Finite Group Databasebeta

← #16822801

Subgroups — #16822801

Order 3712.

1564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×D4×D29C212yesmaximal, complemented
2C2×#11745244C212yesmaximal, complemented
3C2×#11745299C212yesmaximal, complemented
4#14271072C211yesmaximal, characteristic, complemented
5#14271071C212yesmaximal, complemented
6#14271041C212yesmaximal, complemented
7#14271039C212yesmaximal, complemented
8#14271018C211yesmaximal, characteristic, complemented
9C29×#392C211yesmaximal, characteristic, complemented
10C24×D29C2211yescharacteristic
11C24×D29C2212yescomplemented
12C22×D4×C29C2212yescomplemented
13C22×D4×C29C2211yescharacteristic, complemented
14C22×C22⋊D29C2212yes—
15C22×C22⋊D29C2212yescomplemented
16C22×C22⋊D29C2212yescomplemented
17C2×D4×D29—28no—
18C2×D4×D29—28no—
19C2×D4×D29—28no—
20C22×D116C2212yes—
21C22×D116C2211yescharacteristic, complemented
22C22×C4×D29C2212yescomplemented
23C2×C22⋊C4×C29C2212yescomplemented
24C2×C22⋊C4×C29C2211yescharacteristic, complemented
25C2×C22⋊Dic29C2212yescomplemented
26C2×C22⋊Dic29C2211yescharacteristic, complemented
27#11745299—28no—
28C2×#90781C2212yescomplemented
29C2×#90781C2212yescomplemented
30C2×#90781C2212yescomplemented
31C2×#90781C2212yescomplemented
32#11745244—28no—
33C29×#95C2211yescharacteristic, complemented
34#11745179C2211yescharacteristic, complemented
35#11745179C2212yescomplemented
36C24×C29C2311yescharacteristic
37C24×C29C2312yes—
38C23×D29—24no—
39C23×D29—24no—
40C23×D29—24no—
41C23×D29—24no—
42C23×D29—24no—
43C23×D29—24no—
44C23×D29—24no—
45C23×D29—41no—
46C23×D29—41no—
47C23×D29—42no—
48C23×D29—42no—
49C23×D29D414yescomplemented
50C23×D29C2312yes—
51C23×D29C2312yes—
52C2×D4×C29—24no—
53C2×D4×C29—24no—
54C2×D4×C29—24no—
55C2×D4×C29—24no—
56C2×D4×C29—41no—
57C22×C4×C29C2311yescharacteristic
58C22×C4×C29C2311yescharacteristic
59C22×C4×C29C2312yes—
60C2×C22⋊D29—24no—
61C2×C22⋊D29—24no—
62C2×C22⋊D29—24no—
63C2×C22⋊D29—24no—
64C2×C22⋊D29—24no—
65C2×C22⋊D29—24no—
66C2×C22⋊D29—24no—
67C2×C22⋊D29—42no—
68C2×C22⋊D29D418yescomplemented
69C22×Dic29C2312yes—
70C22×Dic29C2312yes—
71D4×D29—264no—
72C2×D116—24no—
73C2×D116—24no—
74C2×D116—24no—
75C2×D116—41no—
76C2×D116D418yescomplemented
77C2×C4×D29—24no—
78C2×C4×D29—24no—
79C2×C4×D29—24no—
80C22⋊C4×C29—24no—
81C22⋊C4×C29—41no—
82C22⋊Dic29—24no—
83C22⋊Dic29—41no—
84#90781—24no—
85#90781—24no—
86#90781—24no—
87#90781—24no—
88C23×C29—22no—
89C23×C29—22no—
90C23×C29—22no—
91C23×C29—22no—
92C23×C29—22no—
93C23×C29—22no—
94C23×C29—21no—
95C23×C29—21no—
96C23×C29—22no—
97C23×C29—22no—
98C23×C29C2×D412yes—
99C23×C29C2×D414yescomplemented
100C23×C29C2411yescharacteristic
101C22×D29—22no—
102C22×D29—22no—
103C22×D29—22no—
104C22×D29—22no—
105C22×D29—216no—
106C22×D29—22no—
107C22×D29—22no—
108C22×D29—22no—
109C22×D29—22no—
110C22×D29—22no—
111C22×D29—22no—
112C22×D29—216no—
113C22×D29—44no—
114C22×D29—44no—
115C22×D29—48no—
116C22×D29—44no—
117C22×D29—48no—
118C22×D29—48no—
119C22×D29—48no—
120C22×D29—48no—
121C22×D29—48no—
122C22×D29—48no—
123C22×D29C2×D414yes—
