Finite Group Databasebeta

← C2×#23782368 #35228062

Subgroups — C2×#23782368 #35228062

Order 20864.

1252 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782368C2116yesmaximal, direct product factor
2C2×#19104730C212yesmaximal, complemented
3C2×#19104720C212yesmaximal, complemented
4C2×#19104710C211yesmaximal, characteristic, complemented
5C2×#19104710C211yesmaximal, characteristic, complemented
6C2×#19104710C212yesmaximal, complemented
7C2×C4×D652C211yesmaximal, characteristic, complemented
8C2×C4×D652C211yesmaximal, characteristic, complemented
9C2×#19104697C212yesmaximal, complemented
10C2×#19104693C211yesmaximal, characteristic, complemented
11C2×C4⋊C4×D163C211yesmaximal, characteristic, complemented
12C2×#125×C163C211yesmaximal, characteristic, complemented
13C22×D652C2212yes—
14C22×D652C2211yescharacteristic
15C22×D652C2211yescharacteristic
16C22×D652C2212yes—
17C22×C4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2212yes—
20C22×C4×D163C2212yes—
21#19104730C22116yescomplemented
22#19104720C22116yescomplemented
23#19104710C2218yescomplemented
24#19104710C2218yescomplemented
25#19104710C22116yescomplemented
26C4×D652C2218yescomplemented
27C4×D652C2218yescomplemented
28#19104697C22116yescomplemented
29#19104693C2218yescomplemented
30C2×C4⋊Dic163C2211yescharacteristic
31C2×C4⋊Dic163C2211yescharacteristic
32C2×Dic163⋊C4C2212yes—
33C4⋊C4×D163C2218yescomplemented
34C2×(C2×D163)⋊C4C2212yes—
35C2×(C2×D163)⋊C4C2212yes—
36C2×(C2×D163)⋊C4C2212yes—
37C2×(C2×D163)⋊C4C2212yes—
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×C4×Dic163C2212yes—
41#125×C163C2218yescomplemented
42C2×C4⋊C4×C163C2212yes—
43C2×C4⋊C4×C163C2211yescharacteristic
44C2×C4⋊C4×C163C2211yescharacteristic
45C2×C4⋊C4×C163C2212yes—
46C2×C42×C163C2211yescharacteristic
47C23×D163C2311yescharacteristic
48C23×D163C2312yes—
49C23×D163C2311yescharacteristic
50C2×D652—28no—
51C2×D652—24no—
52C2×D652—24no—
53C2×D652—44no—
54C2×D652C2318yes—
55C2×D652C2314yes—
56C2×D652C2314yes—
57C2×D652C2318yes—
58C22×Dic163C2312yes—
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C2×C4×D163—24no—
62C2×C4×D163—24no—
63C2×C4×D163—44no—
64C2×C4×D163—44no—
65C2×C4×D163C2314yes—
66C2×C4×D163C2314yes—
67C2×C4×D163C2318yes—
68C2×C4×D163C2318yes—
69C4⋊Dic163C2314yes—
70C4⋊Dic163C2314yes—
71Dic163⋊C4C2318yes—
72(C2×D163)⋊C4C2318yes—
73(C2×D163)⋊C4C2314yes—
74(C2×D163)⋊C4C2318yes—
75(C2×D163)⋊C4C2318yes—
76(C2×D163)⋊C4C2314yes—
77(C2×D163)⋊C4C2318yes—
78C4×Dic163C2318yes—
79C22×C4×C163C2311yescharacteristic
80C22×C4×C163C2312yes—
81C22×C4×C163C2311yescharacteristic
82C22×C4×C163C2312yes—
83C22×C4×C163C2311yescharacteristic
84C4⋊C4×C163C2314yes—
85C4⋊C4×C163C2314yes—
86C4⋊C4×C163C2318yes—
87C4⋊C4×C163C2318yes—
88C42×C163C2314yes—
89C22×D163—22no—
90C22×D163—22no—
91C22×D163—22no—
