Finite Group Databasebeta

← #469×D163 #35224128

Subgroups — #469×D163 #35224128

Order 20864.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D163C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#121×D163C212yesmaximal, complemented
4#23781919C211yesmaximal, characteristic, complemented
5#23781918C211yesmaximal, characteristic, complemented
6#23781917C212yesmaximal, complemented
7#23781916C212yesmaximal, complemented
8#23781915C211yesmaximal, characteristic, complemented
9#106×D163C211yesmaximal, characteristic, complemented
10#102×D163C212yesmaximal, complemented
11#469×C163C211yesmaximal, characteristic, complemented
12C2×D4×D163—22no—
13C2×D4×D163C2212yes—
14C22×C4×D163C2211yescharacteristic
15C2×D1304C2211yescharacteristic, complemented
16C2×#14579684C2212yescomplemented
17D8×D163—24no—
18C2×#14579683C2211yescharacteristic, complemented
19C2×#14579682C2211yescharacteristic, complemented
20C2×#14579682C2211yescharacteristic, complemented
21M16×D163—24no—
22C2×C8×D163C2211yescharacteristic
23C2×C8×D163C2211yescharacteristic
24#19104710C2212yescomplemented
25#19104709C2212yescomplemented
26#19104708—22no—
27#19104707—22no—
28#19104706C2212yescomplemented
29C4⋊C4×D163C2212yes—
30C22⋊C4×D163—22no—
31#19104667C2211yescharacteristic, complemented
32#19104666C2212yescomplemented
33#19104657C2212yescomplemented
34#19104656C2212yescomplemented
35#19104655C2212yescomplemented
36C2×D8×C163C2211yescharacteristic, complemented
37C2×M16×C163C2211yescharacteristic, complemented
38#121×C163C2212yescomplemented
39#106×C163C2211yescharacteristic, complemented
40#102×C163C2212yescomplemented
41C23×D163—22no—
42C23×D163D411yescharacteristic
43C2×D652—22no—
44C2×D652C2312yes—
45C2×C22⋊D163—22no—
46C2×C22⋊D163—22no—
47C2×C22⋊D163—22no—
48D4×D163—24no—
49D4×D163—24no—
50D4×D163—48no—
51C22×Dic163D411yescharacteristic
52C2×C4×D163—22no—
53C2×C4×D163—22no—
54C2×C4×D163—22no—
55C2×C4×D163D411yescharacteristic
56C2×C4×D163D411yescharacteristic
57C2×C4×D163C2311yescharacteristic
58C2×C4×D163C2311yescharacteristic
59D1304—22no—
60#14579684—24no—
61#14579683—22no—
62#14579682—22no—
63#14579682—22no—
64C2×C163⋊C8C2311yescharacteristic
65C2×C163⋊C8C2311yescharacteristic
66C8×D163—22no—
67C8×D163D414yescomplemented
68C4⋊Dic163C2312yes—
69Dic163⋊C4—22no—
70C22⋊Dic163—22no—
71(C2×D163)⋊C4—22no—
72(C2×D163)⋊C4—22no—
73C2×D4×C163—22no—
74C2×D4×C163C2312yes—
75C22×C4×C163C2311yescharacteristic
76D8×C163—22no—
77M16×C163—22no—
78C2×C8×C163C2311yescharacteristic
79C2×C8×C163C2311yescharacteristic
80C4⋊C4×C163C2312yes—
81C22⋊C4×C163—22no—
82C22×D163—22no—
83C22×D163—24no—
84C22×D163—22no—
85C22×D163—44no—
86C22×D163—44no—
87C22×D163—44no—
88C22×D163C2×D411yescharacteristic
89C22×D163C2×D411yescharacteristic
90D652—22no—
91D652—22no—