124C22×D29C2×D414yes—
125C22×D29C2×D414yes—
126C22×D29C2×D414yescomplemented
127D4×C29—216no—
128D4×C29—44no—
129C2×C4×C29—22no—
130C2×C4×C29—22no—
131C2×C4×C29—22no—
132C2×C4×C29—22no—
133C2×C4×C29—21no—
134C2×C4×C29—21no—
135C2×C4×C29C2×D414yescomplemented
136C22⋊D29—216no—
137C22⋊D29—216no—
138C22⋊D29—48no—
139C2×Dic29—22no—
140C2×Dic29—22no—
141C2×Dic29—22no—
142C2×Dic29—22no—
143C2×Dic29—22no—
144C2×Dic29C2×D414yescomplemented
145D116—216no—
146D116—44no—
147C4×D29—216no—
148#2403—291nomaximal, Sylow
149C22×C29—24no—
150C22×C29—24no—
151C22×C29—22no—
152C22×C29—24no—
153C22×C29—24no—
154C22×C29—42no—
155C22×C29—42no—
156C22×C29—41no—
157C22×C29—42no—
158C22×C29—42no—
159C22×C29—42no—
160C22×C29—41no—
161C22×C29C22×D411yescharacteristic
162C22×C29C22×D412yes—
163C22×C29C22×D412yes—
164C22×C29H(5,𝔽2)11yescharacteristic
165C22×C29H(5,𝔽2)11yescharacteristic
166C2×D29—24no—
167C2×D29—24no—
168C2×D29—24no—
169C2×D29—24no—
170C2×D29—24no—
171C2×D29—24no—
172C2×D29—24no—
173C2×D29—24no—
174C2×D29—24no—
175C2×D29—416no—
176C2×D29—432no—
177C2×D29—42no—
178C2×D29—42no—
179C2×D29—42no—
180C2×D29—42no—
181C2×D29#12018yescomplemented
182C4×C29—24no—
183C4×C29—41no—
184C4×C29—41no—
185Dic29—24no—
186Dic29—42no—
187C23×D4—292no—
188C2×#120—292no—
189C2×#120—292no—
190#392—292no—
191#392—291no—
192#392D2911yescharacteristic, complemented
193#386—292no—
194#386—292no—
195#379—291no—
196C2×C29—42no—
197C2×C29—44no—
198C2×C29C2×#12012yes—
199C2×C29#53411yescharacteristic
200C2×C29D4211yescharacteristic
201C2×C29D4212yes—
202C2×C29D4211yescharacteristic
203D29—28no—
204D29—44no—
205C25—291no—
206C25—292no—
207C22×D4—292no—
208C22×D4—292no—
209C22×D4—292no—
210C22×D4—292no—
211C22×D4—291no—
212C22×D4—588no—
213C22×D4—588no—
214C22×D4—588no—
215C22×D4C2×D2912yescomplemented
216C22×D4C2×D2911yescharacteristic, complemented
217C23×C4—292no—
218#120—588no—
219#120—588no—
220C2×C22⋊C4—292no—
221C2×C22⋊C4—292no—
222C2×C22⋊C4—292no—
223C2×C22⋊C4—292no—
224C2×C22⋊C4—292no—
225C2×C22⋊C4—291no—
226C2×C22⋊C4C2×D2912yescomplemented
227C2×C22⋊C4C2×D2911yescharacteristic, complemented
228#95—291no—
229#95—292no—
230#95C2×D2911yescharacteristic, complemented
231C29#240311yescharacteristic, complemented, Sylow
232C24—292no—
233C24—292no—
234C24—294no—
235C24—584no—
236C24—584no—
237C24—584no—
238C24—584no—
239C24—584no—
240C24—584no—
241C24—584no—
242C24—1161no—
243C24—1162no—
244C24—1161no—
245C24—1162no—
246C24C22×D2911yescharacteristic
247C24C22×D2912yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—298no—
254C2×D4—298no—
255C2×D4—584no—
256C2×D4—584no—
257C2×D4—584no—
258C2×D4—584no—
259C2×D4—5864no—
260C2×D4—584no—
261C2×D4—584no—
262C2×D4—584no—
263C2×D4—584no—
264C2×D4—584no—
265C2×D4—584no—
266C2×D4—1161no—
267C2×D4—1162no—
268C22×C4—292no—
269C22×C4—292no—
270C22×C4—584no—
271C22×C4—584no—
272C22×C4—584no—
273C22×C4C22×D2911yescharacteristic
274C22×C4C22×D2911yescharacteristic
275C22×C4C22×D2912yes—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—584no—
279C22⋊C4—584no—
280C22⋊C4—584no—
281C22⋊C4—584no—
