Finite Group Databasebeta

← #496×D163 #35224966

Subgroups — #496×D163 #35224966

Order 20864.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×SD16×D163C212yesmaximal, complemented
2C2×D8×D163C212yesmaximal, complemented
3#127×D163C211yesmaximal, characteristic, complemented
4#124×D163C211yesmaximal, characteristic, complemented
5#23782055C211yesmaximal, characteristic, complemented
6#23782054C211yesmaximal, characteristic, complemented
7#23782053C212yesmaximal, complemented
8#23782052C211yesmaximal, characteristic, complemented
9#23782051C212yesmaximal, complemented
10#97×D163C211yesmaximal, characteristic, complemented
11#496×C163C211yesmaximal, characteristic, complemented
12C2×Q8×D163C2211yescharacteristic
13C2×D4×D163—22no—
14C2×D4×D163C2211yescharacteristic
15C2×D4×D163C2212yes—
16C2×#14579688C2212yescomplemented
17C2×#14579687C2212yescomplemented
18C2×#14579686C2212yescomplemented
19SD16×D163—28no—
20C2×D1304C2212yescomplemented
21C2×#14579684C2212yescomplemented
22C2×#14579684C2212yescomplemented
23D8×D163—28no—
24C2×C8×D163C2212yes—
25#19104732C2211yescharacteristic, complemented
26#19104731—22no—
27#19104731C2212yescomplemented
28#19104724C2211yescharacteristic, complemented
29#19104723C2211yescharacteristic, complemented
30#19104722C2211yescharacteristic, complemented
31#19104721—22no—
32C22⋊C4×D163—22no—
33C42×D163C2211yescharacteristic
34#19104646C2211yescharacteristic, complemented
35#19104645C2212yescomplemented
36C2×SD16×C163C2212yescomplemented
37C2×D8×C163C2212yescomplemented
38#127×C163C2211yescharacteristic, complemented
39#124×C163C2211yescharacteristic, complemented
40#97×C163C2211yescharacteristic, complemented
41C23×D163—22no—
42C23×D163—21no—
43C2×Dic326—21no—
44C2×Dic326C2311yescharacteristic
45Q8×D163—24no—
46C2×D652—22no—
47C2×D652C2311yescharacteristic
48C2×D652C2312yes—
49C2×C22⋊D163—24no—
50C2×C22⋊D163—21no—
51C2×C22⋊D163—22no—
52D4×D163—24no—
53D4×D163—24no—
54D4×D163—24no—
55D4×D163—48no—
56C2×C4×D163—21no—
57C2×C4×D163D412yes—
58C2×C4×D163C2311yescharacteristic
59#14579688—24no—
60#14579687—24no—
61#14579686—24no—
62D1304—24no—
63#14579684—24no—
64#14579684—24no—
65C2×C163⋊C8C2312yes—
66C8×D163D418yescomplemented
67C22⋊Dic163—22no—
68(C2×D163)⋊C4—22no—
69(C2×D163)⋊C4—22no—
70C4×Dic163D412yes—
71C4×Dic163C2311yescharacteristic
72C2×Q8×C163C2311yescharacteristic
73C2×D4×C163—22no—
74C2×D4×C163C2311yescharacteristic
75C2×D4×C163C2312yes—
76SD16×C163—24no—
77D8×C163—24no—
78C2×C8×C163C2312yes—
79C22⋊C4×C163—22no—
80C42×C163C2311yescharacteristic
81C22×D163—21no—
82C22×D163—22no—
83C22×D163—42no—
84C22×D163—44no—
85C22×D163—48no—
86C22×D163—44no—
87C22×D163C2×D411yescharacteristic
88Dic326—22no—
89Dic326—42no—
90D652—22no—
91D652—22no—
92D652—22no—
93D652—44no—
94C22⋊D163—48no—
95C22⋊D163—42no—
96C22⋊D163—44no—
97C2×Dic163—21no—
98C2×Dic163C2×D412yes—
99C2×Dic163C2×D411yescharacteristic
100C4×D163—24no—
101C4×D163—42no—
102C4×D163C2×D414yes—
103C163⋊C8C2×D414yescomplemented
104C23×C163—21no—
105C23×C163—22no—
106Q8×C163—22no—
107D4×C163—22no—
108D4×C163—22no—
109D4×C163—22no—
110D4×C163—44no—
111C2×C4×C163—21no—
112C2×C4×C163C2×D412yes—
113C2×C4×C163C2411yescharacteristic
114C8×C163C2×D414yescomplemented
115C22×C163—41no—
116C22×C163—42no—
117C22×C163—44no—
118C22×C163—42no—
119C22×C163C22×D411yescharacteristic
120C2×D163—41no—
121C2×D163—42no—
122C2×D163—44no—
123C2×D163—42no—
124C2×D163—88no—
125C2×D163—84no—
126C2×D163#12712yes—
127C2×D163AGL(1,ℤ/8)14yes—
128C4×C163—22no—
129C4×C163—41no—
130C4×C163C22×D412yes—
131Dic163—22no—
132Dic163—41no—
133Dic163#12712yes—
134C2×C163—82no—
135C2×C163—81no—
136C2×C163C2×#12711yescharacteristic
137C2×C163C2×AGL(1,ℤ/8)12yes—
138D163—82no—
139D163—81no—
140D163#49614yesdirect product factor
141C163C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1631nomaximal, Sylow
143C22×SD16—1632no—
144C22×D8—1632no—
145C2×#127—1631no—
