Finite Group Databasebeta

← #35226623

Subgroups — #35226623

Order 20864.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782373C211yesmaximal, characteristic, complemented
2#23782306C211yesmaximal, characteristic
3#23782026C211yesmaximal, characteristic, complemented
4#23781998C211yesmaximal, characteristic, complemented
5#23781980C211yesmaximal, characteristic, complemented
6#23781944C211yesmaximal, characteristic, complemented
7#23781914C211yesmaximal, characteristic, complemented
8#23781889C211yesmaximal, characteristic, complemented
9#23781826C211yesmaximal, characteristic, complemented
10#23781815C211yesmaximal, characteristic, complemented
11#23781768C211yesmaximal, characteristic, complemented
12#23781736C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15#467×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579682C2211yescharacteristic, complemented
21#19104734C2211yescharacteristic, complemented
22#19104725—21no—
23#19104724C2211yescharacteristic
24#19104720—21no—
25#19104715—21no—
26#19104712—21no—
27#19104711C2211yescharacteristic, complemented
28#19104710—21no—
29#19104710C2211yescharacteristic
30#19104705—21no—
31Q8×Dic163C2211yescharacteristic, complemented
32#19104697C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34(C22⋊D163)⋊C4—21no—
35C2×C4⋊Dic163C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic, complemented
37C22⋊C4×D163—21no—
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C2×Q16×C163C2211yescharacteristic, complemented
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#103×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579700—22no—
61#14579700—41no—
62C2×Dic326—21no—
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—21no—
66C2×D652—21no—
67C2×D652—41no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163D411yescharacteristic
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163C2311yescharacteristic
76#14579686—22no—
77#14579686—41no—
78#14579682—41no—
79C2×C163⋊C8C2311yescharacteristic
80C2×C163⋊C8C2311yescharacteristic
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89C22⋊Dic163D411yescharacteristic
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
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101Q16×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117Dic326—81no—
118D652—21no—
119D652—21no—
120D652—21no—
121D652—42no—
122D652—42no—
123D652—81no—
124D652#4112yes—
125C22⋊D163—22no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—81no—
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C2×Dic163C2×D411yescharacteristic
132C4×D163—21no—
133C4×D163—21no—
134C4×D163—42no—
135C4×D163—81no—
136C4×D163—81no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C23×C163C2×D411yescharacteristic
140Q8×C163—21no—
141Q8×C163—21no—
142Q8×C163—21no—
143Q8×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163#13712yes—
157C22×C163C22×D411yescharacteristic
158C2×D163—21no—
159C2×D163—22no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163C2×#4111yescharacteristic
174C4×C163H(5,𝔽2)11yescharacteristic
175Dic163—21no—
176Dic163—21no—
177Dic163—21no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—22no—
181C2×C163#54611yescharacteristic
182C2×C163C2×#13711yescharacteristic
183C2×C163#57611yescharacteristic
184D163—41no—
185D163—42no—
186D163—81no—
187C163#331411yescharacteristic, complemented, Sylow
188#3314—1631nomaximal, Sylow
189#554—1631no—
190#546—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#474—1631no—
195#468—1631no—
196#467D16311yescharacteristic, complemented
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102—1631no—
204C2×#41—1631no—
205C22×D4—1631no—
206C2×Q16C2×D16311yescharacteristic, complemented
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×M16—1631no—
210C22×C8C2×D16311yescharacteristic
211C4⋊Q8—1631no—
212#125—3261no—
213#124—1631no—
214#123—3261no—
215C22⋊Q8—3261no—
216C22⋊Q8—3261no—
217C22⋊Q8—1631no—
218C22⋊Q8C2×D16311yescharacteristic
219C22⋊Q8C2×D16311yescharacteristic
220#121—3261no—
221#121—1631no—
222#120—3261no—
223C4×Q8—1631no—
224C4×D4—3261no—
225C4×D4—1631no—
226C4×D4—1631no—
227C2×C4⋊C4—1631no—
228C2×C4⋊C4—1631no—
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107—1631no—
232#107C2×D16311yescharacteristic, complemented
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103C2×D16311yescharacteristic, complemented
237#103C2×D16311yescharacteristic, complemented
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247C24—3261no—
248#41—3262no—
249#41—6521no—
250C2×Q8—3261no—
251C2×Q8C22×D16311yescharacteristic
252C2×Q8C22×D16311yescharacteristic
253C2×D4—3261no—
254C2×D4—3262no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4—1631no—
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266C22×C4C22×D16311yescharacteristic
267Q16—41no—
268SD16—3262no—
269SD16—6521no—
270M16—6521no—
271C2×C8—41no—
272C2×C8—1631no—
273C2×C8—1631no—
274C2×C8C22×D16311yescharacteristic
275C2×C8C22×D16311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4C22×D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—1631no—
296C22⋊C4—1631no—
297C42—3261no—
298C42—1631no—
299C23—13041no—
300C23—3261no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23—1631no—
305C23D4×D16311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—13041no—
311D4—13041no—
312D4—3262no—
313D4—3261no—
314D4—3261no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4—1632no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—13041no—
326C2×C4—13041no—
327C2×C4—13041no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6522no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4—1631no—
336C2×C4D4×D16311yescharacteristic
337C2×C4C23×D16311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3261no—
341C8—3261no—
342C22—21no—
343C22—21no—
344C22—13044no—
345C22—3261no—
346C22—3261no—
347C22—3261no—
348C22—3262no—
349C22—6521no—
350C22—6521no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22#1910476712yes—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3261no—
362C4—3261no—
363C4—3261no—
364C4—6521no—
365C4—6521no—
366C4#41×D16311yescharacteristic
367C4#1910478011yescharacteristic
368C2—22no—
369C2—13041no—
370C2—6522no—
371C2—6521no—
372C2#2378230411yescharacteristic
373C2C2×#1910476711yescharacteristic
374C2#2378246711yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#467×C163C2
socleC22×C163C22×D4
prime radical#23782306C2
hypercenter#467D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23782373C2semidirect
3. #23782026C2semidirect
4. #23781998C2semidirect
5. #23781980C2semidirect
6. #23781944C2semidirect
7. #23781914C2semidirect
8. #23781889C2semidirect
9. #23781826C2semidirect
10. #23781815C2semidirect
11. #23781768C2semidirect
12. #23781736C2semidirect
13. #23781665C2semidirect
14. C2×#19104657C2semidirect
15. #467×C163C2semidirect
20. C2×#14579682C22semidirect
21. #19104734C22semidirect
27. #19104711C22semidirect
31. Q8×Dic163C22semidirect
36. C4⋊C4×D163C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. Dic163⋊C8C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
52. C2×Q16×C163C22semidirect
56. #107×C163C22semidirect
57. #103×C163C22semidirect
58. #103×C163C22semidirect
187. C163#3314semidirect
196. #467D163semidirect
206. C2×Q16C2×D163semidirect
232. #107C2×D163semidirect
236. #103C2×D163semidirect
237. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823732
#237823062
#237820262
#237819982
#237819802
#237819442
#237819142
#237818892
#237818262
#237818152
#237817682
#237817362
#237816652
C2×#191046572
#467×C1632
#3314163