Finite Group Databasebeta

← #35223802

Subgroups — #35223802

Order 20864.

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

normal subgroups only

This group has 784 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×#19104781C211yesmaximal, characteristic, complemented
2C2×#19104761C211yesmaximal, characteristic, complemented
3C22×#14579684C211yesmaximal, characteristic, complemented
4#23781808C214yesmaximal, complemented
5#23781798C214yesmaximal, complemented
6#23781669C211yesmaximal, characteristic, complemented
7#23781659C211yesmaximal, characteristic, complemented
8#23781632C211yesmaximal, characteristic, complemented
9#535×C163C211yesmaximal, characteristic, complemented
10#19104781—24no—
11#19104781—24no—
12C2×#14579700—21no—
13C2×#14579699C2211yescharacteristic
14C2×#14579699C2212yescomplemented
15C22×D652C2211yescharacteristic
16C22×D652C2212yescomplemented
17C2×D4×D163—22no—
18C2×D4×D163—21no—
19#19104761—24no—
20C2×#14579687C2212yescomplemented
21C2×#14579684—24no—
22C2×#14579684C2214yescomplemented
23C2×#14579684C2212yescomplemented
24C2×#14579683C2211yescharacteristic, complemented
25C22×C163⋊C8C2211yescharacteristic, complemented
26#19104658C2212yescomplemented
27#19104655C2212yescomplemented
28#19104655C2214yescomplemented
29C22⋊(C163⋊C8)C2214yescomplemented
30C2×#41×C163C2211yescharacteristic
31C22×D4×C163C2211yescharacteristic, complemented
32#127×C163—21no—
33#124×C163—21no—
34#121×C163C2214yescomplemented
35#120×C163—22no—
36#117×C163C2211yescharacteristic, complemented
37C23×D163—21no—
38C23×D163—22no—
39#14579700—28no—
40#14579699—24no—
41#14579699—24no—
42#14579699—22no—
43#14579699D414yescomplemented
44C2×Dic326C2311yescharacteristic
45C2×D652—24no—
46C2×D652—24no—
47C2×D652—21no—
48C2×D652—22no—
49C2×D652—22no—
50C2×D652D414yescomplemented
51C2×D652C2312yes—
52C2×D652C2311yescharacteristic
53C2×C22⋊D163—21no—
54C2×C22⋊D163—22no—
55D4×D163—216no—
56D4×D163—28no—
57C2×C4×D163—21no—
58C2×C4×D163—22no—
59#14579687—24no—
60#14579684—28no—
61#14579684—24no—
62#14579683—22no—
63C2×C163⋊C8—22no—
64C2×C163⋊C8C2312yes—
65C2×C163⋊C8C2312yes—
66C24×C163—21no—
67#41×C163—22no—
68#41×C163—22no—
69C2×Q8×C163D411yescharacteristic
70C2×D4×C163—24no—
71C2×D4×C163—22no—
72C2×D4×C163—22no—
73C2×D4×C163—21no—
74C2×D4×C163D412yes—
75C2×D4×C163D411yescharacteristic
76C2×D4×C163C2312yes—
77C22×C4×C163C2312yes—
78C22×C4×C163C2311yescharacteristic
79C4⋊C4×C163C2312yes—
80C22⋊C4×C163—21no—
81C22⋊C4×C163—24no—
82C42×C163—21no—
83C22×D163—21no—
84C22×D163—22no—
85C22×D163—28no—
86C22×D163—24no—
87C22×D163—82no—
88C22×D163—84no—
89Dic326—21no—
90Dic326C2×D412yes—
91D652—28no—
92D652—24no—
93D652—22no—
94D652—21no—
95D652C2×D412yes—
96D652C2×D414yes—
97C22⋊D163—28no—
98C22⋊D163—24no—
99C2×Dic163—21no—
100C4×D163—28no—
101C4×D163—24no—
102C163⋊C8—22no—
103C163⋊C8—22no—
104C23×C163—22no—
105C23×C163—42no—
106C23×C163—82no—
107C23×C163C2×D412yes—
108C23×C163C2×D411yescharacteristic
109Q8×C163—22no—
110D4×C163—24no—
111D4×C163—22no—
112D4×C163—22no—
113D4×C163—22no—
114D4×C163—42no—
115D4×C163—84no—
116C2×C4×C163—22no—
117C2×C4×C163—22no—
118C2×C4×C163—22no—
119C2×C4×C163C2×D412yes—
120C2×C4×C163C2×D412yes—
121C2×C4×C163C2×D412yes—
122C2×C4×C163C2×D411yescharacteristic
123C2×C4×C163C2411yescharacteristic
124C22×C163—22no—
125C22×C163—22no—
126C22×C163—24no—
127C22×C163—42no—
128C22×C163—44no—
129C22×C163—84no—
130C22×C163#12012yes—
131C22×C163C22×D411yescharacteristic
132C2×D163—24no—
133C2×D163—22no—
134C2×D163—44no—
135C2×D163—42no—
136C2×D163—84no—
137C2×D163—88no—
138C4×C163—22no—
139C4×C163—42no—
140C4×C163#12012yes—
141C4×C163C22×D411yescharacteristic
142C4×C163C22×D411yescharacteristic
143Dic163—22no—
144C2×C163—22no—
145C2×C163—42no—
146C2×C163—82no—
147C2×C163C2×#12011yescharacteristic
148C2×C163#57612yes—
149D163—44no—
150D163—42no—
