Finite Group Databasebeta

← #16822841

Subgroups — #16822841

Order 3712.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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×D29C211yesmaximal, characteristic, complemented
2C2×#11745301C212yesmaximal, complemented
3C2×#11745299C212yesmaximal, complemented
4C2×#11745243C211yesmaximal, characteristic, complemented
5#14271081C211yesmaximal, characteristic, complemented
6#14271077C211yesmaximal, characteristic, complemented
7#14271072C211yesmaximal, characteristic, complemented
8#14271063C212yesmaximal, complemented
9#14271022C211yesmaximal, characteristic, complemented
10#14271018C212yesmaximal, complemented
11C29×#392C211yesmaximal, characteristic, complemented
12C24×D29C2212yes—
13C22×D4×C29C2211yescharacteristic
14C22×D4×C29C2212yescomplemented
15C22×C22⋊D29C2212yes—
16C22×C22⋊D29C2211yescharacteristic
17C22×C22⋊D29C2212yescomplemented
18C23×Dic29C2211yescharacteristic, complemented
19C2×D4×D29—28no—
20C2×D4×D29—24no—
21C22×D116C2211yescharacteristic
22C22×C4×D29C2211yescharacteristic
23C2×C22⋊C4×C29C2211yescharacteristic, complemented
24C2×C22⋊C4×C29C2212yescomplemented
25C2×C22⋊Dic29C2212yescomplemented
26C2×C22⋊Dic29C2212yescomplemented
27#11745301—28no—
28#11745299—28no—
29C2×#90781C2211yescharacteristic
30C2×#90781C2212yes—
31C2×#90781C2212yescomplemented
32C2×#90781C2212yescomplemented
33C2×#90781C2212yescomplemented
34C2×#90779C2212yescomplemented
35C2×#90779C2211yescharacteristic, complemented
36C2×C4×Dic29C2211yescharacteristic, complemented
37#11745243—24no—
38C29×#95C2211yescharacteristic, complemented
39#11745179C2211yescharacteristic, complemented
40C24×C29C2312yes—
41C24×C29C2311yescharacteristic
42C23×D29—24no—
43C23×D29—24no—
44C23×D29—24no—
45C23×D29—24no—
46C23×D29—24no—
47C23×D29—42no—
48C23×D29—42no—
49C23×D29C2312yes—
50C23×D29C2311yescharacteristic
51C2×D4×C29—24no—
52C2×D4×C29—24no—
53C2×D4×C29—24no—
54C2×D4×C29—22no—
55C2×D4×C29—42no—
56C22×C4×C29C2311yescharacteristic
57C22×C4×C29C2312yes—
58C22×C4×C29C2311yescharacteristic
59C2×C22⋊D29—22no—
60C2×C22⋊D29—22no—
61C2×C22⋊D29—24no—
62C2×C22⋊D29—24no—
63C2×C22⋊D29—24no—
64C2×C22⋊D29—24no—
65C2×C22⋊D29—42no—
66C2×C22⋊D29D418yescomplemented
67C2×C22⋊D29D414yescomplemented
68C22×Dic29—24no—
69C22×Dic29—22no—
70C22×Dic29C2311yescharacteristic
71C22×Dic29C2312yes—
72C22×Dic29C2311yescharacteristic
73C22×Dic29C2311yescharacteristic
74D4×D29—232no—
75C2×D116—24no—
76C2×D116—22no—
77C2×C4×D29—24no—
78C2×C4×D29D414yescomplemented
79C22⋊C4×C29—22no—
80C22⋊C4×C29—42no—
81C22⋊Dic29—24no—
82C22⋊Dic29—24no—
83#90781—24no—
84#90781—22no—
85#90781—24no—
86#90781—24no—
87#90781—42no—
88#90779—22no—
89#90779—24no—
90C4×Dic29—22no—
91C23×C29—22no—
92C23×C29—21no—
93C23×C29—22no—
94C23×C29—22no—
95C23×C29—22no—
96C23×C29—21no—
97C23×C29—22no—
98C23×C29—22no—
99C23×C29—22no—
100C23×C29—22no—
101C23×C29C2×D414yes—
102C23×C29C2×D412yescomplemented
103C23×C29C2411yescharacteristic
104C22×D29—22no—
105C22×D29—22no—
106C22×D29—216no—
107C22×D29—22no—
108C22×D29—22no—
109C22×D29—22no—
110C22×D29—22no—
