Finite Group Databasebeta

← C2×#23782366 #35228060

Subgroups — C2×#23782366 #35228060

Order 20864.

1252 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782366C2116yesmaximal, direct product factor
2C2×#19104730C211yesmaximal, characteristic, complemented
3C2×#19104725C211yesmaximal, characteristic, complemented
4C2×#19104720C211yesmaximal, characteristic, complemented
5C2×#19104719C211yesmaximal, characteristic, complemented
6C2×#19104710C211yesmaximal, characteristic, complemented
7C2×#19104708C211yesmaximal, characteristic, complemented
8C2×#19104708C211yesmaximal, characteristic, complemented
9C2×#19104708C211yesmaximal, characteristic, complemented
10C2×C4×D652C211yesmaximal, characteristic, complemented
11C2×#19104697C211yesmaximal, characteristic, complemented
12C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
13C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
14C2×#19104692C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D163C211yesmaximal, characteristic, complemented
16C2×#126×C163C211yesmaximal, characteristic, complemented
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×D652C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic
23C23×Dic163C2211yescharacteristic
24C22×C4×D163C2211yescharacteristic
25C22×C4×D163C2211yescharacteristic
26C22×C4×D163C2211yescharacteristic
27C22×C4×D163C2211yescharacteristic
28C22×C4×D163C2211yescharacteristic
29#19104730C2218yescomplemented
30#19104725C2218yescomplemented
31#19104720C2218yescomplemented
32#19104719C2218yescomplemented
33#19104710C2218yescomplemented
34#19104708C2218yescomplemented
35#19104708C2218yescomplemented
36#19104708C2218yescomplemented
37C4×D652C2218yescomplemented
38#19104697C2218yescomplemented
39(C22⋊D163)⋊C4C2218yescomplemented
40(C22⋊D163)⋊C4C2218yescomplemented
41#19104692C2218yescomplemented
42C2×C4⋊Dic163C2211yescharacteristic
43C2×C4⋊Dic163C2211yescharacteristic
44C2×Dic163⋊C4C2211yescharacteristic
45C2×Dic163⋊C4C2211yescharacteristic
46C2×Dic163⋊C4C2211yescharacteristic
47C2×Dic163⋊C4C2211yescharacteristic
48C2×Dic163⋊C4C2211yescharacteristic
49C4⋊C4×D163C2218yescomplemented
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×C4×Dic163C2211yescharacteristic
58C2×C4×Dic163C2211yescharacteristic
59#126×C163C2218yescomplemented
60C2×C4⋊C4×C163C2211yescharacteristic
61C2×C4⋊C4×C163C2211yescharacteristic
62C2×C4⋊C4×C163C2211yescharacteristic
63C2×C22⋊C4×C163C2211yescharacteristic
64C2×C22⋊C4×C163C2211yescharacteristic
65C2×C22⋊C4×C163C2211yescharacteristic
66C2×C42×C163C2211yescharacteristic
67C23×D163C2311yescharacteristic
68C23×D163C2311yescharacteristic
69C23×D163C2311yescharacteristic
70C2×D652—24no—
71C2×D652—24no—
72C2×D652—42no—
73C2×D652C2314yes—
74C2×D652C2314yes—
75C2×D652C2314yes—
76C2×C22⋊D163—24no—
77C2×C22⋊D163—24no—
78C2×C22⋊D163—42no—
79C2×C22⋊D163C2314yes—
80C2×C22⋊D163C2314yes—
