Finite Group Databasebeta

← C2×#23781811 #35223738

Subgroups — C2×#23781811 #35223738

Order 20864.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14579698C211yesmaximal, characteristic, complemented
2C22×#14579688C211yesmaximal, characteristic, complemented
3C22×#14579684C211yesmaximal, characteristic, complemented
4C2×#19104710C211yesmaximal, characteristic, complemented
5#23781811C218yesmaximal, direct product factor
6C2×#19104658C211yesmaximal, characteristic, complemented
7C2×(C2×D163)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C163C211yesmaximal, characteristic, complemented
9C2×#14579698—24no—
10C2×#14579698—22no—
11C2×#14579698—24no—
12C2×#14579698C2214yescomplemented
13C22×Dic326C2211yescharacteristic
14C22×D652—21no—
15C22×D652C2211yescharacteristic
16C22×C22⋊D163—21no—
17C23×Dic163—21no—
18C22×C4×D163C2211yescharacteristic
19C2×#14579688—24no—
20C2×#14579688C2214yescomplemented
21C2×#14579684—24no—
22C2×#14579684C2214yescomplemented
23C22×C163⋊C8C2211yescharacteristic
24#19104710—22no—
25#19104710C2214yescomplemented
26C2×(C2×D163)⋊C4—21no—
27#19104658C2214yescomplemented
28(C2×D163)⋊C8C2214yescomplemented
29C22×D4×C163C2211yescharacteristic
30C22×C8×C163C2211yescharacteristic
31C2×C4⋊C4×C163C2211yescharacteristic
32#102×C163C2214yescomplemented
33C23×D163—21no—
34C23×D163D411yescharacteristic
35#14579698—216no—
36#14579698—28no—
37#14579698—28no—
38C2×Dic326—21no—
39C2×Dic326—22no—
40C2×Dic326D414yescomplemented
41C2×Dic326C2312yes—
42C2×D652—24no—
43C2×D652—22no—
44C2×D652—22no—
45C2×D652—21no—
46C2×D652—44no—
47C2×D652C2312yes—
48C2×C22⋊D163—28no—
49C2×C22⋊D163—24no—
50C22×Dic163—28no—
51C22×Dic163—24no—
52C22×Dic163—21no—
53C22×Dic163D411yescharacteristic
54C2×C4×D163—22no—
55C2×C4×D163—22no—
56C2×C4×D163D412yes—
57C2×C4×D163C2312yes—
58#14579688—28no—
59#14579684—28no—
60C2×C163⋊C8—22no—
61C2×C163⋊C8C2312yes—
62(C2×D163)⋊C4—24no—
63C24×C163—21no—
64C2×D4×C163—21no—
65C2×D4×C163—22no—
66C2×D4×C163D414yescomplemented
67C2×D4×C163C2312yes—
68C22×C4×C163—21no—
69C22×C4×C163C2311yescharacteristic
70C2×C8×C163—22no—
71C2×C8×C163C2312yes—
72C4⋊C4×C163—21no—
73C4⋊C4×C163C2312yes—
74C22×D163—22no—
75C22×D163—22no—
76C22×D163—22no—
77C22×D163—22no—
78C22×D163—42no—
79C22×D163—42no—
80C22×D163—42no—
81C22×D163C2×D412yes—
82Dic326—24no—
83Dic326—22no—
84Dic326C2×D414yescomplemented
85D652—22no—
86D652—22no—
87D652—48no—
88D652—42no—
89C22⋊D163—216no—
90C2×Dic163—216no—
91C2×Dic163—22no—
92C2×Dic163—24no—
93C2×Dic163—22no—
94C2×Dic163C2×D412yes—
95C4×D163—24no—
96C4×D163—22no—
97C4×D163—22no—
98C163⋊C8—22no—
99C23×C163—24no—
100C23×C163—22no—
101C23×C163—42no—
102C23×C163—42no—
103C23×C163C2×D411yescharacteristic
104D4×C163—24no—
105D4×C163—22no—
106D4×C163C2×D414yescomplemented
107C2×C4×C163—22no—
108C2×C4×C163—42no—
109C2×C4×C163C2×D411yescharacteristic
110C2×C4×C163C2×D412yes—
111C2×C4×C163C2×D412yes—
112C2×C4×C163C2411yescharacteristic
113C8×C163—22no—
114C22×C163—24no—
115C22×C163—42no—
116C22×C163—42no—
117C22×C163—48no—
118C22×C163#12012yes—
119C22×C163#13512yes—
120C22×C163AGL(1,ℤ/8)12yes—
121C22×C163C22×D411yescharacteristic
122C2×D163—22no—
123C2×D163—22no—
124C2×D163—22no—
125C2×D163—44no—
126C2×D163—42no—
127C2×D163—42no—
128C2×D163—42no—
129C2×D163—84no—
130C4×C163—42no—
131C4×C163#12012yes—
132C4×C163C22×D411yescharacteristic
133C4×C163C22×D411yescharacteristic
134Dic163—22no—
135Dic163—24no—
136C2×C163—44no—
137C2×C163#44914yescomplemented
138C2×C163C2×#12011yescharacteristic
139C2×C163C2×#13511yescharacteristic
140C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
141D163—42no—
142D163—82no—
143C163C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1631nomaximal, Sylow
145C22×#41—1631no—
146C22×SD16—1631no—
147C22×D8—1631no—
