Finite Group Databasebeta

← #35223812

Subgroups — #35223812

Order 20864.

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

normal subgroups only

This group has 752 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
2C22×#14579686C211yesmaximal, characteristic, complemented
3#23782239C211yesmaximal, characteristic, complemented
4#23781826C214yesmaximal, complemented
5#23781814C214yesmaximal, complemented
6#23781671C211yesmaximal, characteristic, complemented
7#23781661C211yesmaximal, characteristic, complemented
8#23781633C211yesmaximal, characteristic, complemented
9C2×#137×C163C211yesmaximal, characteristic, complemented
10#19104781—24no—
11#19104781—24no—
12C2×#14579700C2212yes—
13C2×#14579699—21no—
14C2×#14579699C2211yescharacteristic
15C22×D652—21no—
16C22×D652C2211yescharacteristic
17C2×D4×D163—22no—
18C2×D4×D163C2212yes—
19C2×#14579686—24no—
20C2×#14579686C2214yescomplemented
21C22×C163⋊C8C2211yescharacteristic, complemented
22#19104733—21no—
23#19104724—21no—
24#19104718—22no—
25#19104711C2214yescomplemented
26#19104694C2211yescharacteristic, complemented
27#19104659C2212yescomplemented
28#19104657C2214yescomplemented
29#19104656C2212yescomplemented
30(C2×D163)⋊C8C2214yescomplemented
31C2×#41×C163C2211yescharacteristic
32C22×Q8×C163C2211yescharacteristic, complemented
33#137×C163—24no—
34C2×Q16×C163C2212yescomplemented
35C2×SD16×C163C2212yescomplemented
36C2×M16×C163C2211yescharacteristic, complemented
37C23×D163—22no—
38C23×D163—21no—
39#14579700—24no—
40#14579700—24no—
41#14579699—28no—
42#14579699—22no—
43#14579699—22no—
44C2×Dic326D411yescharacteristic
45C2×D652—28no—
46C2×D652—22no—
47C2×D652—21no—
48C2×D652—22no—
49C2×D652D411yescharacteristic
50C2×D652D414yes—
51C2×D652C2312yes—
52C2×C22⋊D163—22no—
53C2×C22⋊D163D412yes—
54D4×D163—216no—
55D4×D163—24no—
56D4×D163—24no—
57C2×C4×D163—22no—
58C2×C4×D163C2312yes—
59#14579686—28no—
60C2×C163⋊C8—22no—
61C2×C163⋊C8C2312yes—
62C4⋊Dic163C2312yes—
63Dic163⋊C4—24no—
64C22⋊Dic163—21no—
65(C2×D163)⋊C4—24no—
66C4×Dic163—21no—
67#41×C163—22no—
68#41×C163D414yes—
69C2×Q8×C163—21no—
70C2×Q8×C163—22no—
71C2×Q8×C163—22no—
72C2×Q8×C163C2312yes—
73C2×Q8×C163C2311yescharacteristic
74C2×D4×C163—21no—
75C2×D4×C163C2311yescharacteristic
76C22×C4×C163—21no—
77C22×C4×C163—21no—
78C22×C4×C163C2311yescharacteristic
79Q16×C163—24no—
80SD16×C163—24no—
81M16×C163—22no—
82C2×C8×C163C2312yes—
83C22×D163—22no—
84C22×D163—28no—
85C22×D163—24no—
86C22×D163—84no—
87C22×D163—82no—
88C22×D163C2×D412yes—
89Dic326—22no—
90D652—28no—
91D652—22no—
92D652—24no—
93D652—22no—
94D652C2×D414yes—
95C22⋊D163—28no—
96C22⋊D163—24no—
97C2×Dic163—22no—
98C2×Dic163C2×D412yes—
99C4×D163—28no—
100C4×D163—22no—
101C4×D163—22no—
102C163⋊C8—22no—
103C23×C163—21no—
104C23×C163C2×D411yescharacteristic
105Q8×C163—21no—
106Q8×C163—22no—
107Q8×C163—22no—
108Q8×C163—42no—
109Q8×C163C2×D412yes—
110D4×C163—24no—
111D4×C163—21no—
112D4×C163C2×D412yes—
113C2×C4×C163—21no—
114C2×C4×C163—24no—
115C2×C4×C163—22no—
116C2×C4×C163—42no—
117C2×C4×C163C2×D411yescharacteristic
118C2×C4×C163C2×D412yes—
119C2×C4×C163C2×D412yes—
120C2×C4×C163C2411yescharacteristic
121C8×C163—22no—
122C22×C163—22no—
123C22×C163—22no—
124C22×C163—42no—
125C22×C163#12012yes—
126C22×C163C22×D411yescharacteristic
127C2×D163—22no—
128C2×D163—24no—
129C2×D163—24no—
130C2×D163—44no—
131C2×D163—84no—
132C2×D163—88no—
133C4×C163—22no—
134C4×C163—42no—
135C4×C163#12012yes—
136C4×C163C22×D411yescharacteristic
137C4×C163C22×D411yescharacteristic
138Dic163—22no—
139Dic163—42no—
140C2×C163—22no—
141C2×C163—42no—
142C2×C163C2×#12011yescharacteristic
143C2×C163#57712yes—
144D163—42no—
145D163—44no—
146C163#300911yescharacteristic, complemented, Sylow
147#3009—1631nomaximal, Sylow
