Finite Group Databasebeta

← #16822821

Subgroups — #16822821

Order 3712.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×D116C211yesmaximal, characteristic
2C2×#11745256C212yesmaximal, complemented
3C2×C22⋊C4×D29C211yesmaximal, characteristic, complemented
4#14271081C212yesmaximal, complemented
5#14271080C212yesmaximal, complemented
6#14271041C212yesmaximal, complemented
7#14271036C211yesmaximal, characteristic, complemented
8#14271018C211yesmaximal, characteristic, complemented
9#14271018C211yesmaximal, characteristic, complemented
10#14271018C211yesmaximal, characteristic, complemented
11C29×#385C211yesmaximal, characteristic, complemented
12C24×D29C2211yescharacteristic
13C24×D29C2211yescharacteristic
14C23×C4×C29C2211yescharacteristic
15C22×D116—24no—
16C22×D116—28no—
17C22×D116C2212yes—
18C22×D116C2212yes—
19C22×C4×D29C2211yescharacteristic, complemented
20C22×C4×D29C2211yescharacteristic, complemented
21C2×C4⋊C4×C29C2212yescomplemented
22C2×C22⋊C4×C29C2211yescharacteristic
23C2×C22⋊C4×C29C2211yescharacteristic
24C2×C22⋊Dic29C2211yescharacteristic
25C2×C22⋊Dic29C2211yescharacteristic
26C2×#90781C2212yescomplemented
27C2×#90781C2212yescomplemented
28C2×#90781C2212yescomplemented
29C2×#90781C2211yescharacteristic, complemented
30C2×#90781C2211yescharacteristic, complemented
31C2×#90781C2212yescomplemented
32C2×#90781C2212yescomplemented
33C2×#90781C2212yescomplemented
34C2×#90779C2212yescomplemented
35#11745256—28no—
36C22⋊C4×D29—24no—
37C29×#95C2212yescomplemented
38#11745179C2212yescomplemented
39C24×C29C2311yescharacteristic
40C23×D29—22no—
41C23×D29—22no—
42C23×D29—22no—
43C23×D29—24no—
44C23×D29—24no—
45C23×D29—22no—
46C23×D29—22no—
47C23×D29—42no—
48C23×D29—42no—
49C23×D29D414yes—
50C23×D29C2311yescharacteristic
51C23×D29C2312yes—
52C23×D29C2311yescharacteristic
53C22×C4×C29—22no—
54C22×C4×C29—24no—
55C22×C4×C29C2312yes—
56C22×C4×C29C2311yescharacteristic
57C22×C4×C29C2311yescharacteristic
58C22×C4×C29C2312yes—
59C22×Dic29C2311yescharacteristic
60C22×Dic29C2312yes—
61C22×Dic29C2311yescharacteristic
62C2×D116—232no—
63C2×D116—24no—
64C2×D116—24no—
65C2×D116—24no—
66C2×D116—24no—
67C2×D116—24no—
68C2×D116D418yes—
69C2×C4×D29—22no—
70C2×C4×D29—22no—
71C2×C4×D29—24no—
72C2×C4×D29—42no—
73C4⋊C4×C29—24no—
74C22⋊C4×C29—22no—
75C22⋊C4×C29—22no—
76C22⋊Dic29—22no—
77C22⋊Dic29—22no—
78#90781—22no—
79#90781—22no—
80#90781—24no—
81#90781—24no—
82#90781—24no—
83#90781—42no—
84#90781—42no—
85#90781—42no—
86#90779—24no—
87C23×C29—22no—
88C23×C29—22no—
89C23×C29C2×D412yes—
90C23×C29#4112yes—
91C23×C29#4112yes—
92C23×C29C2411yescharacteristic
93C22×D29—21no—
94C22×D29—21no—
95C22×D29—22no—
96C22×D29—28no—
97C22×D29—28no—
98C22×D29—22no—
99C22×D29—22no—
100C22×D29—22no—
101C22×D29—21no—
102C22×D29—22no—
103C22×D29—21no—
104C22×D29—22no—
105C22×D29—22no—
106C22×D29—22no—
107C22×D29—22no—
108C22×D29—28no—
109C22×D29—44no—
110C22×D29—48no—
111C22×D29—48no—
112C22×D29—48no—