148C2×#121—1631no—
149#449—1638no—
150C2×#103—1631no—
151C2×#102D16311yescharacteristic, complemented
152C2×C22⋊C8—1631no—
153C2×#41—3264no—
154C2×#41—3264no—
155C2×#41—3262no—
156C2×#41—1634no—
157C22×Q8—1631no—
158C22×D4—3261no—
159C22×D4—3261no—
160C22×D4—1631no—
161C22×D4C2×D16311yescharacteristic
162C23×C4—3261no—
163C23×C4—1631no—
164C2×SD16—3264no—
165C2×SD16—1634no—
166C2×D8—3264no—
167C2×D8—1634no—
168C22×C8—1631no—
169C22×C8C2×D16311yescharacteristic
170#121—3262no—
171#121—1634no—
172C2×C4⋊C4C2×D16311yescharacteristic
173C2×C22⋊C4—3261no—
174#103—1634no—
175#102C2×D16314yescomplemented
176C22⋊C8—1634no—
177C24—21no—
178C24—3261no—
179C24—1631no—
180#41—32616no—
181#41—3268no—
182#41—3268no—
183C2×Q8—3261no—
184C2×Q8—3262no—
185C2×Q8—1634no—
186C2×Q8—1632no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—3264no—
190C2×D4—3268no—
191C2×D4—3264no—
192C2×D4—3262no—
193C2×D4—3261no—
194C2×D4—3262no—
195C2×D4—6524no—
196C2×D4—1632no—
197C2×D4D65214yescomplemented
198C2×D4C22×D16312yes—
199C22×C4—21no—
200C22×C4—3268no—
201C22×C4—3261no—
202C22×C4—3264no—
203C22×C4—3262no—
204C22×C4—3262no—
205C22×C4—1631no—
206C22×C4—1632no—
207C22×C4—1632no—
208C22×C4C22×D16311yescharacteristic
209SD16—3268no—
210D8—3268no—
211C2×C8—22no—
212C2×C8—3262no—
213C2×C8—1632no—
214C2×C8C22×D16312yes—
215C4⋊C4—21no—
216C4⋊C4C22×D16312yes—
217C22⋊C4—3264no—
218C23—22no—
219C23—24no—
220C23—42no—
221C23—42no—
222C23—3262no—
223C23—3262no—
224C23—3262no—
225C23—3262no—
226C23—6522no—
227C23—6522no—
228C23—6522no—
229C23—1632no—
230C23D4×D16311yescharacteristic
231Q8—3262no—
232Q8—3264no—
233Q8—1634no—
234D4—22no—
235D4—24no—
236D4—32616no—
237D4—3262no—
238D4—3262no—
239D4—6528no—
240D4—6522no—
241D4C2×D65214yescomplemented
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—32616no—
245C2×C4—3262no—
246C2×C4—3262no—
247C2×C4—3264no—
248C2×C4—3262no—
249C2×C4—3262no—
250C2×C4—3264no—
251C2×C4—1632no—
252C2×C4D4×D16312yes—
253C2×C4D4×D16311yescharacteristic
254C2×C4C2×D65212yes—
255C2×C4C23×D16311yescharacteristic
256C8—22no—
257C8—3262no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—48no—
262C22—13044no—
263C22—3262no—
264C22—3262no—
265C22—3262no—
266C22—6522no—
267C22—6522no—
268C22—6522no—
269C22—6524no—
270C22#1910470512yes—
271C22#1910475612yes—
272C22#1910475912yes—
273C22C2×D4×D16311yescharacteristic
274C4—42no—
275C4—3262no—
276C4—3264no—
277C4#1910470512yes—
278C4C2×D4×D16311yescharacteristic
279C4C22×D65211yescharacteristic
280C2—44no—
281C2—13042no—
282C2—6522no—
283C2#2378181114yesdirect product factor
284C2C2×#1910470511yescharacteristic
285C2C2×#1910475611yescharacteristic
286C2C2×#1910475911yescharacteristic
subgroupG/H
centerC23D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC23×C163C2×D4
hypercenterC2×#102D163
hypocenterC163C2×#449
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14579698C2semidirect
2. C22×#14579688C2semidirect
3. C22×#14579684C2semidirect
4. C2×#19104710C2semidirect
5. #23781811C2direct (N × H)
6. C2×#19104658C2semidirect
7. C2×(C2×D163)⋊C8C2semidirect
8. C2×#102×C163C2semidirect
12. C2×#14579698C22semidirect
20. C2×#14579688C22semidirect
22. C2×#14579684C22semidirect
25. #19104710C22semidirect
27. #19104658C22semidirect
28. (C2×D163)⋊C8C22semidirect
32. #102×C163C22semidirect
40. C2×Dic326D4semidirect
66. C2×D4×C163D4semidirect
84. Dic326C2×D4semidirect
106. D4×C163C2×D4semidirect
137. C2×C163#449semidirect
143. C163C2×#449semidirect
151. C2×#102D163semidirect
175. #102C2×D163semidirect
197. C2×D4D652semidirect
241. D4C2×D652semidirect
283. C2#23781811direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145796982
C22×#145796882
C22×#145796842
C2×#191047102
#237818112
C2×#191046582
C2×(C2×D163)⋊C82
C2×#102×C1632
C2×#449163