Finite Group Databasebeta

← C2×#23782263 #35227888

Subgroups — C2×#23782263 #35227888

Order 20864.

1144 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782263C2116yesmaximal, direct product factor
2C2×#19104730C211yesmaximal, characteristic, complemented
3C2×#19104729C211yesmaximal, characteristic, complemented
4C2×#19104725C211yesmaximal, characteristic, complemented
5C2×#19104720C211yesmaximal, characteristic, complemented
6C2×#19104720C211yesmaximal, characteristic, complemented
7C2×#19104720C211yesmaximal, characteristic, complemented
8C2×#19104720C211yesmaximal, characteristic, complemented
9C2×#19104715C211yesmaximal, characteristic, complemented
10C2×#19104713C211yesmaximal, characteristic, complemented
11C2×#19104713C211yesmaximal, characteristic, complemented
12C2×#19104710C211yesmaximal, characteristic
13C2×C4×D652C211yesmaximal, characteristic, complemented
14C2×#19104697C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D163C211yesmaximal, characteristic, complemented
16C2×#125×C163C211yesmaximal, characteristic, complemented
17C22×Dic326C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×D652C2211yescharacteristic
20C22×D652C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic
22C22×C4×D163C2211yescharacteristic
23C22×C4×D163C2211yescharacteristic
24C22×C4×D163C2211yescharacteristic
25C22×C4×D163C2211yescharacteristic
26#19104730C2218yescomplemented
27#19104729C2218yescomplemented
28#19104725C2218yescomplemented
29#19104720C2218yescomplemented
30#19104720C2218yescomplemented
31#19104720C2218yescomplemented
32#19104720C2218yescomplemented
33#19104715C2218yescomplemented
34#19104713C2218yescomplemented
35#19104713C2218yescomplemented
36#19104710C2218yes—
37C4×D652C2218yescomplemented
38#19104697C2218yescomplemented
39C2×C4⋊Dic163C2211yescharacteristic
40C2×C4⋊Dic163C2211yescharacteristic
41C2×Dic163⋊C4C2211yescharacteristic
42C2×Dic163⋊C4C2211yescharacteristic
43C2×Dic163⋊C4C2211yescharacteristic
44C2×Dic163⋊C4C2211yescharacteristic
45C2×Dic163⋊C4C2211yescharacteristic
46C2×Dic163⋊C4C2211yescharacteristic
47C4⋊C4×D163C2218yescomplemented
48C2×(C2×D163)⋊C4C2211yescharacteristic
49C2×(C2×D163)⋊C4C2211yescharacteristic
50C2×(C2×D163)⋊C4C2211yescharacteristic
51C2×(C2×D163)⋊C4C2211yescharacteristic
52C2×(C2×D163)⋊C4C2211yescharacteristic
53C2×(C2×D163)⋊C4C2211yescharacteristic
54C2×(C2×D163)⋊C4C2211yescharacteristic
55C2×(C2×D163)⋊C4C2211yescharacteristic
56C2×(C2×D163)⋊C4C2211yescharacteristic
57C2×(C2×D163)⋊C4C2211yescharacteristic
58C2×C4×Dic163C2211yescharacteristic
59#125×C163C2218yescomplemented
60C2×C4⋊C4×C163C2211yescharacteristic
61C2×C4⋊C4×C163C2211yescharacteristic
62C2×C4⋊C4×C163C2211yescharacteristic
63C2×C4⋊C4×C163C2211yescharacteristic
64C2×C4⋊C4×C163C2211yescharacteristic
65C2×C4⋊C4×C163C2211yescharacteristic
