Finite Group Databasebeta

← #541×D163 #35227898

Subgroups — #541×D163 #35227898

Order 20864.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782278C211yesmaximal, characteristic, complemented
2#23782277C211yesmaximal, characteristic, complemented
3#23782276C211yesmaximal, characteristic, complemented
4#23782275C212yesmaximal, complemented
5#23782274C214yesmaximal, complemented
6#23782273C212yesmaximal, complemented
7#23782272C214yesmaximal, complemented
8C4⋊Q8×D163C211yesmaximal, characteristic, complemented
9#126×D163C214yesmaximal
10#125×D163C214yesmaximal, complemented
11#125×D163C211yesmaximal, characteristic, complemented
12C22⋊Q8×D163C212yesmaximal
13C4×Q8×D163C212yesmaximal, complemented
14#117×D163C211yesmaximal, characteristic
15#541×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2212yes—
17C22×C4×D163C2211yescharacteristic
18#19104735C2211yescharacteristic, complemented
19#19104734C2214yescomplemented
20#19104733C2212yescomplemented
21#19104730C2214yes—
22#19104730C2214yes—
23#19104730C2214yes—
24#19104729C2214yes—
25#19104728C2214yes—
26#19104728C2214yes—
27#19104728C2214yes—
28#19104727C2214yescomplemented
29#19104727C2211yescharacteristic, complemented
30#19104726C2214yescomplemented
31#19104726C2212yescomplemented
32#19104726C2214yescomplemented
33#19104725C2214yescomplemented
34#19104725C2214yescomplemented
35#19104725C2214yescomplemented
36#19104725C2214yescomplemented
37#19104725C2214yescomplemented
38#19104715C2212yes—
39#19104714C2212yes—
40#19104713C2214yes—
41#19104712C2214yes—
42#19104711C2212yes—
43Q8×Dic163C2212yescomplemented
44C4×Dic326C2214yescomplemented
45C4×Dic326C2212yescomplemented
46#19104700C2214yescomplemented
47#19104700C2212yescomplemented
48#19104694C2211yescharacteristic
49#19104693C2212yes—
50#19104692C2211yescharacteristic
51#19104692C2211yescharacteristic
52#19104691C2212yes—
53C4⋊C4×D163C2214yes—
54C4⋊C4×D163C2212yes—
55C4⋊C4×D163C2214yes—
56C4⋊C4×D163C2212yes—
57C4⋊C4×D163C2214yes—
58C4⋊C4×D163C2214yes—
59C22⋊C4×D163C2214yes—
60C22⋊C4×D163C2212yes—
61C42×D163C2214yes—
62C42×D163C2211yescharacteristic
63C42×D163C2211yescharacteristic
64C4⋊Q8×C163C2211yescharacteristic, complemented
65#126×C163C2214yes—
66#125×C163C2214yescomplemented
67#125×C163C2211yescharacteristic, complemented
68C22⋊Q8×C163C2212yes—
69C4×Q8×C163C2212yescomplemented
70#117×C163C2211yescharacteristic
71C23×D163C2311yescharacteristic
72C2×Dic326C2314yes—
73C2×Dic326C2312yes—
74Q8×D163—28no—
75C22×Dic163C2311yescharacteristic
76C2×C4×D163—24no—
77C2×C4×D163C2311yescharacteristic
78C2×C4×D163C2311yescharacteristic
79C2×C4×D163C2311yescharacteristic
80C2×C4×D163C2314yes—
81C2×C4×D163C2312yes—
82C2×C4×D163C2314yes—
83C2×C4×D163C2312yes—
84C2×C4×D163C2311yescharacteristic
85C4⋊Dic163C2312yes—
86C4⋊Dic163C2312yes—
87C4⋊Dic163C2314yes—
88C4⋊Dic163C2314yes—
89C4⋊Dic163C2314yes—
90C4⋊Dic163C2314yes—
91Dic163⋊C4C2312yes—
92Dic163⋊C4C2312yes—
93Dic163⋊C4C2314yes—
94Dic163⋊C4C2314yes—
95Dic163⋊C4C2314yes—
96Dic163⋊C4C2314yes—
97Dic163⋊C4C2314yes—
98Dic163⋊C4C2314yes—
99Dic163⋊C4C2314yes—
100Dic163⋊C4C2314yes—
101Dic163⋊C4C2314yes—
102C22⋊Dic163C2312yes—
103C22⋊Dic163C2314yes—
104(C2×D163)⋊C4C2312yes—
105(C2×D163)⋊C4C2312yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108C4×Dic163C2312yes—
109C4×Dic163C2312yes—
110C4×Dic163C2311yescharacteristic
