Finite Group Databasebeta

← C2×#23782403 #35228651

Subgroups — C2×#23782403 #35228651

Order 20864.

1344 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782403C2116yesmaximal, direct product factor
2C2×#19104731C211yesmaximal, characteristic, complemented
3C2×#19104730C211yesmaximal, characteristic, complemented
4C2×#19104728C211yesmaximal, characteristic, complemented
5C2×#19104720C211yesmaximal, characteristic, complemented
6C2×#19104719C211yesmaximal, characteristic, complemented
7C2×#19104718C211yesmaximal, characteristic, complemented
8C2×#19104709C211yesmaximal, characteristic, complemented
9C2×#19104708C211yesmaximal, characteristic, complemented
10C2×#19104707C211yesmaximal, characteristic, complemented
11C2×#19104707C211yesmaximal, characteristic, complemented
12C2×#19104706C211yesmaximal, characteristic, complemented
13C2×#19104705C211yesmaximal, characteristic, complemented
14C2×#19104703C211yesmaximal, characteristic, complemented
15C2×#19104703C211yesmaximal, characteristic, complemented
16C2×#121×C163C211yesmaximal, characteristic, complemented
17C24×D163C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×D652C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic
23C22×C22⋊D163C2211yescharacteristic
24C22×C22⋊D163C2211yescharacteristic
25C22×C22⋊D163C2211yescharacteristic, complemented
26C22×C22⋊D163C2211yescharacteristic, complemented
27C23×Dic163C2211yescharacteristic, complemented
28C22×C4×D163C2211yescharacteristic, complemented
29#19104731C2218yescomplemented
30#19104730C2218yescomplemented
31#19104728C2218yescomplemented
32#19104720C2218yescomplemented
33#19104719C2218yescomplemented
34#19104718C2218yescomplemented
35#19104709C2218yescomplemented
36#19104708C2218yescomplemented
37#19104707C2218yescomplemented
38#19104707C2218yescomplemented
39#19104706C2218yescomplemented
40#19104705C2218yescomplemented
41#19104703C2218yescomplemented
42#19104703C2218yescomplemented
43C2×C4⋊Dic163C2211yescharacteristic, complemented
44C2×C4⋊Dic163C2211yescharacteristic, complemented
45C2×Dic163⋊C4C2211yescharacteristic, complemented
46C2×Dic163⋊C4C2211yescharacteristic, complemented
47C2×Dic163⋊C4C2211yescharacteristic, complemented
48C2×C22⋊Dic163C2211yescharacteristic
49C2×C22⋊Dic163C2211yescharacteristic
50C2×C22⋊Dic163C2211yescharacteristic, complemented
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, complemented
58C2×C4×Dic163C2211yescharacteristic, complemented
59C22×D4×C163C2211yescharacteristic
60C22×D4×C163C2211yescharacteristic
61C22×D4×C163C2211yescharacteristic, complemented
62C23×C4×C163C2211yescharacteristic, complemented
63#121×C163C2218yescomplemented
64C2×C4⋊C4×C163C2211yescharacteristic, complemented
65C2×C22⋊C4×C163C2211yescharacteristic
66C2×C22⋊C4×C163C2211yescharacteristic, complemented