66C2×C42×C163C2211yescharacteristic
67C23×D163C2311yescharacteristic
68C23×D163C2311yescharacteristic
69C23×D163C2311yescharacteristic
70C2×Dic326—42no—
71C2×Dic326C2314yes—
72C2×D652—24no—
73C2×D652—24no—
74C2×D652—42no—
75C2×D652C2314yes—
76C2×D652C2314yes—
77C2×D652C2314yes—
78C22×Dic163C2311yescharacteristic
79C22×Dic163C2311yescharacteristic
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C2×C4×D163—24no—
84C2×C4×D163—42no—
85C2×C4×D163—42no—
86C2×C4×D163—42no—
87C2×C4×D163—42no—
88C2×C4×D163C2314yes—
89C2×C4×D163C2314yes—
90C2×C4×D163C2314yes—
91C2×C4×D163C2314yes—
92C2×C4×D163C2314yes—
93C4⋊Dic163C2314yes—
94C4⋊Dic163C2314yes—
95Dic163⋊C4C2314yes—
96Dic163⋊C4C2314yes—
97Dic163⋊C4C2314yes—
98Dic163⋊C4C2314yes—
99Dic163⋊C4C2314yes—
100Dic163⋊C4C2314yes—
101(C2×D163)⋊C4C2314yes—
102(C2×D163)⋊C4C2314yes—
103(C2×D163)⋊C4C2314yes—
104(C2×D163)⋊C4C2314yes—
105(C2×D163)⋊C4C2314yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108(C2×D163)⋊C4C2314yes—
109(C2×D163)⋊C4C2314yes—
110(C2×D163)⋊C4C2314yes—
111C4×Dic163C2314yes—
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114C22×C4×C163C2311yescharacteristic
115C22×C4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C4⋊C4×C163C2314yes—
120C4⋊C4×C163C2314yes—
121C4⋊C4×C163C2314yes—
122C4⋊C4×C163C2314yes—
123C4⋊C4×C163C2314yes—
124C4⋊C4×C163C2314yes—
125C42×C163C2314yes—
126C22×D163—22no—
127C22×D163—22no—
128C22×D163—22no—
129C22×D163—22no—
130C22×D163—22no—
131C22×D163—22no—
132C22×D163—22no—
133C22×D163—22no—
134C22×D163#4114yes—
135C22×D163C2412yes—
136C22×D163C2412yes—
137C22×D163C2412yes—
138Dic326—44no—
139D652—28no—
140D652—28no—
141D652—44no—
142C2×Dic163—22no—
143C2×Dic163—22no—
144C2×Dic163—22no—
145C2×Dic163—22no—
146C2×Dic163—22no—
147C2×Dic163C2412yes—
148C2×Dic163C2412yes—
149C2×Dic163C2412yes—
150C2×Dic163C2412yes—
151C2×Dic163C2412yes—
152C4×D163—28no—
153C4×D163—44no—
154C4×D163—44no—
155C4×D163—44no—
156C4×D163—44no—
157C23×C163C2411yescharacteristic
158C2×C4×C163—22no—
159C2×C4×C163—22no—
160C2×C4×C163—22no—
161C2×C4×C163—22no—
162C2×C4×C163—22no—
163C2×C4×C163—22no—
164C2×C4×C163—22no—
165C2×C4×C163C2412yes—
166C2×C4×C163C2412yes—
167C2×C4×C163C2412yes—
168C2×C4×C163C2412yes—
169C2×C4×C163C2412yes—
170C2×C4×C163C2412yes—
171C2×C4×C163C2412yes—
172C22×C163C2×#4112yes—
173C22×C163H(5,𝔽2)12yes—
174C22×C1632−1+412yes—
175C22×C163C2511yescharacteristic
176C2×D163—28no—
177C2×D163—22no—
178C2×D163—22no—
179C2×D163—22no—
180C2×D163—22no—
181C2×D163—22no—
182C2×D163—22no—
183C2×D163—22no—
184C2×D163—22no—
185C2×D163—44no—
186C2×D163—44no—
187C2×D163C2×#4114yes—
188C4×C163—22no—
189C4×C163—22no—
190C4×C163—22no—
191C4×C163—22no—
192C4×C163—22no—
193C4×C163—22no—
194C4×C163—22no—
195Dic163—22no—
196Dic163—22no—
197Dic163—22no—
198Dic163—22no—
199Dic163—22no—
