Finite Group Databasebeta

← #35218170

Subgroups — #35218170

Order 20864.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×D4×D163C211yesmaximal, characteristic, complemented
2C2×#19104731C211yesmaximal, characteristic, complemented
3C2×#19104706C211yesmaximal, characteristic, complemented
4C2×#19104703C214yesmaximal, complemented
5#23781573C214yesmaximal, complemented
6#23781568C211yesmaximal, characteristic, complemented
7#23781567C211yesmaximal, characteristic, complemented
8C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
9C2×#127×C163C211yesmaximal, characteristic, complemented
10C24×D163C2212yes—
11C22×D652C2211yescharacteristic
12C22×C22⋊D163C2212yes—
13C22×C22⋊D163C2212yes—
14C22×C22⋊D163C2214yescomplemented
15C2×D4×D163—24no—
16C2×D4×D163—24no—
17C2×D4×D163—24no—
18C22×C4×D163C2211yescharacteristic
19#19104731—24no—
20#19104706—24no—
21#19104703—216no—
22C2×C4⋊Dic163C2211yescharacteristic, complemented
23C2×C22⋊Dic163C2214yescomplemented
24C2×C22⋊Dic163C2212yescomplemented
25C2×(C2×D163)⋊C4C2212yes—
26C2×(C2×D163)⋊C4C2214yescomplemented
27C2×C4×Dic163C2211yescharacteristic, complemented
28C163⋊#95C2212yescomplemented
29C22×D4×C163C2211yescharacteristic
30C22×D4×C163C2211yescharacteristic, complemented
31C22×D4×C163C2214yescomplemented
32#127×C163—24no—
33C2×C42×C163C2211yescharacteristic, complemented
34C23×D163—24no—
35C23×D163—24no—
36C23×D163—28no—
37C23×D163—24no—
38C23×D163—44no—
39C23×D163C2312yes—
40C23×D163C2311yescharacteristic
41C2×D652—22no—
42C2×D652—22no—
43C2×D652D414yescomplemented
44C2×C22⋊D163—24no—
45C2×C22⋊D163—28no—
46C2×C22⋊D163—24no—
47C2×C22⋊D163—28no—
48C2×C22⋊D163—24no—
49C2×C22⋊D163—28no—
50C2×C22⋊D163—24no—
51C2×C22⋊D163—44no—
52D4×D163—232no—
53C22×Dic163C2312yes—
54C22×Dic163C2311yescharacteristic
55C22×Dic163C2312yes—
56C2×C4×D163—22no—
57C2×C4×D163—22no—
58C2×C4×D163D414yescomplemented
59C4⋊Dic163—22no—
60C22⋊Dic163—24no—
61C22⋊Dic163—28no—
62(C2×D163)⋊C4—28no—
63(C2×D163)⋊C4D418yescomplemented
64C4×Dic163—22no—
65C24×C163C2312yes—
66C24×C163C2312yes—
67C2×D4×C163—28no—
68C2×D4×C163—22no—
69C2×D4×C163—22no—
70C2×D4×C163—22no—
71C2×D4×C163—22no—
72C2×D4×C163—22no—
73C2×D4×C163—28no—
74C2×D4×C163—44no—
75C2×D4×C163D414yescomplemented
76C22×C4×C163C2312yes—
77C22×C4×C163C2311yescharacteristic
78C42×C163—22no—
79C22×D163—21no—
80C22×D163—21no—
81C22×D163—22no—
82C22×D163—216no—
83C22×D163—24no—
84C22×D163—22no—
85C22×D163—22no—
86C22×D163—24no—
87C22×D163—416no—
88C22×D163—416no—
89C22×D163—48no—
90C22×D163—48no—
91C22×D163C2×D414yes—
92C22×D163C2×D414yes—
93C22×D163C2×D412yes—
94D652—28no—
95C22⋊D163—216no—
96C22⋊D163—416no—
97C22⋊D163—48no—
98C2×Dic163—22no—
99C2×Dic163—21no—
100C2×Dic163—21no—
101C2×Dic163—22no—
102C2×Dic163—24no—
103C2×Dic163—22no—
104C2×Dic163C2×D412yes—
105C2×Dic163C2×D414yescomplemented
