Finite Group Databasebeta

← #35223778

Subgroups — #35223778

Order 20864.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19104766C211yesmaximal, characteristic, complemented
2C2×#19104761C211yesmaximal, characteristic, complemented
3#23782176C211yesmaximal, characteristic, complemented
4#23781827C214yesmaximal, complemented
5#23781810C214yesmaximal, complemented
6#23781674C211yesmaximal, characteristic, complemented
7#23781670C211yesmaximal, characteristic, complemented
8#23781629C211yesmaximal, characteristic, complemented
9C22×#41×C163C211yesmaximal, characteristic, complemented
10C2×#14579699C2211yescharacteristic
11#19104766—24no—
12#19104761—24no—
13C2×#14579689C2212yescomplemented
14C2×#14579687C2212yescomplemented
15C2×#14579686C2212yescomplemented
16C2×#14579684C2212yescomplemented
17C2×#14579683C2211yescharacteristic
18C2×#14579683C2211yescharacteristic
19#19104718—21no—
20#19104714C2212yescomplemented
21#19104709C2212yescomplemented
22#19104704—21no—
23C4×C22⋊D163—22no—
24#19104694C2211yescharacteristic
25#19104659C2214yescomplemented
26#19104656C2214yescomplemented
27C22⋊(C163⋊C8)C2214yescomplemented
28C2×#41×C163—24no—
29C2×#41×C163—24no—
30C2×#41×C163—22no—
31C2×#41×C163C2214yes—
32C22×Q8×C163C2211yescharacteristic
33C22×D4×C163—21no—
34C22×D4×C163C2211yescharacteristic
35C23×C4×C163—21no—
36C23×C4×C163C2211yescharacteristic
37#14579699—22no—
38#14579699—22no—
39C2×Dic326C2311yescharacteristic
40C2×D652C2311yescharacteristic
41C2×C22⋊D163—21no—
42C2×C22⋊D163—22no—
43C2×C4×D163—21no—
44#14579689—24no—
45#14579687—24no—
46#14579686—24no—
47#14579684—24no—
48#14579683—22no—
49#14579683—22no—
50C2×C163⋊C8C2312yes—
51C2×C163⋊C8C2312yes—
52C4⋊Dic163C2312yes—
53Dic163⋊C4—22no—
54C22⋊Dic163—21no—
55C22⋊Dic163—22no—
56(C2×D163)⋊C4—22no—
57C4×Dic163—21no—
58C24×C163—21no—
59C24×C163D411yescharacteristic
60#41×C163—216no—
61#41×C163—28no—
62#41×C163—28no—
63C2×Q8×C163—22no—
64C2×Q8×C163—21no—
65C2×Q8×C163D414yescomplemented
66C2×Q8×C163C2312yes—
67C2×D4×C163—28no—
68C2×D4×C163—24no—
69C2×D4×C163—21no—
70C2×D4×C163—22no—
71C2×D4×C163D414yescomplemented
72C2×D4×C163C2312yes—
73C22×C4×C163—28no—
74C22×C4×C163—21no—
75C22×C4×C163—24no—
76C22×C4×C163—22no—
77C22×C4×C163—22no—
78C22×C4×C163D411yescharacteristic
79C22×C4×C163D412yes—
80C22×C4×C163C2312yes—
81C22×C4×C163C2311yescharacteristic
82C22×D163—21no—
83Dic326—21no—
84Dic326—21no—
85D652—21no—
86D652—21no—
87C22⋊D163—44no—
88C22⋊D163—42no—
89C2×Dic163—22no—
90C2×Dic163—21no—
91C4×D163—42no—
92C163⋊C8—22no—
93C163⋊C8—22no—
94C23×C163—22no—
95C23×C163—24no—
96C23×C163—22no—
97C23×C163—42no—
98C23×C163—44no—
99C23×C163C2×D412yes—
100C23×C163C2×D411yescharacteristic
101Q8×C163—24no—
102Q8×C163—22no—
103Q8×C163C2×D414yes—
104D4×C163—216no—
105D4×C163—24no—
106D4×C163—22no—
107D4×C163C2×D414yes—
108C2×C4×C163—216no—
109C2×C4×C163—22no—
110C2×C4×C163—24no—
111C2×C4×C163—22no—
112C2×C4×C163—24no—
113C2×C4×C163—22no—
114C2×C4×C163—22no—
115C2×C4×C163C2×D412yes—
116C2×C4×C163C2×D411yescharacteristic
117C2×C4×C163C2×D412yes—
118C2×C4×C163C2×D412yes—
119C2×C4×C163C2411yescharacteristic
120C22×C163—22no—
121C22×C163—22no—
122C22×C163—24no—
123C22×C163—22no—
124C22×C163—22no—
125C22×C163—44no—
126C22×C163—48no—
127C22×C163—44no—
128C22×C163#12012yes—
129C22×C163C22×D411yescharacteristic
130C2×D163—41no—
131C2×D163—42no—
132C4×C163—22no—
133C4×C163—24no—
134C4×C163#12012yes—
135C4×C163C22×D411yescharacteristic
136C4×C163C22×D411yescharacteristic
137Dic163—41no—
138Dic163—42no—
139C2×C163—22no—
140C2×C163—42no—
141C2×C163—44no—
142C2×C163C2×#12011yescharacteristic
