Finite Group Databasebeta

← #35226299

Subgroups — #35226299

Order 20864.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782477C211yesmaximal, characteristic, complemented
2#23782475C211yesmaximal, characteristic, complemented
3#23782469C211yesmaximal, characteristic, complemented
4#23782466C211yesmaximal, characteristic, complemented
5#23782053C211yesmaximal, characteristic, complemented
6#23781934C211yesmaximal, characteristic, complemented
7#23781929C211yesmaximal, characteristic, complemented
8#23781927C211yesmaximal, characteristic, complemented
9#23781845C211yesmaximal, characteristic, complemented
10#23781836C211yesmaximal, characteristic, complemented
11#23781835C211yesmaximal, characteristic, complemented
12#23781831C211yesmaximal, characteristic, complemented
13#23781786C211yesmaximal, characteristic, complemented
14#23781785C211yesmaximal, characteristic, complemented
15#472×C163C211yesmaximal, characteristic, complemented
16#19104782C2211yescharacteristic, complemented
17#19104781C2211yescharacteristic, complemented
18#19104780C2211yescharacteristic
19#19104780C2211yescharacteristic, complemented
20#19104768—21no—
21#19104768C2211yescharacteristic, complemented
22#19104768C2211yescharacteristic, complemented
23#19104764—21no—
24#19104763—21no—
25#19104763C2211yescharacteristic
26#19104763C2211yescharacteristic, complemented
27#19104762—21no—
28#19104761C2211yescharacteristic
29#19104761C2211yescharacteristic, complemented
30#19104759—21no—
31#19104759—21no—
32#19104759—21no—
33#19104759—21no—
34#19104757—21no—
35#19104757C2211yescharacteristic
36#19104756—21no—
37#19104755C2211yescharacteristic
38#19104754—21no—
39#19104754—21no—
40C2×#14579688—21no—
41C2×#14579688C2211yescharacteristic, complemented
42C2×#14579687C2211yescharacteristic, complemented
43SD16×D163—21no—
44C2×#14579684C2211yescharacteristic, complemented
45D8×D163—21no—
46D8×D163—21no—
47#19104743C2211yescharacteristic, complemented
48#19104743C2211yescharacteristic, complemented
49#19104741C2211yescharacteristic, complemented
50#19104731C2211yescharacteristic, complemented
51#19104722C2211yescharacteristic, complemented
52#19104670C2211yescharacteristic, complemented
53#19104662C2211yescharacteristic, complemented
54#19104661C2211yescharacteristic, complemented
55#19104661C2211yescharacteristic, complemented
56#19104661C2211yescharacteristic, complemented
57#19104653C2211yescharacteristic, complemented
58#19104651C2211yescharacteristic, complemented
59#19104645C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C163C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C163C2211yescharacteristic, complemented
62C2×D8×C163C2211yescharacteristic, complemented
63#131×C163C2211yescharacteristic, complemented
64#108×C163C2211yescharacteristic, complemented
65#100×C163C2211yescharacteristic, complemented
66#100×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579699—21no—
69#14579699—21no—
70#14579699C2311yescharacteristic
71#14579699C2311yescharacteristic
72#14579699C2311yescharacteristic
73#14579698—21no—
74#14579698—21no—
75#14579698—21no—
76#14579698—21no—
77#14579698—21no—
78#14579698—41no—
79C2×Dic326—21no—