113C22×D29—48no—
114C22×D29—44no—
115C22×D29—44no—
116C22×D29C2×D412yes—
117C22×D29C2×D414yes—
118C22×D29C2×D414yes—
119C22×D29C2×D412yes—
120C2×C4×C29—28no—
121C2×C4×C29—22no—
122C2×C4×C29—22no—
123C2×C4×C29—22no—
124C2×C4×C29—21no—
125C2×C4×C29—21no—
126C2×C4×C29—22no—
127C2×C4×C29—22no—
128C2×C4×C29—22no—
129C2×C4×C29—22no—
130C2×C4×C29C2×D414yes—
131C2×Dic29—22no—
132C2×Dic29—22no—
133C2×Dic29—21no—
134C2×Dic29—21no—
135C2×Dic29—22no—
136C2×Dic29—22no—
137C2×Dic29—22no—
138D116—216no—
139D116—216no—
140C4×D29—44no—
141C4×D29—44no—
142#2405—291nomaximal, Sylow
143C22×C29—22no—
144C22×C29—22no—
145C22×C29—24no—
146C22×C29—24no—
147C22×C29#12314yescomplemented
148C22×C29C22×D411yescharacteristic
149C22×C29C22×D412yes—
150C22×C29C2×#4111yescharacteristic
151C22×C29C2×#4111yescharacteristic
152C22×C29H(5,𝔽2)12yes—
153C2×D29—22no—
154C2×D29—22no—
155C2×D29—24no—
156C2×D29—24no—
157C2×D29—22no—
158C2×D29—24no—
159C2×D29—22no—
160C2×D29—22no—
161C2×D29—24no—
162C2×D29—24no—
163C2×D29—416no—
164C2×D29—416no—
165C2×D29—42no—
166C2×D29—42no—
167C2×D29—42no—
168C2×D29—42no—
169C2×D29—42no—
170C2×D29—42no—
171C2×D29#12014yes—
172C4×C29—24no—
173C4×C29—41no—
174C4×C29—41no—
175C4×C29—42no—
176Dic29—41no—
177Dic29—41no—
178Dic29—42no—
179C23×D4—291no—
180C2×#123—292no—
181C22×C22⋊C4—291no—
182#394—292no—
183#394—292no—
184#386—292no—
185#385—291no—
186#385D2911yescharacteristic, complemented
187#379—291no—
188#379—291no—
189#379—291no—
190C2×C29—24no—
191C2×C29C2×#12011yescharacteristic
192C2×C29C2×#12311yescharacteristic
193C2×C29#53411yescharacteristic
194C2×C29#54612yes—
195C2×C29#54612yes—
196D29—24no—
197D29—42no—
198D29—44no—
199C25—291no—
200C25—291no—
201C22×D4—292no—
202C22×D4—292no—
203C22×D4—588no—
204C22×D4—584no—
205C23×C4—291no—
206C23×C4—291no—
207C23×C4C2×D2911yescharacteristic
208#123—588no—
209C2×C4⋊C4—292no—
210C2×C4⋊C4C2×D2912yescomplemented
211C2×C22⋊C4—291no—
212C2×C22⋊C4—291no—
213C2×C22⋊C4—291no—
214C2×C22⋊C4—291no—
215C2×C22⋊C4—292no—
216C2×C22⋊C4—292no—
217C2×C22⋊C4—292no—
218C2×C22⋊C4—292no—
219C2×C22⋊C4—292no—
220C2×C22⋊C4—292no—
221C2×C22⋊C4—584no—
222C2×C22⋊C4C2×D2911yescharacteristic
223C2×C22⋊C4C2×D2911yescharacteristic
224#95—292no—
225#95C2×D2912yescomplemented
226C29#240511yescharacteristic, complemented, Sylow
227C24—291no—
228C24—292no—
229C24—291no—
230C24—294no—
231C24—584no—
232C24—584no—
233C24—582no—
234C24—582no—
235C24—582no—
236C24—582no—
237C24—582no—
238C24—1162no—
239C24—1162no—
240C24C22×D2911yescharacteristic
241C2×D4—298no—
242C2×D4—584no—
243C2×D4—584no—
244C2×D4—584no—
245C2×D4—584no—
246C2×D4—5832no—
247C2×D4—584no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—291no—
251C22×C4—292no—
252C22×C4—291no—
253C22×C4—582no—
254C22×C4—582no—
255C22×C4—584no—
256C22×C4—1162no—
257C22×C4C22×D2911yescharacteristic
258C22×C4C22×D2912yes—
259C22×C4C22×D2911yescharacteristic
260C22×C4C22×D2912yes—
261C4⋊C4—24no—