151C163#300811yescharacteristic, complemented, Sylow
152#3008—1631nomaximal, Sylow
153C2×H(5,𝔽2)—1631no—
154C2×AGL(1,ℤ/8)—1631no—
155C22×D8—1631no—
156#535D16311yescharacteristic, complemented
157#449—1634no—
158#447—1634no—
159#418—1631no—
160#416—1631no—
161#408—1631no—
162H(5,𝔽2)—3264no—
163H(5,𝔽2)—3264no—
164C2×#41—3261no—
165C2×#41—1632no—
166C2×#41—1631no—
167C2×#41C2×D16311yescharacteristic
168C22×D4—3262no—
169C22×D4—3261no—
170C22×D4—1632no—
171C22×D4—1631no—
172C22×D4C2×D16311yescharacteristic, complemented
173AGL(1,ℤ/8)—3264no—
174C2×SD16—1632no—
175C2×D8—3264no—
176C2×D8—1634no—
177C2×D8—1632no—
178C2×M16—1631no—
179C22×C8—1631no—
180#127—21no—
181#124—21no—
182#121C2×D16314yescomplemented
183#120—22no—
184#117C2×D16311yescharacteristic, complemented
185#103—1632no—
186#102—1632no—
187#102—1634no—
188C22⋊C8—1634no—
189C24—21no—
190C24—3262no—
191C24—3261no—
192#41—22no—
193#41—22no—
194#41—3264no—
195#41—3264no—
196#41—3268no—
197#41—3262no—
198#41—1634no—
199C2×Q8—1631no—
200C2×Q8C22⋊D16311yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—3264no—
206C2×D4—3264no—
207C2×D4—32616no—
208C2×D4—3268no—
209C2×D4—3262no—
210C2×D4—3261no—
211C2×D4—3262no—
212C2×D4—3262no—
213C2×D4—3261no—
214C2×D4—1634no—
215C2×D4—1631no—
216C2×D4—1632no—
217C2×D4C22⋊D16312yes—
218C2×D4C22⋊D16311yescharacteristic
219C2×D4C22×D16312yes—
220C22×C4—3262no—
221C22×C4—3261no—
222C22×C4C22×D16312yes—
223C22×C4C22×D16311yescharacteristic
224SD16—3264no—
225D8—3268no—
226D8—3264no—
227M16—3262no—
228C2×C8—3262no—
229C2×C8—1632no—
230C2×C8—1632no—
231C4⋊C4C22×D16312yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—13042no—
239C23—13044no—
240C23—3268no—
241C23—3264no—
242C23—3262no—
243C23—3261no—
244C23C2×C22⋊D16312yes—
245C23C2×C22⋊D16311yescharacteristic
246Q8—22no—
247Q8—3261no—
248Q8—1632no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—3268no—
256D4—3268no—
257D4—3264no—
258D4—3262no—
259D4—3261no—
260D4—3264no—
261D4—1632no—
262D4—1634no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3268no—
267C2×C4—3261no—
268C2×C4—3264no—
269C2×C4D4×D16312yes—
270C2×C4D4×D16312yes—
271C2×C4C2×C22⋊D16312yes—
272C2×C4C2×C22⋊D16311yescharacteristic
273C2×C4C23×D16311yescharacteristic
274C8—3262no—
275C8—3262no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—13044no—
283C22—13048no—
284C22—3264no—
285C22—3262no—
286C22—6524no—
287C22—6522no—
288C22#1910470312yes—
289C22C22×C22⋊D16311yescharacteristic
290C4—22no—
291C4—42no—
292C4—3262no—
293C4#1910470312yes—
294C4C2×D4×D16311yescharacteristic
295C4C2×D4×D16311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—6524no—
300C2—6522no—
301C2C2×#1910470311yescharacteristic
302C2#2378246812yes—
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#535×C163C2
socleC22×C163C22×D4
hypercenter#535D163
hypocenterC163#3008
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19104781C2semidirect
2. C2×#19104761C2semidirect
3. C22×#14579684C2semidirect
4. #23781808C2semidirect
5. #23781798C2semidirect
6. #23781669C2semidirect
7. #23781659C2semidirect
8. #23781632C2semidirect
9. #535×C163C2semidirect
14. C2×#14579699C22semidirect
16. C22×D652C22semidirect
20. C2×#14579687C22semidirect
22. C2×#14579684C22semidirect
23. C2×#14579684C22semidirect
24. C2×#14579683C22semidirect
25. C22×C163⋊C8C22semidirect
26. #19104658C22semidirect
27. #19104655C22semidirect
28. #19104655C22semidirect
29. C22⋊(C163⋊C8)C22semidirect
31. C22×D4×C163C22semidirect
34. #121×C163C22semidirect
36. #117×C163C22semidirect
43. #14579699D4semidirect
50. C2×D652D4semidirect
151. C163#3008semidirect
156. #535D163semidirect
172. C22×D4C2×D163semidirect
182. #121C2×D163semidirect
184. #117C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047812
C2×#191047612
C22×#145796842
#237818082
#237817982
#237816692
#237816592
#237816322
#535×C1632
#3008163