67C23×D163—42no—
68C23×D163—42no—
69C23×D163—42no—
70C23×D163C2311yescharacteristic
71C23×D163C2311yescharacteristic
72C23×D163C2311yescharacteristic
73C23×D163C2314yes—
74C2×D652—42no—
75C2×D652—42no—
76C2×D652C2314yes—
77C2×D652C2314yes—
78C2×C22⋊D163—42no—
79C2×C22⋊D163—42no—
80C2×C22⋊D163—42no—
81C2×C22⋊D163—42no—
82C2×C22⋊D163—42no—
83C2×C22⋊D163—42no—
84C2×C22⋊D163—42no—
85C2×C22⋊D163C2314yes—
86C2×C22⋊D163C2314yes—
87C2×C22⋊D163C2314yes—
88C2×C22⋊D163C2314yes—
89C2×C22⋊D163C2314yes—
90C2×C22⋊D163C2314yescomplemented
91C2×C22⋊D163C2314yescomplemented
92C22×Dic163—42no—
93C22×Dic163C2311yescharacteristic
94C22×Dic163C2311yescharacteristic
95C22×Dic163C2311yescharacteristic
96C22×Dic163C2311yescharacteristic
97C22×Dic163C2311yescharacteristic
98C22×Dic163C2314yescomplemented
99C2×C4×D163—42no—
100C2×C4×D163C2314yescomplemented
101C4⋊Dic163C2314yescomplemented
102C4⋊Dic163C2314yescomplemented
103Dic163⋊C4C2314yescomplemented
104Dic163⋊C4C2314yescomplemented
105Dic163⋊C4C2314yescomplemented
106C22⋊Dic163C2314yes—
107C22⋊Dic163C2314yes—
108C22⋊Dic163C2314yescomplemented
109(C2×D163)⋊C4C2314yes—
110(C2×D163)⋊C4C2314yes—
111(C2×D163)⋊C4C2314yes—
112(C2×D163)⋊C4C2314yes—
113(C2×D163)⋊C4C2314yes—
114(C2×D163)⋊C4C2314yes—
115(C2×D163)⋊C4C2314yescomplemented
116C4×Dic163C2314yescomplemented
117C24×C163C2311yescharacteristic
118C24×C163C2311yescharacteristic
119C24×C163C2311yescharacteristic
120C2×D4×C163—42no—
121C2×D4×C163—42no—
122C2×D4×C163—42no—
123C2×D4×C163C2314yes—
124C2×D4×C163C2314yes—
125C2×D4×C163C2314yescomplemented
126C22×C4×C163—42no—
127C22×C4×C163C2311yescharacteristic
128C22×C4×C163C2311yescharacteristic
129C22×C4×C163C2311yescharacteristic
130C22×C4×C163C2311yescharacteristic
131C22×C4×C163C2314yescomplemented
132C4⋊C4×C163C2314yescomplemented
133C22⋊C4×C163C2314yes—
134C22⋊C4×C163C2314yescomplemented
135C22×D163—22no—
136C22×D163—22no—
137C22×D163—22no—
138C22×D163—22no—
139C22×D163—22no—
140C22×D163—22no—
141C22×D163—22no—
142C22×D163—22no—
143C22×D163—22no—
144C22×D163—44no—
145C22×D163—44no—
146C22×D163—44no—
147C22×D163—416no—
148C22×D163C2412yes—
149C22×D163C2412yes—
150C22×D163C2412yes—
151D652—44no—
152D652—44no—
153C22⋊D163—44no—
154C22⋊D163—44no—
155C22⋊D163—44no—
156C22⋊D163—44no—
157C22⋊D163—44no—
158C22⋊D163—44no—
159C22⋊D163—44no—
160C2×Dic163—22no—
161C2×Dic163—22no—
162C2×Dic163—22no—
163C2×Dic163—22no—
164C2×Dic163—22no—
165C2×Dic163—44no—
166C2×Dic163C2412yes—
167C2×Dic163C2412yes—
168C2×Dic163C2412yes—
169C2×Dic163C2412yes—
170C2×Dic163C2412yes—
171C4×D163—44no—
172C23×C163—22no—
173C23×C163—22no—
174C23×C163—22no—
175C23×C163—22no—
176C23×C163—22no—
177C23×C163—22no—
178C23×C163—22no—
179C23×C163—22no—
180C23×C163—22no—
181C23×C163C2412yes—
182C23×C163C2412yes—
183C23×C163C2412yes—
