Finite Group Databasebeta

← #16822806

Subgroups — #16822806

Order 3712.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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×#90807C211yesmaximal, characteristic, complemented
2C2×#11745299C212yesmaximal, complemented
3C2×C22⋊C4×D29C211yesmaximal, characteristic, complemented
4#14271076C211yesmaximal, characteristic, complemented
5#14271075C211yesmaximal, characteristic, complemented
6#14271071C211yesmaximal, characteristic, complemented
7#14271043C212yesmaximal, complemented
8#14271040C212yesmaximal, complemented
9#14271039C212yesmaximal, complemented
10#14271025C211yesmaximal, characteristic, complemented
11C29×#392C211yesmaximal, characteristic, complemented
12C24×D29C2211yescharacteristic
13C22×D4×C29C2212yes—
14C22×D4×C29C2211yescharacteristic, complemented
15C22×C22⋊D29C2212yes—
16C22×C22⋊D29C2212yes—
17C23×Dic29C2212yescomplemented
18C2×#90807—28no—
19C2×#90807—24no—
20C22×C4×D29C2211yescharacteristic
21C22×C4×D29C2211yescharacteristic
22C22×Dic58C2211yescharacteristic, complemented
23C2×C22⋊C4×C29C2211yescharacteristic
24C2×C22⋊C4×C29C2212yescomplemented
25C2×C22⋊Dic29C2211yescharacteristic
26C2×C22⋊Dic29C2212yes—
27C2×C22⋊Dic29C2212yescomplemented
28#11745299—28no—
29C2×#90781C2212yes—
30C2×#90781C2211yescharacteristic
31C2×#90781C2212yes—
32C2×#90781C2211yescharacteristic
33C22⋊C4×D29—24no—
34C29×#95C2211yescharacteristic, complemented
35#11745179C2212yescomplemented
36#11745179C2211yescharacteristic, complemented
37#11745179C2212yescomplemented
38#11745179C2212yescomplemented
39C24×C29C2312yes—
40C24×C29C2311yescharacteristic
41C23×D29—22no—
42C23×D29—22no—
43C23×D29—22no—
44C23×D29—24no—
45C23×D29—42no—
46C23×D29C2311yescharacteristic
47C23×D29C2311yescharacteristic
48C2×D4×C29—24no—
49C2×D4×C29—24no—
50C2×D4×C29—24no—
51C2×D4×C29—22no—
52C2×D4×C29—42no—
53C22×C4×C29C2312yes—
54C22×C4×C29C2311yescharacteristic
55C22×C4×C29C2311yescharacteristic
56C2×C22⋊D29—24no—
57C2×C22⋊D29—24no—
58C2×C22⋊D29—24no—
59C2×C22⋊D29—24no—
60C2×C22⋊D29—42no—
61C2×C22⋊D29D418yescomplemented
62C22×Dic29—24no—
63C22×Dic29—24no—
64C22×Dic29—24no—
65C22×Dic29C2311yescharacteristic
66C22×Dic29C2312yes—
67C22×Dic29C2312yes—
68C22×Dic29C2311yescharacteristic
69#90807—232no—
70C2×C4×D29—24no—
71C2×C4×D29—22no—
72C2×C4×D29—42no—
73C2×C4×D29D414yescomplemented
74C2×Dic58—22no—
75C2×Dic58—24no—
76C22⋊C4×C29—22no—
77C22⋊C4×C29—42no—
78C22⋊Dic29—24no—
79C22⋊Dic29—22no—
80C22⋊Dic29—42no—
81#90781—24no—
82#90781—22no—
83#90781—24no—
84#90781—22no—
85C23×C29—21no—
86C23×C29—22no—
87C23×C29—22no—
88C23×C29—22no—
89C23×C29—22no—
90C23×C29—22no—
91C23×C29—21no—
92C23×C29—22no—
93C23×C29—22no—
94C23×C29—22no—
95C23×C29C2×D414yes—
96C23×C29#4112yescomplemented
97C23×C29C2411yescharacteristic
98C22×D29—21no—
99C22×D29—21no—
100C22×D29—22no—
101C22×D29—28no—
