Finite Group Databasebeta

← #35226569

Subgroups — #35226569

Order 20864.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782389C211yesmaximal, characteristic, complemented
2#23782310C211yesmaximal, characteristic
3#23782027C211yesmaximal, characteristic, complemented
4#23782009C211yesmaximal, characteristic, complemented
5#23781991C211yesmaximal, characteristic, complemented
6#23781924C211yesmaximal, characteristic, complemented
7#23781895C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781828C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781765C211yesmaximal, characteristic, complemented
12#23781735C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic, complemented
14#23781658C211yesmaximal, characteristic, complemented
15#476×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×Dic652C2211yescharacteristic, complemented
19C2×#14579688C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579682C2211yescharacteristic
22C2×C8×D163C2211yescharacteristic, complemented
23#19104730—21no—
24#19104730—21no—
25#19104724—21no—
26#19104723C2211yescharacteristic
27#19104719—21no—
28#19104715C2211yescharacteristic
29#19104715C2211yescharacteristic
30#19104715C2211yescharacteristic, complemented
31#19104710C2211yescharacteristic
32#19104709—21no—
33#19104705—21no—
34Q8×Dic163—21no—
35C4×Dic326C2211yescharacteristic, complemented
36C4×D652C2211yescharacteristic
37#19104693C2211yescharacteristic
38#19104693C2211yescharacteristic, complemented
39C2×(C2×D163)⋊C4—21no—
40#19104668C2211yescharacteristic, complemented
41#19104660C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104646C2211yescharacteristic, complemented
52C4⋊Q8×C163C2211yescharacteristic, complemented
53C4×D4×C163C2211yescharacteristic
54#106×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60#14579698—22no—
61#14579698—41no—
62C2×Dic326C2311yescharacteristic
63C2×Dic326C2311yescharacteristic
64C2×D652—21no—
65C2×D652D411yescharacteristic
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69D4×D163—22no—
70D4×D163—41no—
71C22×Dic163—21no—
72C2×C4×D163C2311yescharacteristic
73C2×C4×D163C2311yescharacteristic
74Dic652—41no—
75#14579688—41no—
76#14579687—22no—
77#14579682—41no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163—41no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163—21no—
84C4⋊Dic163D411yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163—21no—
98C4×Dic163—21no—
99C4×Dic163—21no—
100C2×Q8×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116C22×D163C2×D411yescharacteristic
117Dic326—21no—
118Dic326—21no—
119Dic326—21no—
120Dic326—21no—
121D652—21no—
122D652—21no—
123D652—41no—
124D652—81no—
125C22⋊D163—42no—
126C22⋊D163—42no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—42no—
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C4×D163—21no—
134C4×D163—21no—
135C4×D163—21no—
136C4×D163—21no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C23×C163—21no—
140Q8×C163—42no—
141D4×C163—21no—
142D4×C163#4112yes—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—81no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163—21no—
152C22×C163—22no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—84no—
166C4×C163—21no—
167C4×C163—21no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163C2×#4111yescharacteristic
172C4×C163H(5,𝔽2)11yescharacteristic
173Dic163—21no—
174Dic163—21no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178C2×C163—42no—
179C2×C163#54611yescharacteristic
180C2×C163#57511yescharacteristic
181C2×C163#57711yescharacteristic
182D163—41no—
183D163—41no—
184D163—81no—
185C163#331111yescharacteristic, complemented, Sylow
186#3311—1631nomaximal, Sylow
187#556—1631no—
188#546—1631no—
189#489—1631no—
190#485—1631no—
191#482—1631no—
192#476D16311yescharacteristic, complemented
193#470—1631no—
194#465—1631no—
195#464—1631no—
196#452—1631no—
197#450—1631no—
198#440—1631no—
199#435—1631no—
200#418—1631no—
201#416—1631no—
202C2×#41—1631no—
203C22×D4—1631no—
204C2×Q16—1631no—
205C2×SD16—1631no—
