23×C29—22no—
89C23×C29—22no—
90C23×C29—22no—
91C23×C29—48no—
92C23×C29—48no—
93C23×C29—44no—
94C23×C29—44no—
95C23×C29C2×D412yes—
96C23×C29C2×D412yes—
97C23×C29#4114yes—
98C23×C29#4112yes—
99C23×C29C2411yescharacteristic
100C22×D29—21no—
101C22×D29—21no—
102C22×D29—22no—
103C22×D29—21no—
104C22×D29—22no—
105C2×C4×C29—28no—
106C2×C4×C29—21no—
107C2×C4×C29—21no—
108C2×C4×C29—21no—
109C2×C4×C29—21no—
110C2×C4×C29—22no—
111C2×C4×C29—22no—
112C2×C4×C29—22no—
113C2×C4×C29—44no—
114C2×C4×C29C2×D412yes—
115C2×C4×C29C2×D412yes—
116C22⋊D29—48no—
117C22⋊D29—44no—
118C2×Dic29—22no—
119C2×Dic29—22no—
120C2×Dic29—22no—
121C2×Dic29—21no—
122C2×Dic29—21no—
123C2×Dic29—21no—
124C2×Dic29—21no—
125C2×Dic29—22no—
126C2×Dic29—22no—
127C2×Dic29—21no—
128C2×Dic29—22no—
129C2×Dic29—22no—
130C2×Dic29C2×D414yes—
131#2411—291nomaximal, Sylow
132C22×C29—22no—
133C22×C29—22no—
134C22×C29—24no—
135C22×C29—24no—
136C22×C29—24no—
137C22×C29—24no—
138C22×C29—24no—
139C22×C29—416no—
140C22×C29—42no—
141C22×C29—42no—
142C22×C29—42no—
143C22×C29—42no—
144C22×C29#12114yes—
145C22×C29C22×D411yescharacteristic
146C22×C29C22×D411yescharacteristic
147C22×C29C2×#4112yes—
148C22×C29C2×#4111yescharacteristic
149C22×C29H(5,𝔽2)12yes—
150C2×D29—41no—
151C2×D29—41no—
152C2×D29—41no—
153C2×D29—42no—
154C2×D29—42no—
155C4×C29—22no—
156C4×C29—22no—
157C4×C29—42no—
158Dic29—24no—
159Dic29—41no—
160Dic29—41no—
161Dic29—41no—
162Dic29—42no—
163C2×#123—291no—
164C2×#120—291no—
165C2×#117—291no—
166C22×C22⋊C4D2911yescharacteristic, complemented
167#396—292no—
168#394—292no—
169#390—292no—
170#388—292no—
171#386—291no—
172#385—291no—
173#379—291no—
174C2×C29—24no—
175C2×C29—44no—
176C2×C29C2×#12111yescharacteristic
177C2×C29#52511yescharacteristic
178C2×C29#53511yescharacteristic
179C2×C29#54612yes—
180C2×C29#55112yes—
181D29—81no—
182C25C2×D2911yescharacteristic
183C22×D4—291no—
184C22×D4—292no—
185C23×C4C2×D2911yescharacteristic
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⋊C4—292no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—291no—
196C2×C22⋊C4—291no—
197C2×C22⋊C4—292no—
198C2×C22⋊C4—291no—
199C2×C22⋊C4—292no—
200C2×C22⋊C4—291no—
201C2×C22⋊C4—291no—
202C2×C22⋊C4—292no—
203C2×C22⋊C4—292no—
204C2×C22⋊C4C2×D2912yes—
205C2×C22⋊C4C2×D2912yes—
206C2×C42—292no—
207#95—292no—
208#95—292no—
209C29#241111yescharacteristic, complemented, Sylow
210C24—24no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—291no—
216C24C22×D2912yes—
217C24C22×D2911yescharacteristic
218C2×D4—584no—
219C2×D4—584no—
220C2×D4—582no—
221C2×D4—582no—
222C2×D4—582no—
223C2×D4—1162no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—292no—
229C22×C4—291no—
230C22×C4—291no—
231C22×C4—292no—
232C22×C4—291no—
233C22×C4C22⋊D2914yescomplemented
234C22×C4C22×D2912yes—
235C22×C4C22×D2911yescharacteristic
236C22×C4C22×D2911yescharacteristic
237C4⋊C4—584no—
238C4⋊C4—582no—
239C4⋊C4—582no—
240C4⋊C4—1162no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—294no—
244C22⋊C4—584no—
245C22⋊C4—584no—
246C22⋊C4—582no—
247C22⋊C4—582no—
248C22⋊C4—582no—
249C22⋊C4—582no—
250C22⋊C4—1162no—
251C22⋊C4—1162no—
252C42—584no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—582no—
267C23—581no—
268C23—581no—
269C23—582no—
270C23—581no—
271C23#9080514yes—
272C23D4×D2912yes—
273C23#9080712yes—
274C23C2×C22⋊D2912yes—
275C23C23×D2911yescharacteristic
276D4—1168no—
277D4—1164no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—294no—
288C2×C4—582no—
289C2×C4—582no—
290C2×C4—582no—
291C2×C4—582no—
292C2×C4—582no—
293C2×C4—581no—
294C2×C4—581no—
295C2×C4—582no—
296C2×C4—582no—
297C2×C4—581no—
298C2×C4—581no—
299C2×C4—581no—
300C2×C4C2×C22⋊D2912yes—
301C2×C4C2×C22⋊D2912yes—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—1162no—
315C22—1162no—
316C22—1161no—
317C22—1161no—
318C22—1161no—
319C22#1174530114yes—
320C22C2×#9080512yes—
321C22C2×D4×D2911yescharacteristic
322C22C2×#9080711yescharacteristic
323C22#1174535912yes—
324C22C22×C22⋊D2911yescharacteristic
325C4—22no—
326C4—22no—
327C4—42no—
328C4—584no—
329C4—1162no—
330C4—1161no—
331C4—1161no—
332C4—1161no—
333C2—24no—
334C2—44no—
335C2—2321no—
336C2C2×#1174530111yescharacteristic
337C2#1427167411yescharacteristic
338C2#1427173011yescharacteristic
339C2#1427180312yes—
340C2#1427183912yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC22×C22⋊C4×C29C2
prime radicalC2×#11745315C2
hypercenterC22×C22⋊C4D29
hypocenterC29#2411
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11745291C2semidirect
3. C2×#11745288C2semidirect
4. #14271086C2semidirect
5. #14271079C2semidirect
6. #14271063C2semidirect
7. #14271057C2semidirect
8. #14271042C2semidirect
9. #14271036C2semidirect
10. #14271019C2semidirect
11. C22×C22⋊C4×C29C2semidirect
28. C2×#90781C22semidirect
29. C2×#90781C22semidirect
30. C2×#90781C22semidirect
32. C2×C4⋊Dic29C22semidirect
33. C2×#90779C22semidirect
34. C2×#90779C22semidirect
35. C2×#90779C22semidirect
36. C2×C4×Dic29C22semidirect
37. #11745179C22semidirect
38. #11745179C22semidirect
51. C22×C4×C29D4semidirect
166. C22×C22⋊C4D29semidirect
209. C29#2411semidirect
233. C22×C4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452912
C2×#117453152
C2×#117452882
#142710862
#142710792
#142710632
#142710572
#142710422
#142710362
#142710192
C22×C22⋊C4×C292
#241129