80C2×Dic326C2311yescharacteristic
81Q8×D163—21no—
82C2×D652—21no—
83C2×D652C2311yescharacteristic
84C2×C22⋊D163—41no—
85C2×C22⋊D163—41no—
86C2×C22⋊D163—41no—
87D4×D163—21no—
88D4×D163—21no—
89D4×D163—21no—
90D4×D163—21no—
91D4×D163—21no—
92D4×D163—41no—
93Dic652—21no—
94Dic652—21no—
95Dic652D411yescharacteristic, complemented
96#14579689D411yescharacteristic, complemented
97#14579688—21no—
98#14579688—21no—
99#14579688—21no—
100#14579688—21no—
101#14579688—21no—
102#14579688—21no—
103#14579688D411yescharacteristic, complemented
104#14579688D411yescharacteristic, complemented
105#14579687—21no—
106#14579687—21no—
107#14579687—21no—
108#14579687—21no—
109#14579687D411yescharacteristic, complemented
110#14579686D411yescharacteristic, complemented
111D1304—21no—
112D1304—21no—
113D1304D411yescharacteristic, complemented
114#14579684—21no—
115#14579684—21no—
116#14579684—21no—
117#14579684—21no—
118#14579684D411yescharacteristic, complemented
119#14579683C2311yescharacteristic
120#14579682—21no—
121#14579682—21no—
122#14579682—21no—
123C2×C163⋊C8C2311yescharacteristic
124C8×D163—21no—
125C8×D163—21no—
126C8×D163—21no—
127C22⋊Dic163—41no—
128C4×Dic163C2311yescharacteristic
129#41×C163C2311yescharacteristic
130C2×D4×C163C2311yescharacteristic
131C2×D4×C163C2311yescharacteristic
132SD16×C163—21no—
133SD16×C163—21no—
134D8×C163—21no—
135D8×C163—21no—
136D8×C163—21no—
137D8×C163—21no—
138M16×C163—21no—
139M16×C163C2311yescharacteristic
140M16×C163C2311yescharacteristic
141M16×C163C2311yescharacteristic
142C2×C8×C163—21no—
143C2×C8×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—41no—
146C22×D163—41no—
147C22×D163—41no—
148Dic326—21no—
149Dic326—21no—
150Dic326—41no—
151Dic326C2×D411yescharacteristic
152Dic326C2×D411yescharacteristic
153Dic326C2×D411yescharacteristic
154D652—21no—
155D652—21no—
156D652—41no—
157D652C2×D411yescharacteristic
158D652C2×D411yescharacteristic
159D652C2×D411yescharacteristic
160C22⋊D163—21no—
161C22⋊D163—21no—
162C22⋊D163—21no—
163C22⋊D163—41no—
164C22⋊D163—41no—
165C22⋊D163—41no—
166C22⋊D163—41no—
167C22⋊D163—41no—
168C22⋊D163—41no—
169C22⋊D163—82no—
170C2×Dic163—21no—
171C2×Dic163—21no—
172C2×Dic163—41no—
173C2×Dic163—41no—
174C2×Dic163—41no—
175C4×D163—21no—
176C4×D163—21no—
177C4×D163—21no—
178C4×D163—41no—
179C4×D163—41no—
180C163⋊C8—21no—
181C163⋊C8C2×D411yescharacteristic
182C163⋊C8C2×D411yescharacteristic
183C23×C163—21no—
184C23×C163—21no—
185Q8×C163C2×D411yescharacteristic
186D4×C163—21no—
187D4×C163—21no—
188D4×C163—21no—
189D4×C163—21no—
190D4×C163—21no—
191D4×C163C2×D411yescharacteristic
192C2×C4×C163—21no—
193C2×C4×C163C2411yescharacteristic
194C8×C163—21no—
195C8×C163—21no—
196C8×C163—21no—
197C8×C163C2×D411yescharacteristic
198C8×C163C2×D411yescharacteristic
199C22×C163—21no—
200C22×C163—41no—
201C22×C163—41no—
202C22×C163—81no—
203C22×C163—81no—
204C22×C163H(5,𝔽2)11yescharacteristic
205C2×D163—21no—
206C2×D163—21no—
207C2×D163—21no—
208C2×D163—41no—
209C2×D163—81no—
210C2×D163—161no—
211C2×D163—161no—
212C2×D163—161no—
213C4×C163—21no—
214C4×C163C22×D411yescharacteristic
215C4×C163C22×D411yescharacteristic
216Dic163—21no—
217Dic163—21no—
218Dic163—21no—
219Dic163—41no—
220Dic163—41no—