92C22×D163—22no—
93C22×D163—24no—
94C22×D163—24no—
95C22×D163—24no—
96C22×D163—22no—
97C22×D163#4114yes—
98C22×D163C2412yes—
99C22×D163C2414yes—
100C22×D163C2412yes—
101D652—216no—
102D652—28no—
103D652—28no—
104D652—48no—
105C2×Dic163—22no—
106C2×Dic163—22no—
107C2×Dic163—24no—
108C2×Dic163C2412yes—
109C2×Dic163C2414yes—
110C2×Dic163C2412yes—
111C4×D163—28no—
112C4×D163—28no—
113C4×D163—48no—
114C4×D163—48no—
115C23×C163C2411yescharacteristic
116C2×C4×C163—22no—
117C2×C4×C163—24no—
118C2×C4×C163—22no—
119C2×C4×C163—24no—
120C2×C4×C163#4114yes—
121C2×C4×C163C2412yes—
122C2×C4×C163C2412yes—
123C2×C4×C163C2414yes—
124C2×C4×C163C2414yes—
125C2×C4×C163C2412yes—
126C22×C163C2×#4112yes—
127C22×C163C2×#4112yes—
128C22×C163H(5,𝔽2)12yes—
129C22×C163C2511yescharacteristic
130C2×D163—28no—
131C2×D163—22no—
132C2×D163—22no—
133C2×D163—22no—
134C2×D163—22no—
135C2×D163—24no—
136C2×D163—24no—
137C2×D163—24no—
138C2×D163—22no—
139C2×D163—44no—
140C2×D163—48no—
141C2×D163C2×#4114yes—
142C4×C163—22no—
143C4×C163—24no—
144C4×C163—22no—
145C4×C163—24no—
146C4×C163C2×#4114yes—
147Dic163—22no—
148Dic163—22no—
149Dic163—24no—
150C2×C163#55314yescomplemented
151C2×C163C22×#4111yescharacteristic
152C2×C163C22×#4111yescharacteristic
153C2×C163C2×H(5,𝔽2)11yescharacteristic
154D163—24no—
155D163—42no—
156D163—44no—
157C163C2×#55311yescharacteristic, complemented, Sylow
158C2×#553—1631nomaximal, Sylow
159#553—16316no—
160C2×#126—1632no—
161C2×#125D16311yescharacteristic, complemented
162C2×#123—1632no—
163C2×#121—1632no—
164C2×#121—1631no—
165C2×#121—1631no—
166C2×C4×D4—1632no—
167C2×C4×D4—1631no—
168C2×C4×D4—1631no—
169C2×#117—1631no—
170C22×C4⋊C4—1631no—
171C22×D4—1632no—
172C22×D4—1631no—
173C22×D4—1631no—
174C22×D4—1632no—
175C23×C4—1631no—
176C23×C4—1631no—
177C23×C4—1632no—
178C23×C4—1632no—
179#126—16316no—
180#125C2×D16318yescomplemented
181#123—16316no—
182#121—16316no—
183#121—1638no—
184#121—1638no—
185C4×D4—16316no—
186C4×D4—1638no—
187C4×D4—1638no—
188#117—1638no—
189C2×C4⋊C4—1631no—
190C2×C4⋊C4—1632no—
191C2×C4⋊C4—1631no—
192C2×C4⋊C4—1638no—
193C2×C4⋊C4C2×D16311yescharacteristic
194C2×C4⋊C4C2×D16311yescharacteristic
195C2×C4⋊C4C2×D16312yes—
196C2×C4⋊C4C2×D16312yes—
197C2×C22⋊C4—1632no—
198C2×C22⋊C4—1631no—
199C2×C22⋊C4—1632no—
200C2×C22⋊C4—1632no—
201C2×C22⋊C4—1631no—
202C2×C22⋊C4—1632no—
203C2×C42—1632no—
204C2×C42C2×D16311yescharacteristic
205C24—1631no—
206C24—1632no—
207C24—1631no—
208C2×D4—3268no—
209C2×D4—3264no—
210C2×D4—3264no—
211C2×D4—6524no—
212C2×D4—1638no—
213C2×D4—1634no—
214C2×D4—1634no—
215C2×D4—1638no—
216C22×C4—3264no—
217C22×C4—3264no—
218C22×C4—6524no—
219C22×C4—6524no—
220C22×C4—1631no—
221C22×C4—1632no—
222C22×C4—1631no—
223C22×C4—1634no—
224C22×C4—1634no—
225C22×C4—1638no—
226C22×C4—1638no—
227C22×C4C22×D16312yes—