81C2×C22⋊D163C2314yes—
82C22×Dic163—42no—
83C22×Dic163C2311yescharacteristic
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2311yescharacteristic
86C22×Dic163C2311yescharacteristic
87C22×Dic163C2314yes—
88C22×Dic163C2311yescharacteristic
89C2×C4×D163—24no—
90C2×C4×D163—24no—
91C2×C4×D163—42no—
92C2×C4×D163—42no—
93C2×C4×D163—42no—
94C2×C4×D163C2314yes—
95C2×C4×D163C2314yes—
96C2×C4×D163C2314yes—
97C2×C4×D163C2314yes—
98C2×C4×D163C2314yes—
99C4⋊Dic163C2314yes—
100C4⋊Dic163C2314yes—
101Dic163⋊C4C2314yes—
102Dic163⋊C4C2314yes—
103Dic163⋊C4C2314yes—
104Dic163⋊C4C2314yes—
105Dic163⋊C4C2314yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108(C2×D163)⋊C4C2314yes—
109(C2×D163)⋊C4C2314yes—
110(C2×D163)⋊C4C2314yes—
111(C2×D163)⋊C4C2314yes—
112(C2×D163)⋊C4C2314yes—
113C4×Dic163C2314yes—
114C4×Dic163C2314yes—
115C24×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2311yescharacteristic
121C22×C4×C163C2311yescharacteristic
122C4⋊C4×C163C2314yes—
123C4⋊C4×C163C2314yes—
124C4⋊C4×C163C2314yes—
125C22⋊C4×C163C2314yes—
126C22⋊C4×C163C2314yes—
127C22⋊C4×C163C2314yes—
128C42×C163C2314yes—
129C22×D163—22no—
130C22×D163—22no—
131C22×D163—22no—
132C22×D163—22no—
133C22×D163—22no—
134C22×D163—22no—
135C22×D163—22no—
136C22×D163—22no—
137C22×D163#4114yes—
138C22×D163C2412yes—
139C22×D163C2412yes—
140C22×D163C2412yes—
141D652—28no—
142D652—28no—
143D652—44no—
144C22⋊D163—28no—
145C22⋊D163—28no—
146C22⋊D163—44no—
147C2×Dic163—22no—
148C2×Dic163—22no—
149C2×Dic163—22no—
150C2×Dic163—22no—
151C2×Dic163—44no—
152C2×Dic163#4114yes—
153C2×Dic163C2412yes—
154C2×Dic163C2412yes—
155C2×Dic163C2412yes—
156C2×Dic163C2412yes—
157C2×Dic163C2412yes—
158C4×D163—28no—
159C4×D163—28no—
160C4×D163—44no—
161C4×D163—44no—
162C4×D163—44no—
163C23×C163—22no—
164C23×C163—22no—
165C23×C163—22no—
166C23×C163C2412yes—
167C23×C163C2411yescharacteristic
168C2×C4×C163—22no—
169C2×C4×C163—22no—
170C2×C4×C163—22no—
171C2×C4×C163—22no—
172C2×C4×C163—22no—
173C2×C4×C163—22no—
174C2×C4×C163C2412yes—
175C2×C4×C163C2412yes—
176C2×C4×C163C2412yes—
177C2×C4×C163C2412yes—
178C2×C4×C163C2412yes—
179C2×C4×C163C2412yes—
180C22×C163—22no—
181C22×C163—22no—
182C22×C163—22no—
183C22×C163—44no—
184C22×C163C2×#4112yes—
185C22×C163C2×#4112yes—
186C22×C163H(5,𝔽2)12yes—
187C22×C163C2511yescharacteristic
188C2×D163—28no—
189C2×D163—22no—
190C2×D163—22no—
191C2×D163—22no—
192C2×D163—22no—
193C2×D163—22no—
194C2×D163—22no—
195C2×D163—22no—
196C2×D163—22no—
197C2×D163—44no—
198C2×D163—44no—
199C2×D163C2×#4114yes—
200C4×C163—22no—
201C4×C163—22no—
202C4×C163—22no—
203C4×C163—22no—
204C4×C163—22no—
205C4×C163—22no—
206Dic163—22no—
207Dic163—22no—
208Dic163—22no—
209Dic163—22no—
210Dic163C2×#4114yes—
211C2×C163—42no—
212C2×C163#55314yescomplemented
213C2×C163C22×#4111yescharacteristic
214C2×C163C22×#4111yescharacteristic
215C2×C163C2×H(5,𝔽2)11yescharacteristic