221C2×C163—21no—
222C2×C163—41no—
223C2×C163—81no—
224C2×C163—81no—
225C2×C163D4211yescharacteristic
226D163—41no—
227D163—41no—
228D163—41no—
229D163—81no—
230C163#328811yescharacteristic, complemented, Sylow
231#3288—1631nomaximal, Sylow
232#577—1631no—
233#577—1631no—
234#576—1631no—
235#576—1631no—
236#496—1631no—
237#472—1631no—
238#472D16311yescharacteristic, complemented
239#471—1631no—
240#471—1631no—
241#455—1631no—
242#454—1631no—
243#453—1631no—
244#453—1631no—
245#444—1631no—
246#444—1631no—
2472−1+4—1631no—
248H(5,𝔽2)—1631no—
249H(5,𝔽2)—1631no—
250H(5,𝔽2)—1631no—
251#137—3261no—
252#137—3261no—
253#137—1631no—
254#137—1631no—
255AGL(1,ℤ/8)—3261no—
256AGL(1,ℤ/8)—3261no—
257AGL(1,ℤ/8)—3261no—
258AGL(1,ℤ/8)—3261no—
259AGL(1,ℤ/8)—3261no—
260AGL(1,ℤ/8)—3261no—
261AGL(1,ℤ/8)—1631no—
262AGL(1,ℤ/8)—1631no—
263AGL(1,ℤ/8)—1631no—
264AGL(1,ℤ/8)—1631no—
265AGL(1,ℤ/8)C2×D16311yescharacteristic, complemented
266AGL(1,ℤ/8)C2×D16311yescharacteristic, complemented
267#135—3261no—
268#135—3261no—
269#135—3261no—
270#135—3261no—
271#135—1631no—
272#135—1631no—
273C2×SD16—3261no—
274C2×SD16—3261no—
275C2×SD16—1631no—
276C2×SD16—1631no—
277C2×D8—3261no—
278C2×D8—3261no—
279C2×D8—1631no—
280C2×D8C2×D16311yescharacteristic, complemented
281#131—1631no—
282#131—1631no—
283#131—1631no—
284#131C2×D16311yescharacteristic, complemented
285#127—1631no—
286#124—1631no—
287#108—1631no—
288#108C2×D16311yescharacteristic, complemented
289#104—1631no—
290#104—1631no—
291#104—1631no—
292#104—1631no—
293#101—1631no—
294#100—1631no—
295#100C2×D16311yescharacteristic, complemented
296#100C2×D16311yescharacteristic, complemented
297#97—1631no—
298#41—3261no—
299#41—3261no—
300#41—3261no—
301#41—3261no—
302#41—3261no—
303#41—3261no—
304#41—3261no—
305#41—3261no—
306#41—6521no—
307#41—1631no—
308#41—1631no—
309#41—1631no—
310#41C22×D16311yescharacteristic
311C2×Q8—3261no—
312C2×Q8—3261no—
313C2×Q8—1631no—
314C2×D4—3261no—
315C2×D4—3261no—
316C2×D4—3261no—
317C2×D4—3261no—
318C2×D4—3261no—
319C2×D4—3261no—
320C2×D4—6521no—
321C2×D4—6521no—
322C2×D4—6521no—
323C2×D4—6521no—
324C2×D4—1631no—
325C2×D4C22×D16311yescharacteristic
326C2×D4C22×D16311yescharacteristic
327Q16—3261no—
328Q16—3261no—
329Q16—1631no—
330Q16—1631no—
331SD16—21no—
332SD16—21no—
333SD16—3261no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338SD16—3261no—
339SD16—3261no—
340SD16—3261no—
341SD16—3261no—
342SD16—3261no—
343SD16—1631no—
344SD16—1631no—
345SD16—1631no—
346SD16—1631no—
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—3261no—
352D8—3261no—
353D8—3261no—
354D8—3261no—
355D8—3261no—
356D8—3261no—
357D8—1631no—
358D8—1631no—
359M16—21no—
360M16—3261no—
361M16—3261no—
362M16—3261no—
363M16—1631no—
364M16C22×D16311yescharacteristic
365M16C22×D16311yescharacteristic
366M16C22×D16311yescharacteristic
367C2×C8—21no—
368C2×C8—3261no—
369C2×C8—3261no—
370C2×C8—3261no—
371C2×C8—1631no—
372C2×C8C22×D16311yescharacteristic
373C22⋊C4—6521no—
374C42—1631no—
375C23—21no—
376C23—21no—
377C23—3261no—
378C23—6521no—
379C23—6521no—
380C23—6521no—
381Q8—3261no—
382Q8—3261no—
383Q8—6521no—
384Q8—1631no—
385Q8—1631no—
386Q8—1631no—
387Q8D4×D16311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—13042no—