206C2×SD16—1631no—
207C2×M16—1631no—
208C22×C8—1631no—
209C4⋊Q8C2×D16311yescharacteristic, complemented
210#126—3261no—
211#126—3261no—
212#124—3261no—
213#124—1631no—
214#123—3261no—
215C22⋊Q8—1631no—
216C22⋊Q8—1631no—
217C22⋊Q8—1631no—
218#121—3261no—
219#121—1631no—
220#120—3261no—
221C4×Q8—3261no—
222C4×Q8—1631no—
223C4×D4—1631no—
224C4×D4C2×D16311yescharacteristic
225#117—1631no—
226#117—1631no—
227C2×C22⋊C4—3261no—
228#107—1631no—
229#106C2×D16311yescharacteristic, complemented
230#106C2×D16311yescharacteristic, complemented
231C4⋊C8C2×D16311yescharacteristic, complemented
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103—1631no—
236#102—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102C2×D16311yescharacteristic
241#102C2×D16311yescharacteristic
242C22⋊C8—1631no—
243C22⋊C8—1631no—
244#97—1631no—
245C24—3261no—
246#41—3262no—
247#41—6521no—
248C2×Q8—21no—
249C2×Q8—1631no—
250C2×Q8—1631no—
251C2×D4—3261no—
252C2×D4—3262no—
253C2×D4—3261no—
254C2×D4—3261no—
255C2×D4—6521no—
256C2×D4—1631no—
257C2×D4—1631no—
258C2×D4C22×D16311yescharacteristic
259C22×C4—21no—
260C22×C4—3261no—
261C22×C4—1631no—
262C22×C4—1631no—
263Q16—6521no—
264SD16—3262no—
265SD16—6521no—
266M16—6521no—
267C2×C8—6521no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—1631no—
278C4⋊C4—1631no—
279C4⋊C4C22×D16311yescharacteristic
280C4⋊C4C22×D16311yescharacteristic
281C4⋊C4C22×D16311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—1631no—
292C22⋊C4—1631no—
293C42—3261no—
294C42—3261no—
295C42—3261no—
296C42C22×D16311yescharacteristic
297C23—21no—
298C23—13041no—
299C23—3261no—
300C23—6522no—
301C23—6522no—
302C23—1631no—
303C23—1631no—
304Q8—42no—
305Q8—3261no—
306Q8—3261no—
307Q8—3261no—
308Q8—3261no—
309D4—21no—
310D4—13041no—
311D4—3261no—
312D4—3261no—
313D4—6521no—
314D4—6522no—
315D4—6522no—
316D4#1457970012yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—6522no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4D4×D16311yescharacteristic
332C2×C4D4×D16311yescharacteristic
333C2×C4C23×D16311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3261no—
337C8—3261no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—13044no—
342C22—3261no—
343C22—3261no—
344C22—3261no—
345C22—3261no—
346C22—3261no—
347C22—3261no—
348C22—6521no—
349C22—6521no—
350C22—6521no—
351C22C2×D4×D16311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3261no—
358C4—3261no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4C2×#1457970011yescharacteristic
363C4#1910478111yescharacteristic
364C2—42no—
365C2—13041no—
366C2—6521no—
367C2—6521no—
368C2#2378230611yescharacteristic
369C2#2378246111yescharacteristic
370C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#476×C163C2
socleC22×C163C22×D4
prime radical#23782310C2
hypercenter#476D163
hypocenterC163#3311
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #23782389C2semidirect
3. #23782027C2semidirect
4. #23782009C2semidirect
5. #23781991C2semidirect
6. #23781924C2semidirect
7. #23781895C2semidirect
8. #23781888C2semidirect
9. #23781828C2semidirect
10. #23781817C2semidirect
11. #23781765C2semidirect
12. #23781735C2semidirect
13. #23781668C2semidirect
14. #23781658C2semidirect
15. #476×C163C2semidirect
18. C2×Dic652C22semidirect
22. C2×C8×D163C22semidirect
30. #19104715C22semidirect
35. C4×Dic326C22semidirect
38. #19104693C22semidirect
40. #19104668C22semidirect
41. #19104660C22semidirect
42. #19104658C22semidirect
43. #19104658C22semidirect
44. #19104658C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104646C22semidirect
52. C4⋊Q8×C163C22semidirect
54. #106×C163C22semidirect
55. #106×C163C22semidirect
56. C4⋊C8×C163C22semidirect
185. C163#3311semidirect
192. #476D163semidirect
209. C4⋊Q8C2×D163semidirect
229. #106C2×D163semidirect
230. #106C2×D163semidirect
231. C4⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823892
#237823102
#237820272
#237820092
#237819912
#237819242
#237818952
#237818882
#237818282
#237818172
#237817652
#237817352
#237816682
#237816582
#476×C1632
#3311163