Finite Group Databasebeta

← C2×#23781800 #35223713

Subgroups — C2×#23781800 #35223713

Order 20864.

1048 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4×D163C211yesmaximal, characteristic, complemented
2C22×D1304C211yesmaximal, characteristic, complemented
3C22×#14579684C211yesmaximal, characteristic, complemented
4C2×#19104710C211yesmaximal, characteristic, complemented
5#23781800C218yesmaximal, direct product factor
6C2×#19104655C211yesmaximal, characteristic, complemented
7C2×(C2×D163)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C163C211yesmaximal, characteristic, complemented
9C24×D163—21no—
10C22×D652—21no—
11C22×D652C2211yescharacteristic
12C22×D652C2211yescharacteristic, complemented
13C22×C22⋊D163—21no—
14C2×D4×D163—24no—
15C2×D4×D163—22no—
16C2×D4×D163—24no—
17C2×D4×D163C2214yescomplemented
18C22×C4×D163C2211yescharacteristic
19C2×D1304—24no—
20C2×D1304C2214yescomplemented
21C2×#14579684—24no—
22C2×#14579684C2214yescomplemented
23C22×C163⋊C8C2211yescharacteristic, complemented
24#19104710—22no—
25#19104710C2214yescomplemented
26C2×(C2×D163)⋊C4—21no—
27#19104655C2214yescomplemented
28(C2×D163)⋊C8C2214yescomplemented
29C22×D4×C163C2211yescharacteristic
30C22×C8×C163C2211yescharacteristic, complemented
31C2×C4⋊C4×C163C2211yescharacteristic, complemented
32#102×C163C2214yescomplemented
33C23×D163—28no—
34C23×D163—24no—
35C23×D163—21no—
36C23×D163—21no—
37C23×D163—44no—
38C23×D163—44no—
39C23×D163D411yescharacteristic
40C2×D652—24no—
41C2×D652—21no—
42C2×D652—22no—
43C2×D652—22no—
44C2×D652—22no—
45C2×D652—21no—
46C2×D652—44no—
47C2×D652D414yescomplemented
48C2×D652C2312yes—
49C2×D652C2312yescomplemented
50C2×C22⋊D163—28no—
51C2×C22⋊D163—24no—
52D4×D163—216no—
53D4×D163—28no—
54D4×D163—28no—
55C22×Dic163D411yescharacteristic
56C2×C4×D163—22no—
57C2×C4×D163—22no—
58C2×C4×D163D412yes—
59C2×C4×D163C2312yes—
60D1304—28no—
61#14579684—28no—
62C2×C163⋊C8—22no—
63C2×C163⋊C8C2312yescomplemented
64(C2×D163)⋊C4—24no—
65C24×C163—21no—
66C2×D4×C163—21no—
67C2×D4×C163—22no—
68C2×D4×C163D414yescomplemented
69C2×D4×C163C2312yes—
70C22×C4×C163—21no—
71C22×C4×C163C2311yescharacteristic
72C2×C8×C163—22no—
73C2×C8×C163C2312yescomplemented
74C4⋊C4×C163—21no—
75C4⋊C4×C163C2312yescomplemented
76C22×D163—22no—
77C22×D163—22no—
78C22×D163—216no—
79C22×D163—24no—
80C22×D163—22no—
81C22×D163—22no—
82C22×D163—48no—
83C22×D163—432no—
84C22×D163—42no—
85C22×D163—42no—
86C22×D163—48no—
87C22×D163—42no—
88C22×D163—42no—
89C22×D163—42no—
90C22×D163D814yescomplemented
91C22×D163C2×D412yes—
92D652—24no—
93D652—22no—
94D652—22no—
95D652—22no—
96D652—48no—
97D652—42no—
98D652C2×D414yescomplemented
99C22⋊D163—216no—
100C2×Dic163—22no—
101C2×Dic163C2×D412yes—
102C4×D163—24no—
103C4×D163—22no—
104C4×D163—22no—
105C163⋊C8—22no—
106C23×C163—24no—
