Finite Group Databasebeta

← #35217893

Subgroups — #35217893

Order 20864.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C23×D652C211yesmaximal, characteristic, complemented
2C2×#19104710C212yesmaximal, complemented
3C2×#19104705C212yesmaximal, complemented
4C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
5#23781583C212yesmaximal, complemented
6#23781575C212yesmaximal, complemented
7#23781566C212yesmaximal, complemented
8#23781520C212yesmaximal, complemented
9#380×C163C211yesmaximal, characteristic, complemented
10C24×D163C2212yes—
11C22×D652—28no—
12C22×D652—24no—
13C22×D652C2212yes—
14C22×D652C2212yes—
15C22×D652C2214yescomplemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic, complemented
18#19104710—28no—
19#19104705—28no—
20C4×C22⋊D163—24no—
21C2×Dic163⋊C4C2211yescharacteristic, complemented
22C2×C22⋊Dic163C2211yescharacteristic
23C2×(C2×D163)⋊C4C2214yescomplemented
24C2×(C2×D163)⋊C4C2214yescomplemented
25C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
26C2×(C2×D163)⋊C4C2214yescomplemented
27C2×C4×Dic163C2211yescharacteristic, complemented
28C23×C4×C163C2211yescharacteristic
29C2×C4⋊C4×C163C2212yescomplemented
30C2×C22⋊C4×C163C2212yes—
31#95×C163C2212yescomplemented
32C23×D163—28no—
33C23×D163—24no—
34C23×D163—24no—
35C23×D163—24no—
36C23×D163—44no—
37C23×D163C2314yes—
38C23×D163C2311yescharacteristic
39C2×D652—232no—
40C2×D652—28no—
41C2×D652—28no—
42C2×D652—28no—
43C2×D652—24no—
44C2×D652—24no—
45C2×D652—24no—
46C2×D652—44no—
47C2×D652D418yescomplemented
48C2×C22⋊D163—22no—
49C2×C22⋊D163—22no—
50C2×C22⋊D163D414yescomplemented
51C22×Dic163C2311yescharacteristic
52C22×Dic163C2311yescharacteristic
53C22×Dic163C2311yescharacteristic
54C2×C4×D163—22no—
55C2×C4×D163—24no—
56Dic163⋊C4—22no—
57C22⋊Dic163—22no—
58(C2×D163)⋊C4—28no—
59(C2×D163)⋊C4—28no—
60(C2×D163)⋊C4—22no—
61(C2×D163)⋊C4—44no—
62C4×Dic163—22no—
63C24×C163C2311yescharacteristic
64C22×C4×C163—24no—
65C22×C4×C163D414yescomplemented
66C22×C4×C163C2314yes—
67C22×C4×C163C2311yescharacteristic
68C22×C4×C163C2311yescharacteristic
69C4⋊C4×C163—24no—
70C22⋊C4×C163—24no—
71C22×D163—216no—
72C22×D163—21no—
73C22×D163—24no—
74C22×D163—22no—
75C22×D163—24no—
76C22×D163—24no—
77C22×D163—21no—
78C22×D163—24no—
79C22×D163—24no—
80C22×D163—22no—
81C22×D163—24no—
82C22×D163—416no—
83C22×D163—416no—
84C22×D163—48no—
85C22×D163—48no—
86C22×D163C2×D418yes—
87C22×D163C2×D412yescomplemented
88D652—216no—
89D652—216no—
90D652—416no—
91C22⋊D163—28no—
92C2×Dic163—21no—
93C2×Dic163—21no—
94C2×Dic163—21no—
95C2×Dic163—22no—
96C2×Dic163—21no—
97C2×Dic163—22no—
98C2×Dic163C2×D412yescomplemented
99C4×D163—44no—
100C23×C163—22no—
101C23×C163—22no—
102C23×C163C2×D412yes—
103C23×C163C2×D412yes—
104C23×C163#4112yes—
105C23×C163C2411yescharacteristic
106C2×C4×C163—22no—
107C2×C4×C163—28no—
108C2×C4×C163—24no—
109C2×C4×C163—24no—
110C2×C4×C163—24no—
111C2×C4×C163C2×D414yes—
112C2×C4×C163C2×D412yes—
113C2×C4×C163C2×D412yes—
114C22×C163—22no—
115C22×C163—22no—
116C22×C163—24no—
117C22×C163—24no—
118C22×C163#12114yescomplemented
119C22×C163C22×D412yes—
120C22×C163C22×D411yescharacteristic
121C22×C163C22×D411yescharacteristic
122C22×C163C2×#4111yescharacteristic
123C22×C163H(5,𝔽2)12yes—
124C2×D163—28no—
125C2×D163—22no—
126C2×D163—24no—
127C2×D163—28no—
128C2×D163—28no—
129C2×D163—432no—
130C2×D163—41no—
131C2×D163—44no—
132C2×D163—41no—
133C2×D163—44no—
134C2×D163—44no—
135C2×D163—42no—
136C2×D163—44no—
137C4×C163—22no—
138C4×C163—44no—
139C4×C163#12014yescomplemented
140Dic163—22no—
141Dic163—41no—
142Dic163—41no—
143C2×C163—24no—
144C2×C163C2×#12011yescharacteristic
145C2×C163C2×#12111yescharacteristic
146C2×C163#53511yescharacteristic
147C2×C163D4212yes—
148C2×C163#54612yes—
149D163—42no—
150D163—48no—
151C163#240811yescharacteristic, complemented, Sylow
152#2408—1631nomaximal, Sylow
153C23×D4—1631no—
154C2×#121—1632no—
155C2×#120—1632no—
156C2×C4×D4—1631no—
157#394—1632no—
158#392—1632no—
159#390—1632no—