228C22×C4C22×D16311yescharacteristic
229C22×C4C22×D16311yescharacteristic
230C22×C4C22×D16312yes—
231C22×C4C22×D16311yescharacteristic
232C4⋊C4—1634no—
233C4⋊C4—1638no—
234C4⋊C4—1634no—
235C4⋊C4C22×D16314yes—
236C4⋊C4C22×D16314yes—
237C4⋊C4C22×D16318yes—
238C4⋊C4C22×D16318yes—
239C22⋊C4—1638no—
240C22⋊C4—1634no—
241C22⋊C4—1638no—
242C22⋊C4—1638no—
243C22⋊C4—1634no—
244C22⋊C4—1638no—
245C42—1638no—
246C42C22×D16314yes—
247C23—3262no—
248C23—3262no—
249C23—3262no—
250C23—3264no—
251C23—3262no—
252C23—3264no—
253C23—3264no—
254C23—3262no—
255C23—1632no—
256C23—1634no—
257C23—1634no—
258C23—1632no—
259C23C23×D16311yescharacteristic
260D4—32616no—
261D4—3268no—
262D4—3268no—
263D4—6528no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—3262no—
269C2×C4—3262no—
270C2×C4—3264no—
271C2×C4—3268no—
272C2×C4—3268no—
273C2×C4—6528no—
274C2×C4—6528no—
275C2×C4—1632no—
276C2×C4—1634no—
277C2×C4—1632no—
278C2×C4#1457970014yes—
279C2×C4C23×D16314yes—
280C2×C4C23×D16312yes—
281C2×C4C23×D16312yes—
282C2×C4C23×D16314yes—
283C2×C4C23×D16312yes—
284C22—3262no—
285C22—3262no—
286C22—3268no—
287C22—3262no—
288C22—3264no—
289C22—3262no—
290C22—3264no—
291C22—3264no—
292C22—3262no—
293C22—6524no—
294C22—6528no—
295C22—1634no—
296C22#41×D16312yes—
297C22C2×#1457970012yes—
298C22#1910478112yes—
299C22C24×D16311yescharacteristic
300C4—24no—
301C4—22no—
302C4—22no—
303C4—24no—
304C4—3262no—
305C4—3262no—
306C4—3264no—
307C4C2×#1457970014yes—
308C2—3264no—
309C2—6522no—
310C2—6524no—
311C2#2378236814yesdirect product factor
312C2C2×#41×D16311yescharacteristic
313C2C22×#1457970011yescharacteristic
314C2C2×#1910478111yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#125×C163C2
socleC23×C163C24
hypercenterC2×#125D163
hypocenterC163C2×#553
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #23782368C2direct (N × H)
2. C2×#19104730C2semidirect
3. C2×#19104720C2semidirect
4. C2×#19104710C2semidirect
5. C2×#19104710C2semidirect
6. C2×#19104710C2semidirect
7. C2×C4×D652C2semidirect
8. C2×C4×D652C2semidirect
9. C2×#19104697C2semidirect
10. C2×#19104693C2semidirect
11. C2×C4⋊C4×D163C2semidirect
12. C2×#125×C163C2semidirect
21. #19104730C22semidirect
22. #19104720C22semidirect
23. #19104710C22semidirect
24. #19104710C22semidirect
25. #19104710C22semidirect
26. C4×D652C22semidirect
27. C4×D652C22semidirect
28. #19104697C22semidirect
29. #19104693C22semidirect
33. C4⋊C4×D163C22semidirect
41. #125×C163C22semidirect
150. C2×C163#553semidirect
157. C163C2×#553semidirect
161. C2×#125D163semidirect
180. #125C2×D163semidirect
311. C2#23782368direct (N × H)

Maximal subgroups

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

subgroupindex
#237823682
C2×#191047302
C2×#191047202
C2×#191047102
C2×C4×D6522
C2×#191046972
C2×#191046932
C2×C4⋊C4×D1632
C2×#125×C1632
C2×#553163