143C2×C163#57512yes—
144D163—81no—
145C163#300511yescharacteristic, complemented, Sylow
146#3005—1631nomaximal, Sylow
147C22×#41D16311yescharacteristic, complemented
148C2×#137—1631no—
149C2×AGL(1,ℤ/8)—1631no—
150#525—1631no—
151#452—1634no—
152#449—1634no—
153#419—1631no—
154#418—1631no—
155#407—1631no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1631no—
160C2×#41C2×D16314yes—
161C22×Q8C2×D16311yescharacteristic
162C22×D4—21no—
163C22×D4C2×D16311yescharacteristic
164C23×C4—21no—
165C23×C4C2×D16311yescharacteristic
166#137—3264no—
167AGL(1,ℤ/8)—3264no—
168C2×Q16—1632no—
169C2×SD16—1632no—
170C2×SD16—1632no—
171C2×D8—1632no—
172C2×M16—1631no—
173C2×M16—1631no—
174#123—3261no—
175C22⋊Q8—1632no—
176#121—1632no—
177#120—3261no—
178C4×D4—3262no—
179#117—1631no—
180#103—1634no—
181#102—1634no—
182C22⋊C8—1634no—
183C24—21no—
184C24C22⋊D16311yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—3262no—
189#41—3262no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1631no—
193C2×Q8C22⋊D16314yescomplemented
194C2×Q8C22×D16312yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—3262no—
200C2×D4—3261no—
201C2×D4—1631no—
202C2×D4C22⋊D16314yescomplemented
203C2×D4C22×D16312yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—3261no—
210C22×C4C22⋊D16312yes—
211C22×C4C22⋊D16311yescharacteristic
212C22×C4C22×D16312yes—
213C22×C4C22×D16311yescharacteristic
214Q16—3264no—
215SD16—3264no—
216SD16—3264no—
217D8—3264no—
218M16—3262no—
219M16—3262no—
220C2×C8—1632no—
221C2×C8—1632no—
222C4⋊C4—3262no—
223C4⋊C4—1632no—
224C22⋊C4—3261no—
225C22⋊C4—3262no—
226C22⋊C4—3262no—
227C42—3261no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—42no—
232C23—44no—
233C23—3261no—
234C23C2×C22⋊D16312yes—
235C23C2×C22⋊D16311yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—3261no—
239Q8—3261no—
240Q8C2×C22⋊D16314yes—
241D4—216no—
242D4—22no—
243D4—24no—
244D4—3261no—
245D4—3261no—
246D4—6524no—
247D4—6522no—
248D4C2×C22⋊D16314yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—3262no—
257C2×C4—3261no—
258C2×C4—6522no—
259C2×C4C2×C22⋊D16312yes—
260C2×C4C2×C22⋊D16312yes—
261C2×C4C2×C22⋊D16312yes—
262C2×C4C2×C22⋊D16311yescharacteristic
263C2×C4C23×D16311yescharacteristic
264C8—3262no—
265C8—3262no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—6521no—
275C22—6522no—
276C22#1910470412yes—
277C22C22×C22⋊D16311yescharacteristic
278C4—22no—
279C4—24no—
280C4—6521no—
281C4—6522no—
282C4#1910470412yes—
283C4C22×C22⋊D16311yescharacteristic
284C4C22×C22⋊D16311yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—13041no—
289C2C2×#1910470411yescharacteristic
290C2#2378246312yes—
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC22×#41×C163C2
socleC22×C163C22×D4
hypercenterC22×#41D163
hypocenterC163#3005
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19104766C2semidirect
2. C2×#19104761C2semidirect
3. #23782176C2semidirect
4. #23781827C2semidirect
5. #23781810C2semidirect
6. #23781674C2semidirect
7. #23781670C2semidirect
8. #23781629C2semidirect
9. C22×#41×C163C2semidirect
13. C2×#14579689C22semidirect
14. C2×#14579687C22semidirect
15. C2×#14579686C22semidirect
16. C2×#14579684C22semidirect
20. #19104714C22semidirect
21. #19104709C22semidirect
25. #19104659C22semidirect
26. #19104656C22semidirect
27. C22⋊(C163⋊C8)C22semidirect
65. C2×Q8×C163D4semidirect
71. C2×D4×C163D4semidirect
145. C163#3005semidirect
147. C22×#41D163semidirect
193. C2×Q8C22⋊D163semidirect
202. C2×D4C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047662
C2×#191047612
#237821762
#237818272
#237818102
#237816742
#237816702
#237816292
C22×#41×C1632
#3005163