216D163—24no—
217D163—42no—
218D163—42no—
219C163C2×#55311yescharacteristic, complemented, Sylow
220C2×#553—1631nomaximal, Sylow
221#553—16316no—
222C2×#126—1631no—
223C2×#126D16311yescharacteristic, complemented
224C2×#125—1631no—
225C2×#123—1631no—
226C2×#123—1631no—
227C2×#121—1631no—
228C2×#121—1631no—
229C2×#121—1631no—
230C2×#121—1631no—
231C2×C4×D4—1631no—
232C2×C4×D4—1631no—
233C2×C4×D4—1631no—
234C2×C4×D4—1631no—
235C2×#117—1631no—
236C22×C4⋊C4—1631no—
237C22×D4—1631no—
238C22×D4—1631no—
239C22×D4—1631no—
240C22×D4—1631no—
241C22×D4—1631no—
242C22×D4—1631no—
243C23×C4—1631no—
244C23×C4—1631no—
245C23×C4—1631no—
246C23×C4—1631no—
247C23×C4—1631no—
248C23×C4—1631no—
249#126—1638no—
250#126C2×D16318yescomplemented
251#125—1638no—
252#123—1638no—
253#123—1638no—
254#121—1638no—
255#121—1638no—
256#121—1638no—
257#121—1638no—
258C4×D4—1638no—
259C4×D4—1638no—
260C4×D4—1638no—
261C4×D4—1638no—
262#117—1638no—
263C2×C4⋊C4—1631no—
264C2×C4⋊C4—1631no—
265C2×C4⋊C4—1631no—
266C2×C4⋊C4—1631no—
267C2×C4⋊C4—1631no—
268C2×C4⋊C4—1631no—
269C2×C4⋊C4—1631no—
270C2×C4⋊C4—1638no—
271C2×C4⋊C4C2×D16311yescharacteristic
272C2×C4⋊C4C2×D16311yescharacteristic
273C2×C4⋊C4C2×D16311yescharacteristic
274C2×C22⋊C4—1631no—
275C2×C22⋊C4—1631no—
276C2×C22⋊C4—1631no—
277C2×C22⋊C4—1631no—
278C2×C22⋊C4—1631no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4C2×D16311yescharacteristic
282C2×C22⋊C4C2×D16311yescharacteristic
283C2×C22⋊C4C2×D16311yescharacteristic
284C2×C42—1631no—
285C2×C42—1631no—
286C2×C42C2×D16311yescharacteristic
287C24—1631no—
288C24—1631no—
289C24—1631no—
290C24C22×D16311yescharacteristic
291C2×D4—3264no—
292C2×D4—3264no—
293C2×D4—3264no—
294C2×D4—3264no—
295C2×D4—6522no—
296C2×D4—6522no—
297C2×D4—1634no—
298C2×D4—1634no—
299C2×D4—1634no—
300C2×D4—1634no—
301C2×D4—1634no—
302C2×D4—1634no—
303C22×C4—3264no—
304C22×C4—3264no—
305C22×C4—6522no—
306C22×C4—6522no—
307C22×C4—6522no—
308C22×C4—6522no—
309C22×C4—1631no—
310C22×C4—1631no—
311C22×C4—1631no—
312C22×C4—1631no—
313C22×C4—1634no—
314C22×C4—1634no—
315C22×C4—1634no—
316C22×C4—1634no—
317C22×C4—1634no—
318C22×C4—1634no—
319C22×C4—1631no—
320C22×C4C22×D16311yescharacteristic
321C22×C4C22×D16311yescharacteristic
322C22×C4C22×D16311yescharacteristic
323C22×C4C22×D16311yescharacteristic
324C22×C4C22×D16311yescharacteristic
325C22×C4C22×D16311yescharacteristic
326C4⋊C4—1634no—
327C4⋊C4—1634no—
328C4⋊C4—1634no—
329C4⋊C4—1634no—
330C4⋊C4—1634no—
331C4⋊C4—1634no—
332C4⋊C4—1634no—
333C4⋊C4C22×D16314yes—
334C4⋊C4C22×D16314yes—
335C4⋊C4C22×D16314yes—
336C22⋊C4—1634no—
337C22⋊C4—1634no—
338C22⋊C4—1634no—
339C22⋊C4—1634no—
340C22⋊C4—1634no—
341C22⋊C4—1634no—
342C22⋊C4—1634no—
343C22⋊C4C22×D16314yes—
344C22⋊C4C22×D16314yes—
345C22⋊C4C22×D16314yes—
346C42—1634no—
347C42—1634no—
348C42C22×D16314yes—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—3262no—