111C4×Dic163C2311yescharacteristic
112C4×Dic163C2314yes—
113C4×Dic163C2314yes—
114C4×Dic163C2314yes—
115C2×Q8×C163C2312yes—
116C22×C4×C163C2311yescharacteristic
117C4⋊C4×C163C2312yes—
118C4⋊C4×C163C2312yes—
119C4⋊C4×C163C2314yes—
120C4⋊C4×C163C2314yes—
121C4⋊C4×C163C2314yes—
122C4⋊C4×C163C2314yes—
123C22⋊C4×C163C2312yes—
124C22⋊C4×C163C2314yes—
125C42×C163C2311yescharacteristic
126C42×C163C2311yescharacteristic
127C42×C163C2314yes—
128C22×D163—22no—
129C22×D163—24no—
130C22×D163C2411yescharacteristic
131C22×D163C2411yescharacteristic
132Dic326—24no—
133Dic326—28no—
134C2×Dic163—22no—
135C2×Dic163C2412yes—
136C2×Dic163C2412yes—
137C2×Dic163C2414yes—
138C2×Dic163C2414yes—
139C2×Dic163C2411yescharacteristic
140C2×Dic163C2411yescharacteristic
141C4×D163—22no—
142C4×D163—28no—
143C4×D163—24no—
144C4×D163—28no—
145C4×D163—24no—
146C4×D163—22no—
147C4×D163—22no—
148C4×D163#4114yes—
149C23×C163C2411yescharacteristic
150Q8×C163—24no—
151C2×C4×C163—22no—
152C2×C4×C163C2412yes—
153C2×C4×C163C2412yes—
154C2×C4×C163C2414yes—
155C2×C4×C163C2414yes—
156C2×C4×C163C2411yescharacteristic
157C2×C4×C163C2411yescharacteristic
158C22×C163—21no—
159C22×C163—22no—
160C22×C163C2511yescharacteristic
161C2×D163—21no—
162C2×D163—22no—
163C2×D163—44no—
164C2×D163C2×#4112yes—
165C2×D1632−1+414yes—
166C4×C163—21no—
167C4×C163—24no—
168C4×C163—22no—
169C4×C163—22no—
170C4×C163—24no—
171C4×C163C2×#4112yes—
172Dic163—21no—
173Dic163—24no—
174Dic163—22no—
175Dic163—22no—
176Dic163—24no—
177Dic163C2×#4112yes—
178C2×C163—41no—
179C2×C163C22×#4111yescharacteristic
180C2×C163C2×2−1+412yes—
181D163—41no—
182D163#54114yesdirect product factor
183C163C2×#54111yescharacteristic, complemented, Sylow
184C2×#541—1631nomaximal, Sylow
185#541—1631no—
186#541—1634no—
187#541—1634no—
188#541—1631no—
189#541—1632no—
190#541—1632no—
191#541—1631no—
192#541D16311yescharacteristic, direct product factor
193C2×C4⋊Q8—1631no—
194C2×#126—1634no—
195C2×#125—1631no—
196C2×#125—1634no—
197C2×C22⋊Q8—1632no—
198C2×C4×Q8—1632no—
199C2×#117—1631no—
200C22×Q8—1632no—
201C23×C4—1631no—
202C4⋊Q8—1632no—
203C4⋊Q8—1634no—
204C4⋊Q8—1631no—
205C4⋊Q8C2×D16311yescharacteristic, complemented
206#126—1634no—
207#126—1634no—
208#126—1634no—
209#126—1634no—
210#126—1634no—
211#126—1634no—
212#126—1634no—
213#126C2×D16314yes—
214#125—1634no—
215#125—1634no—
216#125—1634no—
217#125—1634no—
218#125—1634no—
219#125—1632no—
220#125—1634no—
221#125—1634no—
222#125—1631no—
223#125—1634no—
224#125C2×D16311yescharacteristic, complemented
225#125C2×D16314yescomplemented
226C22⋊Q8—1632no—
227C22⋊Q8—1632no—
228C22⋊Q8—1634no—
229C22⋊Q8—1632no—
230C22⋊Q8—1634no—
231C22⋊Q8C2×D16312yes—
232C4×Q8—1632no—
233C4×Q8—1634no—
234C4×Q8—1632no—
235C4×Q8—1634no—
236C4×Q8—1632no—
237C4×Q8C2×D16312yescomplemented
238#117—1631no—
239#117—1632no—
240#117—1632no—
241#117—1631no—
242#117—1631no—
243#117C2×D16311yescharacteristic
244C2×C4⋊C4—1632no—
245C2×C4⋊C4—1632no—
246C2×C4⋊C4—1634no—
247C2×C4⋊C4—1634no—
248C2×C4⋊C4—1634no—
249C2×C4⋊C4—1634no—
250C2×C22⋊C4—1632no—
251C2×C22⋊C4—1634no—
252C2×C42—1631no—
253C2×C42—1631no—
254C2×C42—1634no—
255C24—1631no—
256C2×Q8—3268no—
257C2×Q8—1632no—
258C2×Q8—1634no—
259C2×Q8C22×D16312yes—
260C22×C4—3264no—
261C22×C4—1631no—
262C22×C4—1632no—
263C22×C4—1632no—
264C22×C4—1634no—
265C22×C4—1634no—
266C22×C4—1631no—
