Finite Group Databasebeta

← #35228677

Subgroups — #35228677

Order 20864.

1058 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782414C212yesmaximal, complemented
2#23782413C212yesmaximal, complemented
3#23782411C214yesmaximal, complemented
4#23782407C211yesmaximal, characteristic, complemented
5#23782406C214yesmaximal, complemented
6#23782405C212yesmaximal, complemented
7#23782216C212yesmaximal, complemented
8#23782214C211yesmaximal, characteristic, complemented
9#23782200C212yesmaximal, complemented
10#23782198C214yesmaximal, complemented
11#23782178C214yesmaximal, complemented
12#23782173C212yesmaximal, complemented
13#563×C163C211yesmaximal, characteristic, complemented
14C2×#14579700C2212yes—
15C2×#14579700C2211yescharacteristic, complemented
16C2×#14579698C2211yescharacteristic
17C22×C4×D163C2212yescomplemented
18#19104733C2211yescharacteristic, complemented
19#19104732C2212yes—
20#19104731C2211yescharacteristic
21#19104730C2214yescomplemented
22#19104729C2214yescomplemented
23#19104725C2214yescomplemented
24#19104724C2212yes—
25#19104724C2212yescomplemented
26#19104723C2212yes—
27#19104723C2211yescharacteristic, complemented
28#19104721C2214yes—
29#19104720C2214yes—
30#19104720C2214yescomplemented
31#19104720C2214yescomplemented
32#19104720C2214yescomplemented
33#19104718C2212yescomplemented
34#19104717C2214yescomplemented
35#19104715C2212yescomplemented
36#19104713C2214yescomplemented
37#19104713C2214yescomplemented
38#19104711C2214yescomplemented
39#19104710C2212yes—
40#19104710C2214yes—
41#19104710C2214yescomplemented
42#19104710C2214yescomplemented
43#19104709C2214yes—
44#19104708C2214yes—
45#19104708C2214yes—
46#19104708C2214yescomplemented
47#19104705C2214yes—
48#19104705C2214yescomplemented
49#19104705C2212yescomplemented
50#19104703C2212yes—
51#19104700C2212yescomplemented
52#19104697C2212yes—
53#19104697C2214yescomplemented
54#19104697C2214yescomplemented
55(C22⋊D163)⋊C4C2214yes—
56(C22⋊D163)⋊C4C2214yescomplemented
57#19104693C2214yescomplemented
58#19104691C2212yescomplemented
59C42×D163C2212yescomplemented
60C42×D163C2211yescharacteristic, complemented
61C4⋊Q8×C163C2212yescomplemented
62#126×C163C2214yescomplemented
63#125×C163C2212yescomplemented
64#124×C163C2211yescharacteristic
65#123×C163C2212yes—
66C22⋊Q8×C163C2214yescomplemented
67C23×D163C2312yes—
68#14579700—44no—
69#14579700—42no—
70#14579698—42no—
71C2×Dic326C2311yescharacteristic
72C2×Dic326C2312yes—
73C2×D652C2312yes—
74C2×D652C2314yes—
75C2×D652C2312yes—
76C2×D652C2311yescharacteristic
77C2×D652C2312yes—
78C2×D652C2314yes—
79C2×D652C2314yes—
80C2×C22⋊D163C2312yes—
81C2×C22⋊D163C2314yes—
82C2×C22⋊D163C2314yes—
83C22×Dic163C2312yes—
84C2×C4×D163—44no—
85C2×C4×D163C2314yes—
86C2×C4×D163C2311yescharacteristic
87C2×C4×D163C2312yes—
88C2×C4×D163C2314yes—
89C2×C4×D163C2314yes—
90C2×C4×D163C2312yes—
91C4⋊Dic163C2314yes—
92Dic163⋊C4C2314yes—
93Dic163⋊C4C2314yes—
94Dic163⋊C4C2314yes—
95Dic163⋊C4C2314yes—
96C22⋊Dic163C2312yes—
97(C2×D163)⋊C4C2314yes—
98(C2×D163)⋊C4C2314yes—
99(C2×D163)⋊C4C2314yes—
100(C2×D163)⋊C4C2314yes—
101(C2×D163)⋊C4C2314yes—
102(C2×D163)⋊C4C2314yes—
103(C2×D163)⋊C4C2314yes—
104(C2×D163)⋊C4C2314yes—
105(C2×D163)⋊C4C2314yes—
106C4×Dic163C2314yes—
107C4×Dic163C2312yes—
108C4×Dic163C2312yes—
109C4×Dic163C2311yescharacteristic
110C2×Q8×C163C2312yes—