102C22×D29—21no—
103C22×D29—22no—
104C22×D29—21no—
105C22×D29—22no—
106C22×D29—48no—
107C22×D29—48no—
108C22×D29—44no—
109C22×D29—44no—
110C22×D29C2×D414yes—
111C22×D29C2×D412yes—
112C22×D29#4112yescomplemented
113D4×C29—28no—
114D4×C29—48no—
115C2×C4×C29—22no—
116C2×C4×C29—22no—
117C2×C4×C29—21no—
118C2×C4×C29—22no—
119C2×C4×C29—22no—
120C2×C4×C29—22no—
121C2×C4×C29C2×D412yes—
122C22⋊D29—216no—
123C22⋊D29—48no—
124C2×Dic29—216no—
125C2×Dic29—22no—
126C2×Dic29—22no—
127C2×Dic29—22no—
128C2×Dic29—21no—
129C2×Dic29—22no—
130C2×Dic29—22no—
131C2×Dic29—22no—
132C2×Dic29—22no—
133C2×Dic29C2×D414yes—
134C2×Dic29C2×D412yes—
135C4×D29—28no—
136C4×D29—44no—
137Dic58—28no—
138#2404—291nomaximal, Sylow
139C22×C29—24no—
140C22×C29—22no—
141C22×C29—24no—
142C22×C29—24no—
143C22×C29—24no—
144C22×C29—41no—
145C22×C29—42no—
146C22×C29—41no—
147C22×C29—42no—
148C22×C29—42no—
149C22×C29—42no—
150C22×C29—42no—
151C22×C29C22×D411yescharacteristic
152C22×C29C22×D412yes—
153C22×C29C2×#4111yescharacteristic
154C22×C29C2×#4111yescharacteristic
155C22×C29H(5,𝔽2)12yes—
156C2×D29—22no—
157C2×D29—22no—
158C2×D29—24no—
159C2×D29—24no—
160C2×D29—22no—
161C2×D29—24no—
162C2×D29—416no—
163C2×D29—41no—
164C2×D29—41no—
165C2×D29—42no—
166C2×D29#12014yes—
167C4×C29—22no—
168C4×C29—41no—
169C4×C29—42no—
170Dic29—24no—
171Dic29—22no—
172Dic29—41no—
173Dic29—42no—
174C22×#41—291no—
175C2×#120—292no—
176C22×C22⋊C4—291no—
177#393—291no—
178#393—291no—
179#392—291no—
180#392D2911yescharacteristic, complemented
181#386—292no—
182#386—292no—
183#386—292no—
184#381—291no—
185C2×C29—42no—
186C2×C29—44no—
187C2×C29C2×#12011yescharacteristic
188C2×C29#52511yescharacteristic
189C2×C29#53511yescharacteristic
190C2×C29#54612yes—
191C2×C29#54612yes—
192D29—24no—
193D29—42no—
194C25—291no—
195C2×#41—584no—
196C2×#41—588no—
197C22×Q8—291no—
198C22×D4—292no—
199C22×D4—292no—
200C22×D4C2×D2912yes—
201C22×D4C2×D2911yescharacteristic, complemented
202C23×C4—291no—
203C23×C4—292no—
204C23×C4—291no—
205#120—588no—
206C2×C22⋊C4—291no—
207C2×C22⋊C4—292no—
208C2×C22⋊C4—291no—
209C2×C22⋊C4—291no—
210C2×C22⋊C4—292no—
211C2×C22⋊C4—292no—
212C2×C22⋊C4—292no—
213C2×C22⋊C4—584no—
214C2×C22⋊C4C2×D2911yescharacteristic
215C2×C22⋊C4C2×D2912yescomplemented
216#95—292no—
217#95—291no—
218#95—292no—
219#95—292no—
220#95C2×D2911yescharacteristic, complemented
221C29#240411yescharacteristic, complemented, Sylow
222C24—291no—
223C24—291no—
224C24—582no—
225C24—582no—
226C24—584no—
227C24—582no—
228C24—1162no—
229C24C22×D2912yes—
230C24C22×D2911yescharacteristic
231#41—5832no—
232C2×Q8—584no—
233C2×Q8—582no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—298no—
240C2×D4—584no—
241C2×D4—584no—
242C2×D4—584no—
243C2×D4—584no—
244C2×D4—1162no—
245C22×C4—291no—
246C22×C4—292no—
247C22×C4—294no—
248C22×C4—292no—
249C22×C4—291no—
250C22×C4—584no—
