Finite Group Databasebeta

← #35225019

Subgroups — #35225019

Order 20864.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104761C211yesmaximal, characteristic, complemented
2C2×#19104760C211yesmaximal, characteristic, complemented
3C2×#19104755C211yesmaximal, characteristic, complemented
4C22×#14579684C211yesmaximal, characteristic, complemented
5#23782234C211yesmaximal, characteristic, complemented
6#23782055C212yesmaximal, complemented
7#23782052C212yesmaximal, complemented
8#23782047C212yesmaximal, complemented
9#23782041C212yesmaximal, complemented
10#23781621C211yesmaximal, characteristic, complemented
11#535×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×D652C2211yescharacteristic
14#19104761—24no—
15#19104760—24no—
16#19104755—24no—
17C2×#14579689C2211yescharacteristic, complemented
18C2×#14579687C2211yescharacteristic, complemented
19C2×#14579687C2212yescomplemented
20C2×#14579686C2211yescharacteristic, complemented
21C2×#14579686C2212yescomplemented
22C2×#14579684—24no—
23C2×#14579684C2212yes—
24C2×#14579684C2212yes—
25C2×#14579684C2211yescharacteristic, complemented
26C2×#14579684C2212yescomplemented
27C2×#14579684C2212yescomplemented
28C2×#14579683C2211yescharacteristic
29C22×C163⋊C8C2211yescharacteristic
30#19104732C2212yescomplemented
31#19104723C2212yescomplemented
32#19104710—22no—
33#19104705—22no—
34#19104646C2212yescomplemented
35C4×C163⋊C8C2212yescomplemented
36C2×#41×C163C2211yescharacteristic
37C22×D4×C163C2211yescharacteristic
38#127×C163C2212yescomplemented
39#124×C163C2212yescomplemented
40#121×C163—22no—
41#120×C163—22no—
42#117×C163C2211yescharacteristic
43C23×D163—21no—
44#14579699—22no—
45#14579699—22no—
46C2×Dic326C2311yescharacteristic
47C2×D652—22no—
48C2×D652—22no—
49C2×D652—21no—
50C2×D652—24no—
51C2×D652C2312yes—
52C2×D652C2311yescharacteristic
53C2×C22⋊D163—21no—
54C2×C4×D163—21no—
55#14579689—22no—
56#14579687—22no—
57#14579687—24no—
58#14579686—22no—
59#14579686—24no—
60#14579684—24no—
61#14579684—24no—
62#14579684—22no—
63#14579684—24no—
64#14579684—24no—
65#14579683D414yescomplemented
66C2×C163⋊C8D414yescomplemented
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2312yes—
69C2×C163⋊C8C2311yescharacteristic
70(C2×D163)⋊C4—24no—
71C24×C163—21no—
72#41×C163—22no—
73#41×C163—22no—
74C2×Q8×C163C2311yescharacteristic
75C2×D4×C163—21no—
76C2×D4×C163—22no—
77C2×D4×C163—22no—
78C2×D4×C163—21no—
79C2×D4×C163—24no—
80C2×D4×C163C2312yes—
81C2×D4×C163C2311yescharacteristic
82C22×C4×C163—21no—
83C22×C4×C163C2311yescharacteristic
84C4⋊C4×C163D412yes—
85C22⋊C4×C163—24no—
86C22⋊C4×C163D412yes—
87C42×C163C2312yes—
88C22×D163—22no—
89C22×D163—21no—
90C22×D163—42no—
91C22×D163—82no—
92Dic326—21no—
93Dic326—21no—
94D652—22no—
95D652—21no—
96D652—22no—
97D652—21no—
98D652—42no—
99D652—84no—
100C22⋊D163—42no—
101C2×Dic163—21no—
102C4×D163—42no—
103C163⋊C8C2×D412yes—
104C163⋊C8C2×D412yes—
105C163⋊C8C2×D414yes—
106C23×C163—22no—
107C23×C163—21no—
108C23×C163—42no—
109C23×C163—82no—
110C23×C163C2×D411yescharacteristic
111Q8×C163—21no—
112Q8×C163—21no—
113D4×C163—22no—
114D4×C163—22no—
115D4×C163—21no—
116D4×C163—21no—
117D4×C163—42no—
118D4×C163—42no—
119D4×C163—84no—
120C2×C4×C163—21no—
121C2×C4×C163—42no—
122C2×C4×C163C2×D412yes—
123C2×C4×C163C2×D412yes—
124C2×C4×C163C2×D414yes—
125C2×C4×C163C2×D411yescharacteristic
126C2×C4×C163C2411yescharacteristic
127C22×C163—22no—
128C22×C163—41no—
129C22×C163—42no—
130C22×C163—42no—
131C22×C163—44no—
132C22×C163—84no—
133C22×C163#12712yes—
134C22×C163C22×D411yescharacteristic
135C2×D163—41no—
136C2×D163—42no—
137C2×D163—42no—
138C2×D163—44no—
139C2×D163—84no—
140C4×C163—24no—
141C4×C163—41no—
142C4×C163#12712yes—
143C4×C163C22×D411yescharacteristic
144C4×C163C22×D411yescharacteristic
145Dic163—41no—
146C2×C163—22no—
147C2×C163—81no—
148C2×C163—82no—
149C2×C163C2×#12711yescharacteristic
150C2×C163#57612yes—
151D163—81no—
152D163—82no—