353C23—3262no—
354C23—3262no—
355C23—3262no—
356C23—3262no—
357C23—3262no—
358C23—3262no—
359C23—3262no—
360C23—1632no—
361C23—1632no—
362C23—1634no—
363C23—1632no—
364C23C23×D16312yes—
365C23C23×D16311yescharacteristic
366D4—3268no—
367D4—3268no—
368D4—3268no—
369D4—3268no—
370D4—6524no—
371D4—6524no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—3268no—
379C2×C4—3268no—
380C2×C4—3262no—
381C2×C4—3262no—
382C2×C4—3262no—
383C2×C4—3262no—
384C2×C4—6524no—
385C2×C4—6524no—
386C2×C4—6524no—
387C2×C4—6524no—
388C2×C4—1632no—
389C2×C4—1632no—
390C2×C4—1632no—
391C2×C4—1632no—
392C2×C4—1634no—
393C2×C4—1632no—
394C2×C4C23×D16312yes—
395C2×C4C23×D16312yes—
396C2×C4C23×D16312yes—
397C2×C4C23×D16312yes—
398C2×C4C23×D16312yes—
399C2×C4C23×D16312yes—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—44no—
404C22—3262no—
405C22—3262no—
406C22—3268no—
407C22—3262no—
408C22—3262no—
409C22—3262no—
410C22—3262no—
411C22—3262no—
412C22—3262no—
413C22—6524no—
414C22—6524no—
415C22—1634no—
416C22#41×D16312yes—
417C22#41×D16312yes—
418C22#1910478112yes—
419C22C24×D16311yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—3262no—
427C4—3262no—
428C4—3262no—
429C4—3262no—
430C4—1634no—
431C2—42no—
432C2—3264no—
433C2—6522no—
434C2—6522no—
435C2#2378236614yesdirect product factor
436C2C2×#41×D16311yescharacteristic
437C2C2×#41×D16311yescharacteristic
438C2C2×#1910478111yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#126×C163C2
socleC23×C163C24
hypercenterC2×#126D163
hypocenterC163C2×#553
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #23782366C2direct (N × H)
2. C2×#19104730C2semidirect
3. C2×#19104725C2semidirect
4. C2×#19104720C2semidirect
5. C2×#19104719C2semidirect
6. C2×#19104710C2semidirect
7. C2×#19104708C2semidirect
8. C2×#19104708C2semidirect
9. C2×#19104708C2semidirect
10. C2×C4×D652C2semidirect
11. C2×#19104697C2semidirect
12. C2×(C22⋊D163)⋊C4C2semidirect
13. C2×(C22⋊D163)⋊C4C2semidirect
14. C2×#19104692C2semidirect
15. C2×C4⋊C4×D163C2semidirect
16. C2×#126×C163C2semidirect
29. #19104730C22semidirect
30. #19104725C22semidirect
31. #19104720C22semidirect
32. #19104719C22semidirect
33. #19104710C22semidirect
34. #19104708C22semidirect
35. #19104708C22semidirect
36. #19104708C22semidirect
37. C4×D652C22semidirect
38. #19104697C22semidirect
39. (C22⋊D163)⋊C4C22semidirect
40. (C22⋊D163)⋊C4C22semidirect
41. #19104692C22semidirect
49. C4⋊C4×D163C22semidirect
59. #126×C163C22semidirect
212. C2×C163#553semidirect
219. C163C2×#553semidirect
223. C2×#126D163semidirect
250. #126C2×D163semidirect
435. C2#23782366direct (N × H)

Maximal subgroups

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

subgroupindex
#237823662
C2×#191047302
C2×#191047252
C2×#191047202
C2×#191047192
C2×#191047102
C2×#191047082
C2×C4×D6522
C2×#191046972
C2×(C22⋊D163)⋊C42
C2×#191046922
C2×C4⋊C4×D1632
C2×#126×C1632
C2×#553163