282C22⋊C4—584no—
283C22⋊C4—1161no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—294no—
295C23—294no—
296C23—294no—
297C23—294no—
298C23—5816no—
299C23—5816no—
300C23—582no—
301C23—582no—
302C23—582no—
303C23—582no—
304C23—582no—
305C23—582no—
306C23—582no—
307C23—582no—
308C23—582no—
309C23—582no—
310C23—1164no—
311C23—1168no—
312C23—1168no—
313C23—1168no—
314C23—1164no—
315C23—1168no—
316C23—1168no—
317C23—1164no—
318C23—1168no—
319C23—1168no—
320C23D4×D2912yes—
321C23D4×D2914yescomplemented
322C23C23×D2911yescharacteristic
323D4—216no—
324D4—44no—
325D4—5816no—
326D4—5816no—
327D4—5816no—
328D4—1164no—
329D4—1168no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—294no—
337C2×C4—5816no—
338C2×C4—582no—
339C2×C4—582no—
340C2×C4—582no—
341C2×C4—582no—
342C2×C4—582no—
343C2×C4D4×D2914yescomplemented
344C22—24no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—22no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—298no—
357C22—584no—
358C22—584no—
359C22—584no—
360C22—584no—
361C22—584no—
362C22—584no—
363C22—584no—
364C22—584no—
365C22—584no—
366C22—11616no—
367C22—11632no—
368C22—1162no—
369C22—1162no—
370C22—1162no—
371C22—1162no—
372C22C2×D4×D2912yes—
373C22C2×D4×D2912yes—
374C22C2×D4×D2911yescharacteristic
375C22#1174535911yescharacteristic
376C22#1174536411yescharacteristic
377C4—24no—
378C4—41no—
379C4—41no—
380C4—584no—
381C4—1162no—
382C2—44no—
383C2—42no—
384C2—588no—
385C2—1164no—
386C2#120×D2912yes—
387C2#1427172111yescharacteristic
388C2#1427179511yescharacteristic
389C2#1427179712yes—
390C2#1427179811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#392C2
hypercenter#392D29
hypocenterC29#2403
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D29C2semidirect
2. C2×#11745244C2semidirect
3. C2×#11745299C2semidirect
4. #14271072C2semidirect
5. #14271071C2semidirect
6. #14271041C2semidirect
7. #14271039C2semidirect
8. #14271018C2semidirect
9. C29×#392C2semidirect
11. C24×D29C22semidirect
12. C22×D4×C29C22semidirect
13. C22×D4×C29C22semidirect
15. C22×C22⋊D29C22semidirect
16. C22×C22⋊D29C22semidirect
21. C22×D116C22semidirect
22. C22×C4×D29C22semidirect
23. C2×C22⋊C4×C29C22semidirect
24. C2×C22⋊C4×C29C22semidirect
25. C2×C22⋊Dic29C22semidirect
26. C2×C22⋊Dic29C22semidirect
28. C2×#90781C22semidirect
29. C2×#90781C22semidirect
30. C2×#90781C22semidirect
31. C2×#90781C22semidirect
33. C29×#95C22semidirect
34. #11745179C22semidirect
35. #11745179C22semidirect
49. C23×D29D4semidirect
68. C2×C22⋊D29D4semidirect
76. C2×D116D4semidirect
99. C23×C29C2×D4semidirect
126. C22×D29C2×D4semidirect
135. C2×C4×C29C2×D4semidirect
144. C2×Dic29C2×D4semidirect
181. C2×D29#120semidirect
192. #392D29semidirect
215. C22×D4C2×D29semidirect
216. C22×D4C2×D29semidirect
226. C2×C22⋊C4C2×D29semidirect
227. C2×C22⋊C4C2×D29semidirect
230. #95C2×D29semidirect
231. C29#2403semidirect
321. C23D4×D29semidirect
343. C2×C4D4×D29semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D292
C2×#117452442
C2×#117452992
#142710722
#142710712
#142710412
#142710392
#142710182
C29×#3922
#240329