111C2×Q8×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C22×C4×C163C2312yes—
114C4⋊C4×C163C2312yes—
115C4⋊C4×C163C2314yes—
116C4⋊C4×C163C2314yes—
117C4⋊C4×C163C2314yes—
118C4⋊C4×C163C2312yes—
119C22⋊C4×C163C2314yes—
120C22⋊C4×C163C2314yes—
121C22⋊C4×C163C2312yes—
122C42×C163C2311yescharacteristic
123C42×C163C2312yes—
124C22×D163—42no—
125C22×D163—44no—
126C22×D163C2414yes—
127C22×D163C2411yescharacteristic
128C22×D163C2412yes—
129Dic326—42no—
130Dic326—41no—
131D652—44no—
132D652—44no—
133D652—42no—
134D652—44no—
135D652—41no—
136D652—42no—
137D652—42no—
138C22⋊D163—44no—
139C22⋊D163—44no—
140C22⋊D163—42no—
141C2×Dic163—24no—
142C2×Dic163C2414yes—
143C2×Dic163C2412yes—
144C2×Dic163C2412yes—
145C2×Dic163C2411yescharacteristic
146C4×D163—28no—
147C4×D163—22no—
148C4×D163—24no—
149C4×D163—44no—
150C4×D163—42no—
151C4×D163—44no—
152C23×C163C2412yes—
153Q8×C163—42no—
154Q8×C163—41no—
155D4×C163—41no—
156C2×C4×C163—42no—
157C2×C4×C163C2412yes—
158C2×C4×C163C2414yes—
159C2×C4×C163C2412yes—
160C2×C4×C163C2411yescharacteristic
161C2×C4×C163C2414yes—
162C22×C163—22no—
163C22×C163—24no—
164C22×C163C2511yescharacteristic
165C2×D163—24no—
166C2×D163—21no—
167C2×D163—22no—
168C2×D163—24no—
169C2×D163—22no—
170C2×D163—24no—
171C2×D163—24no—
172C2×D163—48no—
173C4×C163—24no—
174C4×C163—21no—
175C4×C163—22no—
176C4×C163—24no—
177C4×C163—22no—
178Dic163—22no—
179Dic163—22no—
180Dic163—24no—
181Dic163H(5,𝔽2)12yes—
182C2×C163—42no—
183C2×C163C2×H(5,𝔽2)11yescharacteristic
184C2×C163#58512yes—
185D163—41no—
186D163—42no—
187D163—44no—
188C163#352911yescharacteristic, complemented, Sylow
189#3529—1631nomaximal, Sylow
190#563—1632no—
191#563D16311yescharacteristic, complemented
192#562—1632no—
193#562—1634no—
194H(3,𝔽4)—1631no—
195#560—1634no—
196#560—1632no—
197#532—1632no—
198#532—1631no—
199#529—1634no—
200#529—1632no—
201#525—1632no—
202#525—1634no—
203C2×#41—1632no—
204C2×#41—1631no—
205C2×#41—1631no—
206C23×C4—1632no—
207C4⋊Q8—1631no—
208C4⋊Q8C2×D16312yescomplemented
209#127—1631no—
210#127—1632no—
211#126—1634no—
212#126—1634no—
213#126C2×D16314yescomplemented
214#125—1634no—
215#125C2×D16312yescomplemented
216#124—1632no—
217#124—1632no—
218#124—1634no—
219#124—1632no—
220#124—1631no—
221#124C2×D16311yescharacteristic
222#123—1632no—
223#123—1634no—
224#123—1634no—
225#123—1634no—
226#123—1634no—
227#123—1634no—
228#123C2×D16312yes—
229C22⋊Q8—1634no—
230C22⋊Q8—1634no—
231C22⋊Q8—1634no—
232C22⋊Q8—1632no—
233C22⋊Q8C2×D16314yescomplemented
234#121—1634no—
235#121—1634no—
236#121—1632no—
237#121—1634no—
238#121—1634no—
239#121—1634no—
240#121—1634no—
241#121—1634no—
242#120—1632no—
243#120—1634no—
244#120—1632no—
245#120—1634no—
246C4×Q8—1632no—
247C4×D4—1632no—
248C4×D4—1634no—
249C4×D4—1634no—
250C4×D4—1634no—
251C4×D4—1634no—
252#117—1634no—
253#117—1632no—
254C2×C42—1632no—
255C2×C42—1631no—
256C24—1632no—
257#41—6522no—
258#41—6524no—
259#41—6522no—
260C2×Q8—1632no—
261C2×Q8—1631no—
262C2×Q8C22×D16312yes—
263C2×Q8C22×D16311yescharacteristic
264C2×D4—1634no—
265C2×D4—1634no—
266C2×D4—1632no—
267C2×D4—1632no—
268C2×D4—1634no—
269C2×D4—1634no—
270C2×D4—1632no—
271C2×D4—1634no—
272C2×D4—1631no—
273C2×D4—1632no—
274C2×D4C22×D16311yescharacteristic
275C22×C4—6524no—
276C22×C4—1632no—
277C22×C4—1631no—
278C22×C4—1632no—
279C22×C4—1634no—
280C22×C4—1634no—
281C22×C4—1632no—
282C22×C4—1634no—
283C22×C4C22×D16312yes—
284C4⋊C4—1634no—