153C163#315511yescharacteristic, complemented, Sylow
154#3155—1631nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1631no—
156C2×AGL(1,ℤ/8)—1631no—
157C2×#135—1631no—
158C22×D8—1631no—
159#535—1631no—
160#535D16311yescharacteristic, complemented
161#496—1632no—
162#496—1632no—
163#495—1632no—
164#493—1632no—
165#405—1631no—
166C2×#41—1631no—
167C2×#41C2×D16311yescharacteristic
168C22×D4—1631no—
169C22×D4C2×D16311yescharacteristic
170AGL(1,ℤ/8)—3264no—
171AGL(1,ℤ/8)—3264no—
172#135—3264no—
173C2×Q16—1631no—
174C2×SD16—1631no—
175C2×SD16—1632no—
176C2×SD16—1631no—
177C2×SD16—1632no—
178C2×D8—3264no—
179C2×D8—1632no—
180C2×D8—1632no—
181C2×D8—1631no—
182C2×D8—1632no—
183C2×D8—1632no—
184C2×M16—1631no—
185C22×C8—1631no—
186#127—1632no—
187#127C2×D16312yescomplemented
188#124—1632no—
189#124C2×D16312yescomplemented
190#121—22no—
191#121—3262no—
192#120—22no—
193#120—3262no—
194#117C2×D16311yescharacteristic
195#97—1632no—
196C4×C8—1632no—
197C24—21no—
198C24—3261no—
199#41—22no—
200#41—22no—
201#41—3262no—
202#41—3262no—
203C2×Q8—1631no—
204C2×Q8C22×D16311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—3261no—
211C2×D4—3262no—
212C2×D4—3262no—
213C2×D4—3264no—
214C2×D4—3261no—
215C2×D4—1632no—
216C2×D4—1631no—
217C2×D4C22×D16312yes—
218C2×D4C22×D16311yescharacteristic
219C22×C4—21no—
220C22×C4—3261no—
221C22×C4C22×D16311yescharacteristic
222Q16—3262no—
223SD16—3262no—
224SD16—3264no—
225SD16—3262no—
226SD16—3264no—
227D8—3264no—
228D8—3264no—
229D8—3262no—
230D8—3264no—
231D8—3264no—
232M16—1634no—
233C2×C8—1634no—
234C2×C8—1631no—
235C2×C8—1632no—
236C2×C8—1631no—
237C4⋊C4C22⋊D16312yes—
238C22⋊C4—24no—
239C22⋊C4—3264no—
240C22⋊C4C22⋊D16312yes—
241C42C22×D16312yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—13042no—
247C23—3262no—
248C23—3261no—
249C23—6522no—
250C23C2×C22⋊D16311yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3261no—
254Q8—3261no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—13044no—
263D4—3262no—
264D4—3261no—
265D4—3262no—
266D4—3261no—
267D4—6522no—
268D4—6522no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—3261no—
272C2×C4—6522no—
273C2×C4D4×D16312yes—
274C2×C4D4×D16312yes—
275C2×C4C2×C22⋊D16314yes—
276C2×C4C2×C22⋊D16311yescharacteristic
277C2×C4C23×D16311yescharacteristic
278C8—1632no—
279C8—1632no—
280C8—1634no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—13044no—
288C22—6521no—
289C22—6522no—
290C22—6522no—
291C22—6524no—
292C22#1910473112yes—
293C22C22×C22⋊D16311yescharacteristic
294C4—24no—
295C4—41no—
296C4—6521no—
297C4#1910473112yes—
298C4C2×D4×D16311yescharacteristic
299C4C2×D4×D16311yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—13042no—
304C2—13041no—
305C2C2×#1910473111yescharacteristic
306C2#2378246812yes—
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#535×C163C2
socleC22×C163C22×D4
hypercenter#535D163
hypocenterC163#3155
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19104761C2semidirect
2. C2×#19104760C2semidirect
3. C2×#19104755C2semidirect
4. C22×#14579684C2semidirect
5. #23782234C2semidirect
6. #23782055C2semidirect
7. #23782052C2semidirect
8. #23782047C2semidirect
9. #23782041C2semidirect
10. #23781621C2semidirect
11. #535×C163C2semidirect
17. C2×#14579689C22semidirect
18. C2×#14579687C22semidirect
19. C2×#14579687C22semidirect
20. C2×#14579686C22semidirect
21. C2×#14579686C22semidirect
25. C2×#14579684C22semidirect
26. C2×#14579684C22semidirect
27. C2×#14579684C22semidirect
30. #19104732C22semidirect
31. #19104723C22semidirect
34. #19104646C22semidirect
35. C4×C163⋊C8C22semidirect
38. #127×C163C22semidirect
39. #124×C163C22semidirect
65. #14579683D4semidirect
66. C2×C163⋊C8D4semidirect
153. C163#3155semidirect
160. #535D163semidirect
187. #127C2×D163semidirect
189. #124C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047612
C2×#191047602
C2×#191047552
C22×#145796842
#237822342
#237820552
#237820522
#237820472
#237820412
#237816212
#535×C1632
#3155163