200C2×C163#53914yescomplemented
201C2×C163C22×#4111yescharacteristic
202C2×C163C2×H(5,𝔽2)11yescharacteristic
203C2×C163C2×2−1+411yescharacteristic
204D163—24no—
205D163—42no—
206D163—42no—
207C163C2×#53911yescharacteristic, complemented, Sylow
208C2×#539—1631nomaximal, Sylow
209#539—16316no—
210C2×#126—1631no—
211C2×#126—1631no—
212C2×#125—1631no—
213C2×#125D16311yescharacteristic, complemented
214C2×#123—1631no—
215C2×#123—1631no—
216C2×#123—1631no—
217C2×#123—1631no—
218C2×C22⋊Q8—1631no—
219C2×C22⋊Q8—1631no—
220C2×C22⋊Q8—1631no—
221C2×#121—1631no—
222C2×C4×D4—1631no—
223C2×C4×D4—1631no—
224C22×C4⋊C4—1631no—
225C22×Q8—1631no—
226C22×D4—1631no—
227C22×D4—1631no—
228C22×D4—1631no—
229C23×C4—1631no—
230C23×C4—1631no—
231C23×C4—1631no—
232C23×C4—1631no—
233C23×C4—1631no—
234#126—1638no—
235#126—1638no—
236#125—1638no—
237#125C2×D16318yescomplemented
238#123—1638no—
239#123—1638no—
240#123—1638no—
241#123—1638no—
242C22⋊Q8—1638no—
243C22⋊Q8—1638no—
244C22⋊Q8—1638no—
245#121—1638no—
246C4×D4—1638no—
247C4×D4—1638no—
248C2×C4⋊C4—1631no—
249C2×C4⋊C4—1631no—
250C2×C4⋊C4—1631no—
251C2×C4⋊C4—1631no—
252C2×C4⋊C4—1631no—
253C2×C4⋊C4—1631no—
254C2×C4⋊C4—1631no—
255C2×C4⋊C4—1631no—
256C2×C4⋊C4—1638no—
257C2×C4⋊C4C2×D16311yescharacteristic
258C2×C4⋊C4C2×D16311yescharacteristic
259C2×C4⋊C4C2×D16311yescharacteristic
260C2×C4⋊C4C2×D16311yescharacteristic
261C2×C4⋊C4C2×D16311yescharacteristic
262C2×C4⋊C4C2×D16311yescharacteristic
263C2×C22⋊C4—1631no—
264C2×C22⋊C4—1631no—
265C2×C22⋊C4—1631no—
266C2×C22⋊C4—1631no—
267C2×C22⋊C4—1631no—
268C2×C22⋊C4—1631no—
269C2×C22⋊C4—1631no—
270C2×C22⋊C4—1631no—
271C2×C22⋊C4—1631no—
272C2×C22⋊C4—1631no—
273C2×C42—1631no—
274C2×C42C2×D16311yescharacteristic
275C24—1631no—
276C24—1631no—
277C24—1631no—
278C2×Q8—6522no—
279C2×Q8—1634no—
280C2×D4—3264no—
281C2×D4—3264no—
282C2×D4—6522no—
283C2×D4—1634no—
284C2×D4—1634no—
285C2×D4—1634no—
286C22×C4—3264no—
287C22×C4—6522no—
288C22×C4—6522no—
289C22×C4—6522no—
290C22×C4—6522no—
291C22×C4—1631no—
292C22×C4—1631no—
293C22×C4—1631no—
294C22×C4—1631no—
295C22×C4—1631no—
296C22×C4—1634no—
297C22×C4—1634no—
298C22×C4—1634no—
299C22×C4—1634no—
300C22×C4—1634no—
301C22×C4C22×D16311yescharacteristic
302C22×C4C22×D16311yescharacteristic
303C22×C4C22×D16311yescharacteristic
304C22×C4C22×D16311yescharacteristic
305C22×C4C22×D16311yescharacteristic
306C22×C4C22×D16311yescharacteristic
307C22×C4C22×D16311yescharacteristic
308C4⋊C4—1634no—
309C4⋊C4—1634no—
310C4⋊C4—1634no—
311C4⋊C4—1634no—
312C4⋊C4—1634no—
313C4⋊C4—1634no—
314C4⋊C4—1634no—
315C4⋊C4—1634no—
316C4⋊C4C22×D16314yes—
317C4⋊C4C22×D16314yes—
318C4⋊C4C22×D16314yes—
319C4⋊C4C22×D16314yes—
320C4⋊C4C22×D16314yes—
321C4⋊C4C22×D16314yes—
322C22⋊C4—1634no—
323C22⋊C4—1634no—
324C22⋊C4—1634no—
325C22⋊C4—1634no—
326C22⋊C4—1634no—
327C22⋊C4—1634no—
328C22⋊C4—1634no—