107C23×C163—22no—
108C23×C163—42no—
109C23×C163—42no—
110C23×C163C2×D411yescharacteristic
111D4×C163—24no—
112D4×C163—22no—
113D4×C163C2×D414yescomplemented
114C2×C4×C163—22no—
115C2×C4×C163—42no—
116C2×C4×C163C2×D411yescharacteristic
117C2×C4×C163C2×D412yes—
118C2×C4×C163C2×D412yes—
119C2×C4×C163C2411yescharacteristic
120C8×C163—22no—
121C22×C163—24no—
122C22×C163—42no—
123C22×C163—42no—
124C22×C163—48no—
125C22×C163#12012yes—
126C22×C163C2×D812yes—
127C22×C163AGL(1,ℤ/8)12yes—
128C22×C163C22×D411yescharacteristic
129C2×D163—22no—
130C2×D163—22no—
131C2×D163—28no—
132C2×D163—24no—
133C2×D163—432no—
134C2×D163—42no—
135C2×D163—42no—
136C2×D163—48no—
137C2×D163—42no—
138C2×D163—42no—
139C2×D163—42no—
140C2×D163—84no—
141C2×D163C2×D814yescomplemented
142C4×C163—42no—
143C4×C163#12012yes—
144C4×C163C22×D411yescharacteristic
145C4×C163C22×D411yescharacteristic
146Dic163—22no—
147C2×C163—44no—
148C2×C163#44714yescomplemented
149C2×C163C2×#12011yescharacteristic
150C2×C163C22×D811yescharacteristic
151C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
152D163—24no—
153D163—44no—
154D163—82no—
155C163C2×#44711yescharacteristic, complemented, Sylow
156C2×#447—1631nomaximal, Sylow
157C23×D4—1631no—
158C22×D8—1631no—
159C22×D8—1631no—
160C2×#121—1631no—
161#447—1638no—
162C2×#102—1631no—
163C2×#102D16311yescharacteristic, complemented
164C2×C22⋊C8—1631no—
165C25—3261no—
166C22×D4—3261no—
167C22×D4—3264no—
168C22×D4—3264no—
169C22×D4—3262no—
170C22×D4—3261no—
171C22×D4—1634no—
172C22×D4—1631no—
173C22×D4—1631no—
174C22×D4C2×D16311yescharacteristic
175C23×C4—1631no—
176C2×D8—3264no—
177C2×D8—3264no—
178C2×D8—1634no—
179C2×D8—1634no—
180C22×C8—1631no—
181C22×C8C2×D16311yescharacteristic, complemented
182#121—3262no—
183#121—1634no—
184C2×C4⋊C4C2×D16311yescharacteristic, complemented
185C2×C22⋊C4—3261no—
186#102—1634no—
187#102C2×D16314yescomplemented
188C22⋊C8—1634no—
189C24—21no—
190C24—3268no—
191C24—3264no—
192C24—3261no—
193C24—3261no—
194C24—6524no—
195C24—6524no—
196C24—1631no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—3264no—
200C2×D4—32616no—
201C2×D4—3268no—
202C2×D4—3268no—
203C2×D4—3268no—
204C2×D4—3264no—
205C2×D4—3261no—
206C2×D4—3262no—
207C2×D4—3261no—
208C2×D4—3262no—
209C2×D4—3262no—
210C2×D4—6524no—
211C2×D4—1634no—
212C2×D4—1632no—
213C2×D4—1632no—
214C2×D4D65214yescomplemented
215C2×D4C22×D16312yes—
216C22×C4—21no—
217C22×C4—3262no—
218C22×C4—3262no—
219C22×C4—1631no—
220C22×C4—1632no—
221C22×C4—1632no—
222C22×C4C22×D16311yescharacteristic
223D8—3268no—
224D8—3268no—
225C2×C8—22no—
226C2×C8—3262no—
227C2×C8—1632no—
228C2×C8C22×D16312yescomplemented
229C4⋊C4—21no—
230C4⋊C4C22×D16312yescomplemented
231C22⋊C4—3264no—
232C23—22no—
233C23—24no—
234C23—42no—
235C23—42no—
236C23—32616no—
237C23—3262no—
238C23—3262no—