262C4⋊C4—584no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—582no—
266C22⋊C4—582no—
267C22⋊C4—584no—
268C22⋊C4—584no—
269C22⋊C4—584no—
270C22⋊C4—582no—
271C22⋊C4—582no—
272C22⋊C4—1162no—
273C22⋊C4—1162no—
274C22⋊C4—1162no—
275C23—22no—
276C23—22no—
277C23—294no—
278C23—292no—
279C23—294no—
280C23—292no—
281C23—588no—
282C23—588no—
283C23—581no—
284C23—581no—
285C23—582no—
286C23—582no—
287C23—581no—
288C23—582no—
289C23—582no—
290C23—582no—
291C23—582no—
292C23—581no—
293C23—582no—
294C23—582no—
295C23—582no—
296C23—588no—
297C23—1164no—
298C23—1168no—
299C23—1168no—
300C23—1168no—
301C23—1168no—
302C23—1164no—
303C23—1164no—
304C23#9080512yes—
305C23D4×D2912yes—
306C23#9080912yes—
307C23C23×D2911yescharacteristic
308D4—5816no—
309D4—5816no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—582no—
321C2×C4—581no—
322C2×C4—581no—
323C2×C4—582no—
324C2×C4—582no—
325C2×C4—582no—
326C2×C4—582no—
327C2×C4—1164no—
328C2×C4—1164no—
329C2×C4D4×D2914yes—
330C22—24no—
331C22—24no—
332C22—22no—
333C22—22no—
334C22—294no—
335C22—584no—
336C22—584no—
337C22—582no—
338C22—582no—
339C22—584no—
340C22—584no—
341C22—584no—
342C22—582no—
343C22—582no—
344C22—582no—
345C22—11616no—
346C22—11616no—
347C22—1162no—
348C22—1162no—
349C22—1162no—
350C22—1162no—
351C22—1162no—
352C22—1162no—
353C22#1174525614yescomplemented
354C22C2×#9080511yescharacteristic
355C22C2×D4×D2912yes—
356C22C2×D4×D2911yescharacteristic
357C22C2×#9080911yescharacteristic
358C22#1174536412yes—
359C4—24no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—1162no—
364C4—1161no—
365C4—1161no—
366C2—24no—
367C2—584no—
368C2—1164no—
369C2—1162no—
370C2#120×D2911yescharacteristic
371C2C2×#1174525611yescharacteristic
372C2#1427172411yescharacteristic
373C2#1427180412yes—
374C2#1427180912yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#385C2
prime radicalC23×D116C2
hypercenter#385D29
hypocenterC29#2405
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#11745256C2semidirect
3. C2×C22⋊C4×D29C2semidirect
4. #14271081C2semidirect
5. #14271080C2semidirect
6. #14271041C2semidirect
7. #14271036C2semidirect
8. #14271018C2semidirect
9. #14271018C2semidirect
10. #14271018C2semidirect
11. C29×#385C2semidirect
19. C22×C4×D29C22semidirect
20. C22×C4×D29C22semidirect
21. C2×C4⋊C4×C29C22semidirect
26. C2×#90781C22semidirect
27. C2×#90781C22semidirect
28. C2×#90781C22semidirect
29. C2×#90781C22semidirect
30. C2×#90781C22semidirect
31. C2×#90781C22semidirect
32. C2×#90781C22semidirect
33. C2×#90781C22semidirect
34. C2×#90779C22semidirect
37. C29×#95C22semidirect
38. #11745179C22semidirect
147. C22×C29#123semidirect
186. #385D29semidirect
210. C2×C4⋊C4C2×D29semidirect
225. #95C2×D29semidirect
226. C29#2405semidirect
353. C22#11745256semidirect

Maximal subgroups

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

subgroupindex
C23×D1162
C2×#117452562
C2×C22⋊C4×D292
#142710812
#142710802
#142710412
#142710362
#142710182
C29×#3852
#240529