106C4×D163—28no—
107C23×C163—24no—
108C23×C163—22no—
109C23×C163—22no—
110C23×C163—24no—
111C23×C163—22no—
112C23×C163—22no—
113C23×C163—22no—
114C23×C163—24no—
115C23×C163—24no—
116C23×C163C2×D414yes—
117C23×C163C2411yescharacteristic
118D4×C163—28no—
119D4×C163—416no—
120D4×C163—44no—
121C2×C4×C163—22no—
122C2×C4×C163—22no—
123C2×C4×C163C2×D412yes—
124C2×C4×C163C2×D412yes—
125C2×C4×C163C2×D412yes—
126C2×C4×C163C2×D414yescomplemented
127C22×C163—24no—
128C22×C163—28no—
129C22×C163—42no—
130C22×C163—42no—
131C22×C163—44no—
132C22×C163—42no—
133C22×C163—42no—
134C22×C163—42no—
135C22×C163—44no—
136C22×C163—44no—
137C22×C163C22×D411yescharacteristic
138C22×C163C22×D411yescharacteristic
139C22×C163C22×D412yes—
140C22×C163C22×D411yescharacteristic
141C22×C163H(5,𝔽2)12yes—
142C2×D163—22no—
143C2×D163—22no—
144C2×D163—24no—
145C2×D163—24no—
146C2×D163—24no—
147C2×D163—28no—
148C2×D163—432no—
149C2×D163—42no—
150C2×D163—42no—
151C2×D163—44no—
152C2×D163#12714yescomplemented
153C4×C163—24no—
154C4×C163#12014yescomplemented
155Dic163—22no—
156Dic163—24no—
157Dic163—42no—
158C2×C163—44no—
159C2×C163—82no—
160C2×C163C2×#12011yescharacteristic
161C2×C163C2×#12711yescharacteristic
162C2×C163#53511yescharacteristic
163C2×C163D4212yes—
164C2×C163D4212yes—
165D163—24no—
166D163—44no—
167C163#243311yescharacteristic, complemented, Sylow
168#2433—1631nomaximal, Sylow
169C23×D4—1631no—
170C2×#127—1631no—
171C2×#127D16311yescharacteristic, complemented
172C2×#121—1631no—
173C2×#120—1634no—
174#392—1634no—
175#390—1631no—
176#390—1631no—
177C4×C22⋊C4—1631no—
178C25—1632no—
179C22×D4—3264no—
180C22×D4—3264no—
181C22×D4—3264no—
182C22×D4—1632no—
183C22×D4—1632no—
184C22×D4—1634no—
185C22×D4—1631no—
186C22×D4C2×D16311yescharacteristic
187C22×D4C2×D16311yescharacteristic, complemented
188C22×D4C2×D16314yescomplemented
189C23×C4—1631no—
190#127—24no—
191#127—3264no—
192#121—3264no—
193#120—32616no—
194C2×C4⋊C4—1631no—
195C2×C22⋊C4—1634no—
196C2×C22⋊C4—1632no—
197C2×C22⋊C4—1634no—
198C2×C22⋊C4—1632no—
199C2×C42—1631no—
200C2×C42C2×D16311yescharacteristic, complemented
201#95—1632no—
202C24—3264no—
203C24—3264no—
204C24—3268no—
205C24—3264no—
206C24—6524no—
207C24—1632no—
208C24—1631no—
209C24C22×D16312yes—
210C24C22×D16312yes—
211C2×D4—28no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—28no—
217C2×D4—22no—
218C2×D4—44no—
219C2×D4—32632no—
220C2×D4—3264no—
221C2×D4—3264no—
222C2×D4—3268no—
223C2×D4—3264no—
224C2×D4—3268no—
225C2×D4—3264no—
226C2×D4—3268no—
227C2×D4—3262no—
228C2×D4—3262no—
229C2×D4—6524no—
230C2×D4—1634no—
231C2×D4C22⋊D16314yescomplemented
232C22×C4—3262no—
233C22×C4—3262no—
234C22×C4—1632no—
235C22×C4—1631no—
236C22×C4—1632no—
237C22×C4—1634no—
238C22×C4C22×D16312yes—
239C22×C4C22×D16311yescharacteristic
240C4⋊C4—3262no—
241C22⋊C4—3268no—
242C22⋊C4—3264no—
243C22⋊C4—3268no—