329C22⋊C4—1634no—
330C22⋊C4—1634no—
331C22⋊C4—1634no—
332C42—1634no—
333C42C22×D16314yes—
334C23—3262no—
335C23—3262no—
336C23—3262no—
337C23—3262no—
338C23—3262no—
339C23—3262no—
340C23—3262no—
341C23—3262no—
342C23—1632no—
343C23—1632no—
344C23—1634no—
345C23—1632no—
346C23C23×D16311yescharacteristic
347Q8—6524no—
348D4—3268no—
349D4—3268no—
350D4—6524no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—3268no—
359C2×C4—3262no—
360C2×C4—3262no—
361C2×C4—3262no—
362C2×C4—3262no—
363C2×C4—3262no—
364C2×C4—6524no—
365C2×C4—6524no—
366C2×C4—6524no—
367C2×C4—6524no—
368C2×C4—1632no—
369C2×C4—1632no—
370C2×C4—1632no—
371C2×C4—1632no—
372C2×C4—1632no—
373C2×C4C23×D16312yes—
374C2×C4C23×D16312yes—
375C2×C4C23×D16312yes—
376C2×C4C23×D16312yes—
377C2×C4C23×D16312yes—
378C2×C4C23×D16312yes—
379C2×C4C23×D16312yes—
380C22—3262no—
381C22—3262no—
382C22—3268no—
383C22—3262no—
384C22—3262no—
385C22—3262no—
386C22—3262no—
387C22—3262no—
388C22—3262no—
389C22—6524no—
390C22—6524no—
391C22—1634no—
392C22#41×D16312yes—
393C22#1910478112yes—
394C22#1910478312yes—
395C22C24×D16311yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—3262no—
404C4—3262no—
405C4—3262no—
406C4—3262no—
407C4—3262no—
408C2—3264no—
409C2—6522no—
410C2—6522no—
411C2#2378226314yesdirect product factor
412C2C2×#41×D16311yescharacteristic
413C2C2×#1910478111yescharacteristic
414C2C2×#1910478311yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#125×C163C2
socleC23×C163C24
prime radicalC2×#19104710C2
hypercenterC2×#125D163
hypocenterC163C2×#539
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #23782263C2direct (N × H)
2. C2×#19104730C2semidirect
3. C2×#19104729C2semidirect
4. C2×#19104725C2semidirect
5. C2×#19104720C2semidirect
6. C2×#19104720C2semidirect
7. C2×#19104720C2semidirect
8. C2×#19104720C2semidirect
9. C2×#19104715C2semidirect
10. C2×#19104713C2semidirect
11. C2×#19104713C2semidirect
13. C2×C4×D652C2semidirect
14. C2×#19104697C2semidirect
15. C2×C4⋊C4×D163C2semidirect
16. C2×#125×C163C2semidirect
26. #19104730C22semidirect
27. #19104729C22semidirect
28. #19104725C22semidirect
29. #19104720C22semidirect
30. #19104720C22semidirect
31. #19104720C22semidirect
32. #19104720C22semidirect
33. #19104715C22semidirect
34. #19104713C22semidirect
35. #19104713C22semidirect
37. C4×D652C22semidirect
38. #19104697C22semidirect
47. C4⋊C4×D163C22semidirect
59. #125×C163C22semidirect
200. C2×C163#539semidirect
207. C163C2×#539semidirect
213. C2×#125D163semidirect
237. #125C2×D163semidirect
411. C2#23782263direct (N × H)

Maximal subgroups

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

subgroupindex
#237822632
C2×#191047302
C2×#191047292
C2×#191047252
C2×#191047202
C2×#191047152
C2×#191047132
C2×#191047102
C2×C4×D6522
C2×#191046972
C2×C4⋊C4×D1632
C2×#125×C1632
C2×#539163