239C23—3262no—
240C23—3264no—
241C23—3262no—
242C23—6528no—
243C23—65232no—
244C23—6528no—
245C23—6522no—
246C23—6522no—
247C23—6522no—
248C23—6522no—
249C23—6522no—
250C23—1632no—
251C23—1634no—
252C23D4×D16311yescharacteristic
253D4—22no—
254D4—24no—
255D4—32616no—
256D4—3262no—
257D4—3262no—
258D4—3262no—
259D4—3264no—
260D4—6528no—
261D4—6522no—
262D4—1634no—
263D4C2×D65214yescomplemented
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—3262no—
267C2×C4—3262no—
268C2×C4—3262no—
269C2×C4—3264no—
270C2×C4—1632no—
271C2×C4D4×D16312yes—
272C2×C4D4×D16311yescharacteristic
273C2×C4C2×D65212yes—
274C2×C4C23×D16311yescharacteristic
275C8—22no—
276C8—3262no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—48no—
281C22—13044no—
282C22—3262no—
283C22—3262no—
284C22—3268no—
285C22—3264no—
286C22—65232no—
287C22—6522no—
288C22—6522no—
289C22—6522no—
290C22—6522no—
291C22—6522no—
292C22—6528no—
293C22—1634no—
294C22#1910470512yes—
295C22D8×D16312yes—
296C22#1910476212yes—
297C22C2×D4×D16311yescharacteristic
298C4—42no—
299C4—3262no—
300C4#1910470512yes—
301C4C2×D4×D16311yescharacteristic
302C4C22×D65211yescharacteristic
303C2—44no—
304C2—13042no—
305C2—3264no—
306C2—6524no—
307C2#2378180014yesdirect product factor
308C2C2×#1910470511yescharacteristic
309C2C2×D8×D16311yescharacteristic
310C2C2×#1910476211yescharacteristic
subgroupG/H
centerC23D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC23×C163C2×D4
hypercenterC2×#102D163
hypocenterC163C2×#447
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×D4×D163C2semidirect
2. C22×D1304C2semidirect
3. C22×#14579684C2semidirect
4. C2×#19104710C2semidirect
5. #23781800C2direct (N × H)
6. C2×#19104655C2semidirect
7. C2×(C2×D163)⋊C8C2semidirect
8. C2×#102×C163C2semidirect
12. C22×D652C22semidirect
17. C2×D4×D163C22semidirect
20. C2×D1304C22semidirect
22. C2×#14579684C22semidirect
23. C22×C163⋊C8C22semidirect
25. #19104710C22semidirect
27. #19104655C22semidirect
28. (C2×D163)⋊C8C22semidirect
30. C22×C8×C163C22semidirect
31. C2×C4⋊C4×C163C22semidirect
32. #102×C163C22semidirect
47. C2×D652D4semidirect
49. C2×D652C23semidirect
63. C2×C163⋊C8C23semidirect
68. C2×D4×C163D4semidirect
73. C2×C8×C163C23semidirect
75. C4⋊C4×C163C23semidirect
90. C22×D163D8semidirect
98. D652C2×D4semidirect
113. D4×C163C2×D4semidirect
141. C2×D163C2×D8semidirect
148. C2×C163#447semidirect
155. C163C2×#447semidirect
163. C2×#102D163semidirect
181. C22×C8C2×D163semidirect
184. C2×C4⋊C4C2×D163semidirect
187. #102C2×D163semidirect
214. C2×D4D652semidirect
228. C2×C8C22×D163semidirect
230. C4⋊C4C22×D163semidirect
263. D4C2×D652semidirect
307. C2#23781800direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1632
C22×D13042
C22×#145796842
C2×#191047102
#237818002
C2×#191046552
C2×(C2×D163)⋊C82
C2×#102×C1632
C2×#447163