251C22×C4—582no—
252C22×C4—584no—
253C22×C4—584no—
254C22×C4—584no—
255C22×C4—1162no—
256C22×C4C22×D2912yes—
257C22×C4C22×D2911yescharacteristic
258C22×C4C22×D2911yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—582no—
262C22⋊C4—584no—
263C22⋊C4—584no—
264C22⋊C4—582no—
265C22⋊C4—584no—
266C22⋊C4—582no—
267C22⋊C4—1162no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—294no—
279C23—292no—
280C23—292no—
281C23—588no—
282C23—581no—
283C23—582no—
284C23—581no—
285C23—581no—
286C23—582no—
287C23—581no—
288C23—582no—
289C23—1168no—
290C23—1164no—
291C23—1168no—
292C23—1164no—
293C23D4×D2914yes—
294C23#9080712yescomplemented
295C23C23×D2911yescharacteristic
296Q8—588no—
297D4—28no—
298D4—48no—
299D4—5816no—
300D4—1168no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—292no—
308C2×C4—294no—
309C2×C4—588no—
310C2×C4—5816no—
311C2×C4—582no—
312C2×C4—582no—
313C2×C4—582no—
314C2×C4—581no—
315C2×C4—582no—
316C2×C4—582no—
317C2×C4—582no—
318C2×C4—582no—
319C2×C4—1164no—
320C2×C4D4×D2912yes—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—294no—
334C22—584no—
335C22—584no—
336C22—582no—
337C22—582no—
338C22—584no—
339C22—582no—
340C22—11616no—
341C22—1162no—
342C22—1161no—
343C22—1161no—
344C22C2×D4×D2912yes—
345C22C2×D4×D2911yescharacteristic
346C22C2×#9080711yescharacteristic
347C22#1174535912yes—
348C22#41×D2911yescharacteristic
349C4—22no—
350C4—42no—
351C4—41no—
352C4—582no—
353C4—584no—
354C4—1162no—
355C4—1161no—
356C2—42no—
357C2—44no—
358C2—584no—
359C2—1162no—
360C2#120×D2911yescharacteristic
361C2#1427167311yescharacteristic
362C2#1427172611yescharacteristic
363C2#1427180012yes—
364C2#1427180212yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#392C2
hypercenter#392D29
hypocenterC29#2404
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#90807C2semidirect
2. C2×#11745299C2semidirect
3. C2×C22⋊C4×D29C2semidirect
4. #14271076C2semidirect
5. #14271075C2semidirect
6. #14271071C2semidirect
7. #14271043C2semidirect
8. #14271040C2semidirect
9. #14271039C2semidirect
10. #14271025C2semidirect
11. C29×#392C2semidirect
14. C22×D4×C29C22semidirect
17. C23×Dic29C22semidirect
22. C22×Dic58C22semidirect
24. C2×C22⋊C4×C29C22semidirect
27. C2×C22⋊Dic29C22semidirect
34. C29×#95C22semidirect
35. #11745179C22semidirect
36. #11745179C22semidirect
37. #11745179C22semidirect
38. #11745179C22semidirect
61. C2×C22⋊D29D4semidirect
73. C2×C4×D29D4semidirect
96. C23×C29#41semidirect
112. C22×D29#41semidirect
180. #392D29semidirect
201. C22×D4C2×D29semidirect
215. C2×C22⋊C4C2×D29semidirect
220. #95C2×D29semidirect
221. C29#2404semidirect
294. C23#90807semidirect

Maximal subgroups

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

subgroupindex
C22×#908072
C2×#117452992
C2×C22⋊C4×D292
#142710762
#142710752
#142710712
#142710432
#142710402
#142710392
#142710252
C29×#3922
#240429