111C22×D29—22no—
112C22×D29—22no—
113C22×D29—48no—
114C22×D29—48no—
115C22×D29—48no—
116C22×D29—48no—
117C22×D29—48no—
118C22×D29—48no—
119C22×D29C2×D414yes—
120C22×D29C2×D412yes—
121C22×D29C2×D412yes—
122C22×D29#4112yes—
123D4×C29—28no—
124D4×C29—48no—
125C2×C4×C29—22no—
126C2×C4×C29—21no—
127C2×C4×C29—22no—
128C2×C4×C29—22no—
129C2×C4×C29—22no—
130C2×C4×C29—22no—
131C2×C4×C29C2×D412yes—
132C22⋊D29—28no—
133C22⋊D29—216no—
134C22⋊D29—48no—
135C2×Dic29—21no—
136C2×Dic29—21no—
137C2×Dic29—21no—
138C2×Dic29—22no—
139C2×Dic29—22no—
140C2×Dic29—22no—
141C2×Dic29—22no—
142C2×Dic29—44no—
143C2×Dic29C2×D412yes—
144C2×Dic29C2×D414yes—
145C2×Dic29C2×D412yescomplemented
146D116—28no—
147C4×D29—28no—
148#2408—291nomaximal, Sylow
149C22×C29—24no—
150C22×C29—22no—
151C22×C29—24no—
152C22×C29—24no—
153C22×C29—24no—
154C22×C29—41no—
155C22×C29—42no—
156C22×C29—42no—
157C22×C29—41no—
158C22×C29—42no—
159C22×C29—42no—
160C22×C29—42no—
161C22×C29C22×D411yescharacteristic
162C22×C29C22×D412yes—
163C22×C29C22×D411yescharacteristic
164C22×C29C2×#4111yescharacteristic
165C22×C29H(5,𝔽2)12yes—
166C2×D29—22no—
167C2×D29—22no—
168C2×D29—24no—
169C2×D29—24no—
170C2×D29—24no—
171C2×D29—24no—
172C2×D29—24no—
173C2×D29—432no—
174C2×D29—42no—
175C2×D29—42no—
176C2×D29—42no—
177C2×D29—42no—
178C2×D29#12114yescomplemented
179C4×C29—22no—
180C4×C29—41no—
181C4×C29—42no—
182Dic29—22no—
183Dic29—41no—
184Dic29—42no—
185Dic29#12014yescomplemented
186C23×D4—291no—
187C2×#121—292no—
188C2×#120—292no—
189C2×C4×D4—291no—
190#394—291no—
191#394—291no—
192#392—291no—
193#392D2911yescharacteristic, complemented
194#390—292no—
195#380—291no—
196#379—292no—
197C2×C29—44no—
198C2×C29—42no—
199C2×C29C2×#12011yescharacteristic
200C2×C29C2×#12111yescharacteristic
201C2×C29#53511yescharacteristic
202C2×C29D4212yes—
203C2×C29#54612yes—
204D29—24no—
205D29—44no—
206C25—292no—
207C22×D4—291no—
208C22×D4—292no—
209C22×D4—292no—
210C22×D4—291no—
211C22×D4—584no—
212C22×D4—588no—
213C22×D4C2×D2911yescharacteristic
214C22×D4C2×D2912yescomplemented
215C23×C4—291no—
216C23×C4—291no—
217#121—588no—
218#120—588no—
219C4×D4—584no—
220C2×C4⋊C4—291no—
221C2×C4⋊C4—292no—
222C2×C22⋊C4—292no—
223C2×C22⋊C4—292no—
224C2×C22⋊C4—291no—
225C2×C22⋊C4—292no—
226C2×C22⋊C4—292no—
227C2×C22⋊C4—292no—
228C2×C22⋊C4—292no—
229C2×C22⋊C4C2×D2911yescharacteristic, complemented
230C2×C22⋊C4C2×D2912yescomplemented
231C2×C42—291no—
232#95—291no—
233#95C2×D2911yescharacteristic, complemented
234C29#240811yescharacteristic, complemented, Sylow
235C24—291no—
236C24—292no—
237C24—584no—
238C24—584no—
239C24—584no—
240C24—584no—
241C24—584no—
242C24—1162no—
243C24—1162no—
244C24C22×D2912yes—
245C24C22×D2911yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—298no—
252C2×D4—294no—
253C2×D4—584no—
254C2×D4—584no—
255C2×D4—584no—
256C2×D4—5832no—
257C2×D4—582no—
258C2×D4—582no—
259C2×D4—584no—
260C2×D4—584no—
261C2×D4—582no—
262C2×D4—1162no—
263C22×C4—291no—
264C22×C4—291no—
265C22×C4—294no—
266C22×C4—292no—
267C22×C4—291no—
268C22×C4—584no—
269C22×C4—582no—
270C22×C4—584no—
271C22×C4C22×D2912yes—