394D4—3261no—
395D4—3261no—
396D4—3261no—
397D4—3261no—
398D4—3261no—
399D4—6521no—
400D4—6521no—
401D4—6521no—
402D4—6521no—
403D4—6521no—
404D4—6521no—
405D4—6521no—
406D4—1631no—
407D4—1631no—
408D4—1631no—
409D4D4×D16311yescharacteristic
410C2×C4—21no—
411C2×C4—3261no—
412C2×C4—3261no—
413C2×C4—3261no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—6521no—
417C2×C4—6521no—
418C2×C4—6521no—
419C2×C4—6521no—
420C2×C4—6521no—
421C2×C4C23×D16311yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—3261no—
426C8—1631no—
427C8—1631no—
428C8D4×D16311yescharacteristic
429C8D4×D16311yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—13041no—
436C22—26081no—
437C22—26081no—
438C22—26081no—
439C22—3261no—
440C22—3261no—
441C22—3261no—
442C22—6521no—
443C22#1910478111yescharacteristic
444C4—21no—
445C4—3261no—
446C4—3261no—
447C4—3261no—
448C4—6521no—
449C4—6521no—
450C4C2×D4×D16311yescharacteristic
451C4C2×D4×D16311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—13041no—
457C2—6521no—
458C2—6521no—
459C2—6521no—
460C2#2378229911yescharacteristic
subgroupG/H
centerC2#23782299
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#472×C163C2
socleC2×C163D42
hypercenter#472D163
hypocenterC163#3288
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #23782477C2semidirect
2. #23782475C2semidirect
3. #23782469C2semidirect
4. #23782466C2semidirect
5. #23782053C2semidirect
6. #23781934C2semidirect
7. #23781929C2semidirect
8. #23781927C2semidirect
9. #23781845C2semidirect
10. #23781836C2semidirect
11. #23781835C2semidirect
12. #23781831C2semidirect
13. #23781786C2semidirect
14. #23781785C2semidirect
15. #472×C163C2semidirect
16. #19104782C22semidirect
17. #19104781C22semidirect
19. #19104780C22semidirect
21. #19104768C22semidirect
22. #19104768C22semidirect
26. #19104763C22semidirect
29. #19104761C22semidirect
41. C2×#14579688C22semidirect
42. C2×#14579687C22semidirect
44. C2×#14579684C22semidirect
47. #19104743C22semidirect
48. #19104743C22semidirect
49. #19104741C22semidirect
50. #19104731C22semidirect
51. #19104722C22semidirect
52. #19104670C22semidirect
53. #19104662C22semidirect
54. #19104661C22semidirect
55. #19104661C22semidirect
56. #19104661C22semidirect
57. #19104653C22semidirect
58. #19104651C22semidirect
59. #19104645C22semidirect
60. AGL(1,ℤ/8)×C163C22semidirect
61. AGL(1,ℤ/8)×C163C22semidirect
62. C2×D8×C163C22semidirect
63. #131×C163C22semidirect
64. #108×C163C22semidirect
65. #100×C163C22semidirect
66. #100×C163C22semidirect
95. Dic652D4semidirect
96. #14579689D4semidirect
103. #14579688D4semidirect
104. #14579688D4semidirect
109. #14579687D4semidirect
110. #14579686D4semidirect
113. D1304D4semidirect
118. #14579684D4semidirect
230. C163#3288semidirect
238. #472D163semidirect
265. AGL(1,ℤ/8)C2×D163semidirect
266. AGL(1,ℤ/8)C2×D163semidirect
280. C2×D8C2×D163semidirect
284. #131C2×D163semidirect
288. #108C2×D163semidirect
295. #100C2×D163semidirect
296. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824772
#237824752
#237824692
#237824662
#237820532
#237819342
#237819292
#237819272
#237818452
#237818362
#237818352
#237818312
#237817862
#237817852
#472×C1632
#3288163