184C23×C163C2411yescharacteristic
185D4×C163—44no—
186D4×C163—44no—
187D4×C163—44no—
188C2×C4×C163—22no—
189C2×C4×C163—22no—
190C2×C4×C163—22no—
191C2×C4×C163—22no—
192C2×C4×C163—44no—
193C2×C4×C163C2412yes—
194C2×C4×C163C2412yes—
195C2×C4×C163C2412yes—
196C2×C4×C163C2412yes—
197C22×C163—22no—
198C22×C163—22no—
199C22×C163—22no—
200C22×C163—22no—
201C22×C163—22no—
202C22×C163—22no—
203C22×C163—22no—
204C22×C163—22no—
205C22×C163—22no—
206C22×C163—44no—
207C22×C163—44no—
208C22×C163—44no—
209C22×C163H(5,𝔽2)12yes—
210C22×C163H(5,𝔽2)12yes—
211C22×C163H(5,𝔽2)12yes—
212C22×C163C2511yescharacteristic
213C2×D163—22no—
214C2×D163—22no—
215C2×D163—22no—
216C2×D163—22no—
217C2×D163—22no—
218C2×D163—22no—
219C2×D163—22no—
220C2×D163—22no—
221C2×D163—22no—
222C2×D163—416no—
223C2×D163—44no—
224C2×D163—44no—
225C2×D163—44no—
226C4×C163—22no—
227C4×C163—22no—
228C4×C163—22no—
229C4×C163—22no—
230Dic163—22no—
231Dic163—22no—
232Dic163—22no—
233Dic163—22no—
234Dic163—22no—
235C2×C163—42no—
236C2×C163—42no—
237C2×C163—42no—
238C2×C163#56014yescomplemented
239C2×C163C2×H(5,𝔽2)11yescharacteristic
240C2×C163C2×H(5,𝔽2)11yescharacteristic
241C2×C163C2×H(5,𝔽2)11yescharacteristic
242D163—42no—
243D163—42no—
244D163—42no—
245C163C2×#56011yescharacteristic, complemented, Sylow
246C2×#560—1631nomaximal, Sylow
247#560—16316no—
248C2×#127—1631no—
249C2×#126—1631no—
250C2×#126—1631no—
251C2×#123—1631no—
252C2×#123—1631no—
253C2×#123—1631no—
254C2×#121—1631no—
255C2×#121—1631no—
256C2×#121—1631no—
257C2×#121—1631no—
258C2×#121—1631no—
259C2×#121D16311yescharacteristic, complemented
260C2×#120—1631no—
261C2×#120—1631no—
262C2×#120—1631no—
263C25—1631no—
264C22×D4—1631no—
265C22×D4—1631no—
266C22×D4—1631no—
267C22×D4—1631no—
268C22×D4—1631no—
269C22×D4—1631no—
270C22×D4—1631no—
271C22×D4—1631no—
272C22×D4—1631no—
273C22×D4C2×D16311yescharacteristic
274C22×D4C2×D16311yescharacteristic
275C22×D4C2×D16311yescharacteristic, complemented
276C23×C4—1631no—
277C23×C4—1631no—
278C23×C4C2×D16311yescharacteristic, complemented
279#127—1638no—
280#126—1638no—
281#126—1638no—
282#123—1638no—
283#123—1638no—
284#123—1638no—
285#121—1638no—
286#121—1638no—
287#121—1638no—
288#121—1638no—
289#121—1638no—
290#121C2×D16318yescomplemented
291#120—1638no—
292#120—1638no—
293#120—1638no—
294C2×C4⋊C4—1631no—
295C2×C4⋊C4—1631no—
296C2×C4⋊C4—1631no—
297C2×C4⋊C4—1631no—
298C2×C4⋊C4—1631no—
299C2×C4⋊C4C2×D16311yescharacteristic, complemented
300C2×C22⋊C4—1631no—
301C2×C22⋊C4—1631no—
302C2×C22⋊C4—1631no—
303C2×C22⋊C4—1631no—
304C2×C22⋊C4—1631no—
305C2×C22⋊C4—1631no—
306C2×C22⋊C4—1631no—
307C2×C22⋊C4—1631no—
308C2×C22⋊C4—1631no—
309C2×C22⋊C4—1631no—
310C2×C22⋊C4C2×D16311yescharacteristic
311C2×C22⋊C4C2×D16311yescharacteristic, complemented
312C2×C42—1631no—
313C24—6522no—
314C24—6522no—