285C4⋊C4—1634no—
286C4⋊C4—1634no—
287C4⋊C4—1634no—
288C4⋊C4—1634no—
289C4⋊C4C22×D16314yes—
290C4⋊C4C22×D16314yes—
291C4⋊C4C22×D16314yes—
292C4⋊C4C22×D16312yes—
293C4⋊C4C22×D16312yes—
294C22⋊C4—1632no—
295C22⋊C4—1634no—
296C22⋊C4—1634no—
297C22⋊C4—1634no—
298C22⋊C4—1634no—
299C22⋊C4—1634no—
300C22⋊C4—1634no—
301C22⋊C4—1634no—
302C22⋊C4—1634no—
303C22⋊C4—1634no—
304C22⋊C4C22×D16312yes—
305C22⋊C4C22×D16314yes—
306C22⋊C4C22×D16314yes—
307C42—1634no—
308C42—1632no—
309C42—1632no—
310C42—1631no—
311C42C22×D16311yescharacteristic
312C42C22×D16312yes—
313C23—6522no—
314C23—6524no—
315C23—1634no—
316C23—1632no—
317C23—1631no—
318C23C23×D16312yes—
319Q8—42no—
320Q8—41no—
321Q8—6521no—
322Q8—6522no—
323D4—41no—
324D4—6524no—
325D4—6521no—
326D4—6522no—
327D4—6522no—
328D4—6524no—
329D4—6524no—
330D4—6522no—
331D4—6522no—
332D4—6524no—
333D4—6524no—
334C2×C4—42no—
335C2×C4—3268no—
336C2×C4—3262no—
337C2×C4—3264no—
338C2×C4—3264no—
339C2×C4—6524no—
340C2×C4—6524no—
341C2×C4—6522no—
342C2×C4—1634no—
343C2×C4—1632no—
344C2×C4—1632no—
345C2×C4—1631no—
346C2×C4C23×D16312yes—
347C2×C4C23×D16311yescharacteristic
348C2×C4C23×D16314yes—
349C2×C4C23×D16312yes—
350C2×C4C23×D16314yes—
351C22—22no—
352C22—24no—
353C22—3264no—
354C22—3262no—
355C22—3261no—
356C22—3262no—
357C22—3264no—
358C22—3264no—
359C22—3264no—
360C22—6528no—
361C22C24×D16311yescharacteristic
362C4—24no—
363C4—21no—
364C4—22no—
365C4—24no—
366C4—22no—
367C4—3262no—
368C4—3262no—
369C4—3264no—
370C4—1632no—
371C2—42no—
372C2—6524no—
373C2—6522no—
374C2—6521no—
375C2H(5,𝔽2)×D16311yescharacteristic
376C2#2378250212yes—
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#563×C163C2
hypercenter#563D163
hypocenterC163#3529
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #23782414C2semidirect
2. #23782413C2semidirect
3. #23782411C2semidirect
4. #23782407C2semidirect
5. #23782406C2semidirect
6. #23782405C2semidirect
7. #23782216C2semidirect
8. #23782214C2semidirect
9. #23782200C2semidirect
10. #23782198C2semidirect
11. #23782178C2semidirect
12. #23782173C2semidirect
13. #563×C163C2semidirect
15. C2×#14579700C22semidirect
17. C22×C4×D163C22semidirect
18. #19104733C22semidirect
21. #19104730C22semidirect
22. #19104729C22semidirect
23. #19104725C22semidirect
25. #19104724C22semidirect
27. #19104723C22semidirect
30. #19104720C22semidirect
31. #19104720C22semidirect
32. #19104720C22semidirect
33. #19104718C22semidirect
34. #19104717C22semidirect
35. #19104715C22semidirect
36. #19104713C22semidirect
37. #19104713C22semidirect
38. #19104711C22semidirect
41. #19104710C22semidirect
42. #19104710C22semidirect
46. #19104708C22semidirect
48. #19104705C22semidirect
49. #19104705C22semidirect
51. #19104700C22semidirect
53. #19104697C22semidirect
54. #19104697C22semidirect
56. (C22⋊D163)⋊C4C22semidirect
57. #19104693C22semidirect
58. #19104691C22semidirect
59. C42×D163C22semidirect
60. C42×D163C22semidirect
61. C4⋊Q8×C163C22semidirect
62. #126×C163C22semidirect
63. #125×C163C22semidirect
66. C22⋊Q8×C163C22semidirect
188. C163#3529semidirect
191. #563D163semidirect
208. C4⋊Q8C2×D163semidirect
213. #126C2×D163semidirect
215. #125C2×D163semidirect
233. C22⋊Q8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824142
#237824132
#237824112
#237824072
#237824062
#237824052
#237822162
#237822142
#237822002
#237821982
#237821782
#237821732
#563×C1632
#3529163