148C2×H(5,𝔽2)—1631no—
149C2×#137D16311yescharacteristic, complemented
150C22×SD16—1631no—
151#536—1631no—
152#452—1634no—
153#450—1634no—
154#418—1631no—
155#416—1631no—
156#408—1631no—
157H(5,𝔽2)—3264no—
158H(5,𝔽2)—3264no—
159C2×#41—3261no—
160C2×#41—1632no—
161C2×#41—1631no—
162C2×#41C2×D16311yescharacteristic
163C22×Q8C2×D16311yescharacteristic, complemented
164C22×D4—3261no—
165C22×D4—3262no—
166C22×D4—1632no—
167C22×D4—1631no—
168#137—24no—
169C2×Q16C2×D16312yescomplemented
170C2×SD16—3264no—
171C2×SD16—1634no—
172C2×SD16C2×D16312yescomplemented
173C2×M16C2×D16311yescharacteristic, complemented
174C22×C8—1631no—
175C4⋊Q8—3261no—
176#124—3261no—
177#123—3262no—
178C22⋊Q8—1634no—
179#117—1631no—
180#103—1632no—
181#102—1632no—
182#102—1634no—
183C22⋊C8—1634no—
184C24—3262no—
185C24—3261no—
186#41—22no—
187#41—3268no—
188#41—3264no—
189#41—3264no—
190#41—3262no—
191#41—3262no—
192#41C22⋊D16314yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1631no—
197C2×Q8C22×D16312yes—
198C2×Q8C22×D16311yescharacteristic
199C2×D4—21no—
200C2×D4—3268no—
201C2×D4—3262no—
202C2×D4—32616no—
203C2×D4—3264no—
204C2×D4—3264no—
205C2×D4—3262no—
206C2×D4—3261no—
207C2×D4—3262no—
208C2×D4—1632no—
209C2×D4—1631no—
210C2×D4—1634no—
211C2×D4—1632no—
212C2×D4C22×D16311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3262no—
216C22×C4—1632no—
217C22×C4C22×D16311yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—3268no—
221M16—22no—
222C2×C8—3262no—
223C2×C8—1632no—
224C2×C8C22×D16312yes—
225C4⋊C4—3264no—
226C4⋊C4—1632no—
227C22⋊C4—3264no—
228C22⋊C4—3261no—
229C42—3261no—
230C23—21no—
231C23—13042no—
232C23—13044no—
233C23—3268no—
234C23—3264no—
235C23—3262no—
236C23—1632no—
237C23D4×D16311yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—3262no—
243Q8C2×C22⋊D16312yes—
244D4—24no—
245D4—21no—
246D4—3268no—
247D4—3268no—
248D4—3264no—
249D4—3262no—
250D4—3262no—
251D4—3264no—
252D4—1634no—
253D4C2×C22⋊D16312yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—3268no—
259C2×C4—3262no—
260C2×C4—3262no—
261C2×C4—3262no—
262C2×C4—1632no—
263C2×C4D4×D16312yes—
264C2×C4D4×D16311yescharacteristic
265C2×C4C2×C22⋊D16312yes—
266C2×C4C23×D16311yescharacteristic
267C8—22no—
268C8—3262no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—13044no—
273C22—13048no—
274C22—3262no—
275C22—3264no—
276C22—3264no—
277C22—6524no—
278C22#1910470312yes—
279C22C2×D4×D16311yescharacteristic
280C4—22no—
281C4—42no—
282C4—3262no—
283C4—6522no—
284C4#1910470312yes—
285C4C2×D4×D16311yescharacteristic
286C4C22×C22⋊D16311yescharacteristic
287C2—22no—
288C2—42no—
289C2—6522no—
290C2—6524no—
291C2C2×#1910470311yescharacteristic
292C2#2378247412yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#137×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104781C2
hypercenterC2×#137D163
hypocenterC163#3009
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14579686C2semidirect
3. #23782239C2semidirect
4. #23781826C2semidirect
5. #23781814C2semidirect
6. #23781671C2semidirect
7. #23781661C2semidirect
8. #23781633C2semidirect
9. C2×#137×C163C2semidirect
20. C2×#14579686C22semidirect
21. C22×C163⋊C8C22semidirect
25. #19104711C22semidirect
26. #19104694C22semidirect
27. #19104659C22semidirect
28. #19104657C22semidirect
29. #19104656C22semidirect
30. (C2×D163)⋊C8C22semidirect
32. C22×Q8×C163C22semidirect
34. C2×Q16×C163C22semidirect
35. C2×SD16×C163C22semidirect
36. C2×M16×C163C22semidirect
146. C163#3009semidirect
149. C2×#137D163semidirect
163. C22×Q8C2×D163semidirect
169. C2×Q16C2×D163semidirect
172. C2×SD16C2×D163semidirect
173. C2×M16C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047812
C22×#145796862
#237822392
#237818262
#237818142
#237816712
#237816612
#237816332
C2×#137×C1632
#3009163