267C22×C4—1631no—
268C22×C4—1631no—
269C22×C4—1631no—
270C22×C4C22×D16311yescharacteristic
271C4⋊C4—1632no—
272C4⋊C4—1632no—
273C4⋊C4—1632no—
274C4⋊C4—1634no—
275C4⋊C4—1632no—
276C4⋊C4—1634no—
277C4⋊C4—1634no—
278C4⋊C4—1634no—
279C4⋊C4—1634no—
280C4⋊C4—1634no—
281C4⋊C4—1634no—
282C4⋊C4—1634no—
283C4⋊C4—1634no—
284C4⋊C4—1634no—
285C4⋊C4—1634no—
286C4⋊C4—1634no—
287C4⋊C4—1634no—
288C4⋊C4C22×D16312yes—
289C4⋊C4C22×D16312yes—
290C4⋊C4C22×D16314yes—
291C4⋊C4C22×D16314yes—
292C4⋊C4C22×D16314yes—
293C4⋊C4C22×D16314yes—
294C22⋊C4—1632no—
295C22⋊C4—1634no—
296C22⋊C4—1632no—
297C22⋊C4—1632no—
298C22⋊C4—1634no—
299C22⋊C4—1634no—
300C22⋊C4C22×D16312yes—
301C22⋊C4C22×D16314yes—
302C42—1631no—
303C42—1632no—
304C42—1631no—
305C42—1632no—
306C42—1634no—
307C42—1634no—
308C42—1634no—
309C42C22×D16311yescharacteristic
310C42C22×D16311yescharacteristic
311C42C22×D16314yes—
312C23—3262no—
313C23—3264no—
314C23—1631no—
315C23—1631no—
316C23C23×D16311yescharacteristic
317Q8—24no—
318Q8—3268no—
319Q8—3264no—
320C2×C4—22no—
321C2×C4—3262no—
322C2×C4—3268no—
323C2×C4—3264no—
324C2×C4—3264no—
325C2×C4—3268no—
326C2×C4—3262no—
327C2×C4—3262no—
328C2×C4—3262no—
329C2×C4—1632no—
330C2×C4—1632no—
331C2×C4—1634no—
332C2×C4—1634no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4—1634no—
336C2×C4C23×D16312yes—
337C2×C4C23×D16312yes—
338C2×C4C23×D16314yes—
339C2×C4C23×D16314yes—
340C2×C4C23×D16311yescharacteristic
341C2×C4C23×D16311yescharacteristic
342C22—21no—
343C22—22no—
344C22—3261no—
345C22—3262no—
346C22—6524no—
347C22—1632no—
348C22—1634no—
349C22C24×D16311yescharacteristic
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—24no—
355C4—3261no—
356C4—3264no—
357C4—3262no—
358C4—3262no—
359C4—3264no—
360C4—1632no—
361C4#41×D16312yes—
362C2—41no—
363C2—6521no—
364C2—1634no—
365C2C2×#41×D16311yescharacteristic
366C22−1+4×D16312yes—
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#541×C163C2
prime radicalC23×D163C23
hypercenter#541D163
hypocenterC163C2×#541
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #23782278C2semidirect
2. #23782277C2semidirect
3. #23782276C2semidirect
4. #23782275C2semidirect
5. #23782274C2semidirect
6. #23782273C2semidirect
7. #23782272C2semidirect
8. C4⋊Q8×D163C2semidirect
10. #125×D163C2semidirect
11. #125×D163C2semidirect
13. C4×Q8×D163C2semidirect
15. #541×C163C2semidirect
18. #19104735C22semidirect
19. #19104734C22semidirect
20. #19104733C22semidirect
28. #19104727C22semidirect
29. #19104727C22semidirect
30. #19104726C22semidirect
31. #19104726C22semidirect
32. #19104726C22semidirect
33. #19104725C22semidirect
34. #19104725C22semidirect
35. #19104725C22semidirect
36. #19104725C22semidirect
37. #19104725C22semidirect
43. Q8×Dic163C22semidirect
44. C4×Dic326C22semidirect
45. C4×Dic326C22semidirect
46. #19104700C22semidirect
47. #19104700C22semidirect
64. C4⋊Q8×C163C22semidirect
66. #125×C163C22semidirect
67. #125×C163C22semidirect
69. C4×Q8×C163C22semidirect
182. D163#541direct (N × H)
183. C163C2×#541semidirect
192. #541D163direct (N × H)
205. C4⋊Q8C2×D163semidirect
224. #125C2×D163semidirect
225. #125C2×D163semidirect
237. C4×Q8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822782
#237822772
#237822762
#237822752
#237822742
#237822732
#237822722
C4⋊Q8×D1632
#126×D1632
#125×D1632
C22⋊Q8×D1632
C4×Q8×D1632
#117×D1632
#541×C1632
C2×#541163