272C22×C4C22×D2911yescharacteristic
273C22×C4C22×D2911yescharacteristic
274C4⋊C4—584no—
275C4⋊C4—582no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—584no—
279C22⋊C4—582no—
280C22⋊C4—584no—
281C22⋊C4—584no—
282C22⋊C4—584no—
283C22⋊C4—584no—
284C22⋊C4—1162no—
285C42—582no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—22no—
296C23—292no—
297C23—292no—
298C23—292no—
299C23—294no—
300C23—5816no—
301C23—582no—
302C23—582no—
303C23—582no—
304C23—582no—
305C23—582no—
306C23—582no—
307C23—582no—
308C23—582no—
309C23—1168no—
310C23—1168no—
311C23—1168no—
312C23—1168no—
313C23—1168no—
314C23—1168no—
315C23D4×D2914yes—
316C23D4×D2912yescomplemented
317C23C23×D2911yescharacteristic
318D4—28no—
319D4—48no—
320D4—588no—
321D4—5816no—
322D4—588no—
323D4—1168no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—292no—
331C2×C4—294no—
332C2×C4—292no—
333C2×C4—588no—
334C2×C4—581no—
335C2×C4—581no—
336C2×C4—582no—
337C2×C4—582no—
338C2×C4—581no—
339C2×C4—582no—
340C2×C4—582no—
341C2×C4—1164no—
342C2×C4D4×D2912yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—294no—
356C22—584no—
357C22—584no—
358C22—582no—
359C22—582no—
360C22—584no—
361C22—584no—
362C22—584no—
363C22—11632no—
364C22—1162no—
365C22—1162no—
366C22—1162no—
367C22—1162no—
368C22C2×D4×D2911yescharacteristic
369C22C2×D4×D2912yes—
370C22C2×D4×D2911yescharacteristic
371C22#1174535912yes—
372C22#41×D2911yescharacteristic
373C4—22no—
374C4—42no—
375C4—41no—
376C4—294no—
377C4—582no—
378C4—1162no—
379C4—1161no—
380C2—44no—
381C2—42no—
382C2—584no—
383C2—1164no—
384C2#120×D2911yescharacteristic
385C2#121×D2911yescharacteristic
386C2#1427172611yescharacteristic
387C2#1427179512yes—
388C2#1427180212yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#392C2
hypercenter#392D29
hypocenterC29#2408
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×#11745301C2semidirect
3. C2×#11745299C2semidirect
4. C2×#11745243C2semidirect
5. #14271081C2semidirect
6. #14271077C2semidirect
7. #14271072C2semidirect
8. #14271063C2semidirect
9. #14271022C2semidirect
10. #14271018C2semidirect
11. C29×#392C2semidirect
14. C22×D4×C29C22semidirect
17. C22×C22⋊D29C22semidirect
18. C23×Dic29C22semidirect
23. C2×C22⋊C4×C29C22semidirect
24. C2×C22⋊C4×C29C22semidirect
25. C2×C22⋊Dic29C22semidirect
26. C2×C22⋊Dic29C22semidirect
31. C2×#90781C22semidirect
32. C2×#90781C22semidirect
33. C2×#90781C22semidirect
34. C2×#90779C22semidirect
35. C2×#90779C22semidirect
36. C2×C4×Dic29C22semidirect
38. C29×#95C22semidirect
39. #11745179C22semidirect
66. C2×C22⋊D29D4semidirect
67. C2×C22⋊D29D4semidirect
78. C2×C4×D29D4semidirect
102. C23×C29C2×D4semidirect
145. C2×Dic29C2×D4semidirect
178. C2×D29#121semidirect
185. Dic29#120semidirect
193. #392D29semidirect
214. C22×D4C2×D29semidirect
229. C2×C22⋊C4C2×D29semidirect
230. C2×C22⋊C4C2×D29semidirect
233. #95C2×D29semidirect
234. C29#2408semidirect
316. C23D4×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×#117453012
C2×#117452992
C2×#117452432
#142710812
#142710772
#142710722
#142710632
#142710222
#142710182
C29×#3922
#240829