92D652—44no—
93C22⋊D163—44no—
94C22⋊D163—44no—
95C22⋊D163—44no—
96C2×Dic163—22no—
97C2×Dic163—22no—
98C2×Dic163C2×D411yescharacteristic
99C2×Dic163C2×D411yescharacteristic
100C4×D163—22no—
101C4×D163—21no—
102C4×D163—21no—
103C4×D163—22no—
104C4×D163—44no—
105C4×D163C2×D412yes—
106C4×D163#4112yes—
107C163⋊C8—21no—
108C163⋊C8C2×D412yescomplemented
109C23×C163—22no—
110C23×C163C2×D411yescharacteristic
111D4×C163—22no—
112D4×C163—22no—
113D4×C163—44no—
114C2×C4×C163—21no—
115C2×C4×C163—21no—
116C2×C4×C163—22no—
117C2×C4×C163C2×D411yescharacteristic
118C2×C4×C163C2411yescharacteristic
119C8×C163—21no—
120C8×C163C2×D412yescomplemented
121C22×C163—21no—
122C22×C163—22no—
123C22×C163—42no—
124C22×C163—42no—
125C22×C163—42no—
126C22×C163C22×D411yescharacteristic
127C2×D163—21no—
128C2×D163—22no—
129C2×D163—44no—
130C2×D163—42no—
131C2×D163—42no—
132C2×D163—42no—
133C2×D163—88no—
134C2×D163#12112yes—
135C2×D163AGL(1,ℤ/8)14yes—
136C4×C163—21no—
137C4×C163—42no—
138C4×C163C22×D411yescharacteristic
139C4×C163C2×#4111yescharacteristic
140Dic163—21no—
141Dic163—42no—
142Dic163#12112yes—
143C2×C163—41no—
144C2×C163—82no—
145C2×C163C2×#12111yescharacteristic
146C2×C163C2×AGL(1,ℤ/8)12yes—
147D163—41no—
148D163—82no—
149D163#46914yesdirect product factor
150C163C2×#46911yescharacteristic, complemented, Sylow
151C2×#469—1631nomaximal, Sylow
152C22×D8—1631no—
153C22×M16—1631no—
154C2×#121—1632no—
155#469—1631no—
156#469—1631no—
157#469—1632no—
158#469—1631no—
159#469—1632no—
160#469D16311yescharacteristic, direct product factor
161C2×#106—1631no—
162C2×#102—1632no—
163C22×D4—3262no—
164C22×D4—1632no—
165C23×C4—1631no—
166C2×D8—3264no—
167C2×D8—1632no—
168C2×D8—1631no—
169C2×D8C2×D16311yescharacteristic, complemented
170C2×M16—3264no—
171C2×M16—1631no—
172C2×M16—1631no—
173C2×M16—1631no—
174C2×M16C2×D16311yescharacteristic, complemented
175C22×C8—1631no—
176C22×C8—1631no—
177#121—3262no—
178#121—3262no—
179#121—1632no—
180#121—1632no—
181#121—1632no—
182#121C2×D16312yescomplemented
183C2×C4⋊C4—1632no—
184C2×C22⋊C4—3262no—
185#106—1631no—
186#106—1632no—
187#106C2×D16311yescharacteristic, complemented
188#102—1632no—
189#102—1632no—
190#102—1632no—
191#102C2×D16312yescomplemented
192C24—3262no—
193C24—1631no—
194C2×D4—22no—
195C2×D4—3262no—
196C2×D4—3262no—
197C2×D4—3262no—
198C2×D4—3264no—
199C2×D4—3264no—
200C2×D4—3262no—
201C2×D4—6528no—
202C2×D4—1632no—
203C2×D4C22×D16312yes—
204C22×C4—3262no—
205C22×C4—3262no—
206C22×C4—3262no—
207C22×C4—1631no—
208C22×C4—1631no—
209C22×C4—1631no—
210C22×C4—1631no—
211C22×C4—1631no—
212C22×C4C22×D16311yescharacteristic
213D8—22no—
214D8—3264no—
215D8—3262no—
216M16—22no—
217M16—3262no—
218M16—3262no—
219M16—3262no—
220C2×C8—3262no—
221C2×C8—1631no—
222C2×C8—1631no—
223C2×C8—1634no—