315C24—6522no—
316C24—1631no—
317C24—1631no—
318C24—1631no—
319C24—1634no—
320C24C22×D16311yescharacteristic
321C24C22×D16311yescharacteristic
322C24C22×D16311yescharacteristic
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—6522no—
327C2×D4—6522no—
328C2×D4—6522no—
329C2×D4—6522no—
330C2×D4—6522no—
331C2×D4—6522no—
332C2×D4—6522no—
333C2×D4—6522no—
334C2×D4—6522no—
335C2×D4—1634no—
336C2×D4—1634no—
337C2×D4—1634no—
338C2×D4—1634no—
339C2×D4—1634no—
340C2×D4—1634no—
341C2×D4—1634no—
342C2×D4—1634no—
343C2×D4—1634no—
344C2×D4C22×D16314yes—
345C2×D4C22×D16314yes—
346C2×D4C22×D16314yescomplemented
347C22×C4—42no—
348C22×C4—6522no—
349C22×C4—6522no—
350C22×C4—1631no—
351C22×C4—1631no—
352C22×C4—1631no—
353C22×C4—1631no—
354C22×C4—1631no—
355C22×C4—1634no—
356C22×C4—1634no—
357C22×C4C22×D16311yescharacteristic
358C22×C4C22×D16311yescharacteristic
359C22×C4C22×D16311yescharacteristic
360C22×C4C22×D16311yescharacteristic
361C22×C4C22×D16314yescomplemented
362C4⋊C4—1634no—
363C4⋊C4—1634no—
364C4⋊C4—1634no—
365C4⋊C4—1634no—
366C4⋊C4—1634no—
367C4⋊C4C22×D16314yescomplemented
368C22⋊C4—1634no—
369C22⋊C4—1634no—
370C22⋊C4—1634no—
371C22⋊C4—1634no—
372C22⋊C4—1634no—
373C22⋊C4—1634no—
374C22⋊C4—1634no—
375C22⋊C4—1634no—
376C22⋊C4—1634no—
377C22⋊C4—1634no—
378C22⋊C4C22×D16314yes—
379C22⋊C4C22×D16314yescomplemented
380C42—1634no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—3262no—
391C23—3262no—
392C23—3262no—
393C23—3262no—
394C23—3262no—
395C23—3262no—
396C23—3262no—
397C23—3262no—
398C23—3262no—
399C23—6524no—
400C23—6524no—
401C23—6524no—
402C23—65216no—
403C23—1632no—
404C23—1632no—
405C23—1632no—
406C23C23×D16312yes—
407C23C23×D16312yes—
408C23C23×D16312yes—
409C23C23×D16311yescharacteristic
410D4—44no—
411D4—44no—
412D4—44no—
413D4—6524no—
414D4—6524no—
415D4—6524no—
416D4—6524no—
417D4—6524no—
418D4—6524no—
419D4—6524no—
420D4—6524no—
421D4—6524no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—44no—
427C2×C4—3262no—
428C2×C4—3262no—
429C2×C4—3262no—
430C2×C4—3262no—
431C2×C4—3262no—
432C2×C4—6524no—
433C2×C4—6524no—
434C2×C4—1632no—
435C2×C4—1632no—
436C2×C4—1632no—
437C2×C4—1632no—
438C2×C4—1632no—
439C2×C4C23×D16312yes—
440C2×C4C23×D16312yes—
441C2×C4C23×D16312yes—
442C2×C4C23×D16312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—3262no—
456C22—3262no—
457C22—3262no—
458C22—3262no—
459C22—3262no—
460C22—3262no—
461C22—3262no—
462C22—3262no—
463C22—3262no—
464C22—65216no—
465C22—6524no—
466C22—6524no—
467C22—6524no—
468C22#1910478012yes—
469C22#1910478012yes—
470C22#1910478112yes—
471C22C24×D16311yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—3262no—
477C4—3262no—
478C4—3262no—
479C4—3262no—
480C4—3262no—
481C2—42no—
482C2—42no—
483C2—42no—
484C2—6522no—
485C2—6522no—
486C2—6522no—