244C22⋊C4—1638no—
245C42—22no—
246C42—3262no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—22no—
252C23—22no—
253C23—24no—
254C23—22no—
255C23—24no—
256C23—32616no—
257C23—3261no—
258C23—3261no—
259C23—3262no—
260C23—3264no—
261C23—3262no—
262C23—3262no—
263C23—3264no—
264C23—65216no—
265C23—65216no—
266C23—6528no—
267C23—6528no—
268C23—1634no—
269C23—1634no—
270C23—1632no—
271C23C2×C22⋊D16314yes—
272C23C23×D16311yescharacteristic
273D4—28no—
274D4—416no—
275D4—44no—
276D4—32616no—
277D4—3268no—
278D4—65216no—
279D4—6528no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—3262no—
283C2×C4—3261no—
284C2×C4—3261no—
285C2×C4—3262no—
286C2×C4—3268no—
287C2×C4—3264no—
288C2×C4—3262no—
289C2×C4—1634no—
290C2×C4—1632no—
291C2×C4D4×D16312yes—
292C2×C4D4×D16312yes—
293C2×C4D4×D16314yescomplemented
294C2×C4C2×C22⋊D16312yes—
295C22—24no—
296C22—28no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—3262no—
306C22—3262no—
307C22—3264no—
308C22—3264no—
309C22—3264no—
310C22—3268no—
311C22—65232no—
312C22—6522no—
313C22—6522no—
314C22—6524no—
315C22—1634no—
316C22C2×D4×D16311yescharacteristic
317C22C2×D4×D16311yescharacteristic
318C22C2×D4×D16312yes—
319C22C22×C22⋊D16311yescharacteristic
320C22#1910478012yes—
321C4—24no—
322C4—3262no—
323C4—3264no—
324C4—6522no—
325C4#1910470314yescomplemented
326C2—44no—
327C2—82no—
328C2—3264no—
329C2—6524no—
330C2C2×#1910470311yescharacteristic
331C2#127×D16311yescharacteristic
332C2#2378222811yescharacteristic
333C2D4×C22⋊D16312yes—
334C2#2378230012yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC2×#127×C163C2
hypercenterC2×#127D163
hypocenterC163#2433
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D163C2semidirect
2. C2×#19104731C2semidirect
3. C2×#19104706C2semidirect
4. C2×#19104703C2semidirect
5. #23781573C2semidirect
6. #23781568C2semidirect
7. #23781567C2semidirect
8. C4×(C2×D163)⋊C4C2semidirect
9. C2×#127×C163C2semidirect
14. C22×C22⋊D163C22semidirect
22. C2×C4⋊Dic163C22semidirect
23. C2×C22⋊Dic163C22semidirect
24. C2×C22⋊Dic163C22semidirect
26. C2×(C2×D163)⋊C4C22semidirect
27. C2×C4×Dic163C22semidirect
28. C163⋊#95C22semidirect
30. C22×D4×C163C22semidirect
31. C22×D4×C163C22semidirect
33. C2×C42×C163C22semidirect
43. C2×D652D4semidirect
58. C2×C4×D163D4semidirect
63. (C2×D163)⋊C4D4semidirect
75. C2×D4×C163D4semidirect
105. C2×Dic163C2×D4semidirect
126. C2×C4×C163C2×D4semidirect
152. C2×D163#127semidirect
154. C4×C163#120semidirect
167. C163#2433semidirect
171. C2×#127D163semidirect
187. C22×D4C2×D163semidirect
188. C22×D4C2×D163semidirect
200. C2×C42C2×D163semidirect
231. C2×D4C22⋊D163semidirect
293. C2×C4D4×D163semidirect
325. C4#19104703semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1632
C2×#191047312
C2×#191047062
C2×#191047032
#237815732
#237815682
#237815672
C4×(C2×D163)⋊C42
C2×#127×C1632
#2433163