146C2×#124—1631no—
147#496—1632no—
148#496—1632no—
149#496—1631no—
150#496—1631no—
151#496—1631no—
152#496D16311yescharacteristic, direct product factor
153C2×#97—1631no—
154C22×Q8—1631no—
155C22×D4—3262no—
156C22×D4—1631no—
157C22×D4—1632no—
158C2×SD16—3268no—
159C2×SD16—1632no—
160C2×SD16—1632no—
161C2×SD16—1632no—
162C2×SD16C2×D16312yescomplemented
163C2×D8—3268no—
164C2×D8—1632no—
165C2×D8—1632no—
166C2×D8—1632no—
167C2×D8C2×D16312yescomplemented
168C22×C8—1632no—
169#127—3262no—
170#127—1632no—
171#127—1631no—
172#127C2×D16311yescharacteristic, complemented
173#124—3262no—
174#124—1631no—
175#124—1631no—
176#124—1631no—
177#124C2×D16311yescharacteristic, complemented
178C2×C22⋊C4—3262no—
179C2×C42—1631no—
180#97—1632no—
181#97—1631no—
182#97C2×D16311yescharacteristic, complemented
183C24—3261no—
184C24—3262no—
185C2×Q8—3261no—
186C2×Q8—3264no—
187C2×Q8—1631no—
188C2×Q8C22×D16311yescharacteristic
189C2×D4—22no—
190C2×D4—3264no—
191C2×D4—3264no—
192C2×D4—3264no—
193C2×D4—3261no—
194C2×D4—3262no—
195C2×D4—3264no—
196C2×D4—3262no—
197C2×D4—6528no—
198C2×D4—1631no—
199C2×D4—1632no—
200C2×D4C22×D16311yescharacteristic
201C2×D4C22×D16312yes—
202C22×C4—3261no—
203C22×C4—1632no—
204C22×C4—1631no—
205SD16—24no—
206SD16—3264no—
207SD16—3264no—
208SD16—3264no—
209D8—24no—
210D8—3264no—
211D8—3264no—
212D8—3264no—
213C2×C8—1632no—
214C2×C8—1638no—
215C2×C8C22×D16312yes—
216C22⋊C4—22no—
217C22⋊C4—3262no—
218C22⋊C4—3262no—
219C22⋊C4—3262no—
220C42—1632no—
221C42—1631no—
222C42C22×D16311yescharacteristic
223C23—21no—
224C23—22no—
225C23—3261no—
226C23—3262no—
227C23—6522no—
228C23—6524no—
229C23—6528no—
230C23—6524no—
231C23—1631no—
232Q8—22no—
233Q8—3262no—
234Q8—6522no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—3262no—
240D4—3262no—
241D4—3262no—
242D4—6524no—
243D4—6528no—
244D4—6522no—
245D4—6524no—
246C2×C4—21no—
247C2×C4—3264no—
248C2×C4—3261no—
249C2×C4—6522no—
250C2×C4—1632no—
251C2×C4—1631no—
252C2×C4—1634no—
253C2×C4D4×D16312yes—
254C2×C4C23×D16311yescharacteristic
255C8—1634no—
256C8D4×D16314yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—13048no—
262C22—13044no—
263C22—6524no—
264C22—6522no—
265C22—6521no—
266C22—6522no—
267C22—1634no—
268C22—1632no—
269C22C2×D4×D16311yescharacteristic
270C4—22no—
271C4—41no—
272C4—3262no—
273C4—6521no—
274C4—1632no—
275C4C2×D4×D16312yes—
276C2—82no—
277C2—81no—
278C2—13042no—
279C2—13041no—
280C2—1634no—
281C2#127×D16311yescharacteristic
282C2AGL(1,ℤ/8)×D16312yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#496×C163C2
socleC22×C163C22×D4
hypercenter#496D163
hypocenterC163C2×#496
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. C2×D8×D163C2semidirect
3. #127×D163C2semidirect
4. #124×D163C2semidirect
5. #23782055C2semidirect
6. #23782054C2semidirect
7. #23782053C2semidirect
8. #23782052C2semidirect
9. #23782051C2semidirect
10. #97×D163C2semidirect
11. #496×C163C2semidirect
16. C2×#14579688C22semidirect
17. C2×#14579687C22semidirect
18. C2×#14579686C22semidirect
20. C2×D1304C22semidirect
21. C2×#14579684C22semidirect
22. C2×#14579684C22semidirect
25. #19104732C22semidirect
27. #19104731C22semidirect
28. #19104724C22semidirect
29. #19104723C22semidirect
30. #19104722C22semidirect
34. #19104646C22semidirect
35. #19104645C22semidirect
36. C2×SD16×C163C22semidirect
37. C2×D8×C163C22semidirect
38. #127×C163C22semidirect
39. #124×C163C22semidirect
40. #97×C163C22semidirect
66. C8×D163D4semidirect
103. C163⋊C8C2×D4semidirect
114. C8×C163C2×D4semidirect
140. D163#496direct (N × H)
141. C163C2×#496semidirect
152. #496D163direct (N × H)
162. C2×SD16C2×D163semidirect
167. C2×D8C2×D163semidirect
172. #127C2×D163semidirect
177. #124C2×D163semidirect
182. #97C2×D163semidirect
256. C8D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
C2×D8×D1632
#127×D1632
#124×D1632
#237820552
#237820542
#237820532
#237820522
#237820512
#97×D1632
#496×C1632
C2×#496163