487C2#2378240314yesdirect product factor
488C2C2×#1910478011yescharacteristic
489C2C2×#1910478011yescharacteristic
490C2C2×#1910478111yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#121×C163C2
socleC23×C163C24
hypercenterC2×#121D163
hypocenterC163C2×#560
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #23782403C2direct (N × H)
2. C2×#19104731C2semidirect
3. C2×#19104730C2semidirect
4. C2×#19104728C2semidirect
5. C2×#19104720C2semidirect
6. C2×#19104719C2semidirect
7. C2×#19104718C2semidirect
8. C2×#19104709C2semidirect
9. C2×#19104708C2semidirect
10. C2×#19104707C2semidirect
11. C2×#19104707C2semidirect
12. C2×#19104706C2semidirect
13. C2×#19104705C2semidirect
14. C2×#19104703C2semidirect
15. C2×#19104703C2semidirect
16. C2×#121×C163C2semidirect
25. C22×C22⋊D163C22semidirect
26. C22×C22⋊D163C22semidirect
27. C23×Dic163C22semidirect
28. C22×C4×D163C22semidirect
29. #19104731C22semidirect
30. #19104730C22semidirect
31. #19104728C22semidirect
32. #19104720C22semidirect
33. #19104719C22semidirect
34. #19104718C22semidirect
35. #19104709C22semidirect
36. #19104708C22semidirect
37. #19104707C22semidirect
38. #19104707C22semidirect
39. #19104706C22semidirect
40. #19104705C22semidirect
41. #19104703C22semidirect
42. #19104703C22semidirect
43. C2×C4⋊Dic163C22semidirect
44. C2×C4⋊Dic163C22semidirect
45. C2×Dic163⋊C4C22semidirect
46. C2×Dic163⋊C4C22semidirect
47. C2×Dic163⋊C4C22semidirect
50. C2×C22⋊Dic163C22semidirect
57. C2×(C2×D163)⋊C4C22semidirect
58. C2×C4×Dic163C22semidirect
61. C22×D4×C163C22semidirect
62. C23×C4×C163C22semidirect
63. #121×C163C22semidirect
64. C2×C4⋊C4×C163C22semidirect
66. C2×C22⋊C4×C163C22semidirect
90. C2×C22⋊D163C23semidirect
91. C2×C22⋊D163C23semidirect
98. C22×Dic163C23semidirect
100. C2×C4×D163C23semidirect
101. C4⋊Dic163C23semidirect
102. C4⋊Dic163C23semidirect
103. Dic163⋊C4C23semidirect
104. Dic163⋊C4C23semidirect
105. Dic163⋊C4C23semidirect
108. C22⋊Dic163C23semidirect
115. (C2×D163)⋊C4C23semidirect
116. C4×Dic163C23semidirect
125. C2×D4×C163C23semidirect
131. C22×C4×C163C23semidirect
132. C4⋊C4×C163C23semidirect
134. C22⋊C4×C163C23semidirect
238. C2×C163#560semidirect
245. C163C2×#560semidirect
259. C2×#121D163semidirect
275. C22×D4C2×D163semidirect
278. C23×C4C2×D163semidirect
290. #121C2×D163semidirect
299. C2×C4⋊C4C2×D163semidirect
311. C2×C22⋊C4C2×D163semidirect
346. C2×D4C22×D163semidirect
361. C22×C4C22×D163semidirect
367. C4⋊C4C22×D163semidirect
379. C22⋊C4C22×D163semidirect
487. C2#23782403direct (N × H)

Maximal subgroups

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

subgroupindex
#237824032
C2×#191047312
C2×#191047302
C2×#191047282
C2×#191047202
C2×#191047192
C2×#191047182
C2×#191047092
C2×#191047082
C2×#191047072
C2×#191047062
C2×#191047052
C2×#191047032
C2×#121×C1632
C2×#560163