224C2×C8C22×D16311yescharacteristic
225C2×C8C22×D16311yescharacteristic
226C4⋊C4—3262no—
227C4⋊C4—1632no—
228C4⋊C4C22×D16312yes—
229C22⋊C4—22no—
230C22⋊C4—3262no—
231C22⋊C4—3262no—
232C22⋊C4—3262no—
233C23—22no—
234C23—3262no—
235C23—3264no—
236C23—3262no—
237C23—6524no—
238C23—6524no—
239C23—6524no—
240C23—1631no—
241C23—1631no—
242C23D4×D16311yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—3262no—
247D4—3262no—
248D4—6524no—
249D4—6524no—
250D4—6524no—
251D4—6524no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—3262no—
256C2×C4—3262no—
257C2×C4—3262no—
258C2×C4—3261no—
259C2×C4—3261no—
260C2×C4—3262no—
261C2×C4—6524no—
262C2×C4—1631no—
263C2×C4—1631no—
264C2×C4—1632no—
265C2×C4—1632no—
266C2×C4D4×D16311yescharacteristic
267C2×C4C23×D16311yescharacteristic
268C8—21no—
269C8—3261no—
270C8—1632no—
271C8D4×D16312yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—13048no—
278C22—3261no—
279C22—3262no—
280C22—6524no—
281C22—6522no—
282C22—6522no—
283C22—6522no—
284C22—1632no—
285C22—1634no—
286C22C2×D4×D16311yescharacteristic
287C4—21no—
288C4—42no—
289C4—3261no—
290C4—6522no—
291C4—1632no—
292C4C2×D4×D16311yescharacteristic
293C4#41×D16311yescharacteristic
294C2—41no—
295C2—82no—
296C2—13042no—
297C2—6521no—
298C2—1634no—
299C2#121×D16311yescharacteristic
300C2AGL(1,ℤ/8)×D16312yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#469×C163C2
socleC22×C163C22×D4
hypercenter#469D163
hypocenterC163C2×#469
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D163C2semidirect
2. C2×M16×D163C2semidirect
3. #121×D163C2semidirect
4. #23781919C2semidirect
5. #23781918C2semidirect
6. #23781917C2semidirect
7. #23781916C2semidirect
8. #23781915C2semidirect
9. #106×D163C2semidirect
10. #102×D163C2semidirect
11. #469×C163C2semidirect
15. C2×D1304C22semidirect
16. C2×#14579684C22semidirect
18. C2×#14579683C22semidirect
19. C2×#14579682C22semidirect
20. C2×#14579682C22semidirect
24. #19104710C22semidirect
25. #19104709C22semidirect
28. #19104706C22semidirect
31. #19104667C22semidirect
32. #19104666C22semidirect
33. #19104657C22semidirect
34. #19104656C22semidirect
35. #19104655C22semidirect
36. C2×D8×C163C22semidirect
37. C2×M16×C163C22semidirect
38. #121×C163C22semidirect
39. #106×C163C22semidirect
40. #102×C163C22semidirect
67. C8×D163D4semidirect
108. C163⋊C8C2×D4semidirect
120. C8×C163C2×D4semidirect
149. D163#469direct (N × H)
150. C163C2×#469semidirect
160. #469D163direct (N × H)
169. C2×D8C2×D163semidirect
174. C2×M16C2×D163semidirect
182. #121C2×D163semidirect
187. #106C2×D163semidirect
191. #102C2×D163semidirect
271. C8D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1632
C2×M16×D1632
#121×D1632
#237819192
#237819182
#237819172
#237819162
#237819152
#106×D1632
#102×D1632
#469×C1632
C2×#469163