160#380D16311yescharacteristic, complemented
161#379—1632no—
162C25—1632no—
163C22×D4—3268no—
164C22×D4—3264no—
165C22×D4—1631no—
166C22×D4—1632no—
167C22×D4—1634no—
168C22×D4—1632no—
169C23×C4—1631no—
170C23×C4C2×D16311yescharacteristic
171#121—3268no—
172#120—3268no—
173C4×D4—3264no—
174C2×C4⋊C4—1631no—
175C2×C4⋊C4C2×D16312yescomplemented
176C2×C22⋊C4—1631no—
177C2×C22⋊C4—1631no—
178C2×C22⋊C4—1634no—
179C2×C22⋊C4—1634no—
180C2×C22⋊C4—1634no—
181C2×C22⋊C4C2×D16312yes—
182C2×C42—1631no—
183#95C2×D16312yescomplemented
184C24—3268no—
185C24—3264no—
186C24—3264no—
187C24—3264no—
188C24—6524no—
189C24—1634no—
190C24—1631no—
191C24C22×D16311yescharacteristic
192C2×D4—32632no—
193C2×D4—3268no—
194C2×D4—3268no—
195C2×D4—3268no—
196C2×D4—3262no—
197C2×D4—3262no—
198C2×D4—3264no—
199C2×D4—3264no—
200C2×D4—3264no—
201C2×D4—6524no—
202C2×D4—1634no—
203C2×D4—1638no—
204C22×C4—24no—
205C22×C4—3264no—
206C22×C4—3262no—
207C22×C4—1631no—
208C22×C4—1631no—
209C22×C4—1631no—
210C22×C4D65214yescomplemented
211C22×C4C22×D16314yes—
212C22×C4C22×D16311yescharacteristic
213C22×C4C22×D16311yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—3262no—
216C22⋊C4—24no—
217C22⋊C4—3268no—
218C22⋊C4—3268no—
219C22⋊C4—3262no—
220C22⋊C4—3262no—
221C22⋊C4—6524no—
222C42—3262no—
223C23—22no—
224C23—22no—
225C23—32616no—
226C23—3264no—
227C23—3262no—
228C23—3264no—
229C23—3261no—
230C23—3264no—
231C23—3261no—
232C23—3264no—
233C23—3264no—
234C23—3262no—
235C23—3264no—
236C23—65216no—
237C23—65216no—
238C23—6528no—
239C23—6528no—
240C23—1638no—
241C23—1632no—
242C23D4×D16312yes—
243C23C2×D65212yes—
244C23#1457970012yes—
245C23C23×D16311yescharacteristic
246D4—32616no—
247D4—3268no—
248D4—32616no—
249D4—65216no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—3261no—
256C2×C4—3261no—
257C2×C4—3262no—
258C2×C4—3261no—
259C2×C4—3261no—
260C2×C4—3262no—
261C2×C4—6524no—
262C2×C4—1632no—
263C2×C4D4×D16314yes—
264C2×C4C2×D65212yes—
265C2×C4C2×D65212yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—3268no—
271C22—3262no—
272C22—3264no—
273C22—3268no—
274C22—3268no—
275C22—65232no—
276C22—6521no—
277C22—6524no—
278C22—6524no—
279C22—6521no—
280C22—6524no—
281C22—6524no—
282C22—6522no—
283C22#1910471014yescomplemented
284C22C2×D4×D16312yes—
285C22C2×D4×D16311yescharacteristic
286C22C22×D65211yescharacteristic
287C22C2×#1457970011yescharacteristic
288C22#1910478112yes—
289C4—22no—
290C4—44no—
291C4—3262no—
292C4—6521no—
293C4—6521no—
294C4#1910470514yescomplemented
295C2—24no—
296C2—6528no—
297C2—6522no—
298C2C2×#1910470511yescharacteristic
299C2C2×#1910471011yescharacteristic
300C2#2378223411yescharacteristic
301C2#2378229912yes—
302C2#2378230612yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#380×C163C2
hypercenter#380D163
hypocenterC163#2408
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D652C2semidirect
2. C2×#19104710C2semidirect
3. C2×#19104705C2semidirect
4. C2×C4×C22⋊D163C2semidirect
5. #23781583C2semidirect
6. #23781575C2semidirect
7. #23781566C2semidirect
8. #23781520C2semidirect
9. #380×C163C2semidirect
15. C22×D652C22semidirect
17. C22×C4×D163C22semidirect
21. C2×Dic163⋊C4C22semidirect
23. C2×(C2×D163)⋊C4C22semidirect
24. C2×(C2×D163)⋊C4C22semidirect
25. C2×(C2×D163)⋊C4C22semidirect
26. C2×(C2×D163)⋊C4C22semidirect
27. C2×C4×Dic163C22semidirect
29. C2×C4⋊C4×C163C22semidirect
31. #95×C163C22semidirect
47. C2×D652D4semidirect
50. C2×C22⋊D163D4semidirect
65. C22×C4×C163D4semidirect
87. C22×D163C2×D4semidirect
98. C2×Dic163C2×D4semidirect
118. C22×C163#121semidirect
139. C4×C163#120semidirect
151. C163#2408semidirect
160. #380D163semidirect
175. C2×C4⋊C4C2×D163semidirect
183. #95C2×D163semidirect
210. C22×C4D652semidirect
283. C22#19104710semidirect
294. C4#19104705semidirect

Maximal subgroups

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

subgroupindex
C23×D6522
C2×#191047102
C2×#191047052
C2×C4×C22⋊D1632
#237815832
#237815752
#237815662
#237815202
#380×C1632
#2408163