Finite Group Databasebeta

← #35223656

Subgroups — #35223656

Order 20864.

398 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579698C211yesmaximal, characteristic
2#23781794C211yesmaximal, characteristic, complemented
3#23781793C211yesmaximal, characteristic, complemented
4#23781739C211yesmaximal, characteristic, complemented
5#23781737C211yesmaximal, characteristic, complemented
6#23781734C211yesmaximal, characteristic, complemented
7C8×C22⋊D163C211yesmaximal, characteristic, complemented
8(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
9#23781720C211yesmaximal, characteristic, complemented
10#23781719C211yesmaximal, characteristic, complemented
11#23781715C211yesmaximal, characteristic, complemented
12#23781688C211yesmaximal, characteristic, complemented
13C2×Dic163⋊C8C211yesmaximal, characteristic, complemented
14C4⋊C8×D163C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C411yescharacteristic
17C2×#14579683C2211yescharacteristic
18C22×C163⋊C8C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21C4×Dic326C2211yescharacteristic
22#19104700C411yescharacteristic
23D4×Dic163C411yescharacteristic
24C4×C22⋊D163C2211yescharacteristic
25C4×C22⋊D163C2211yescharacteristic
26(C22⋊D163)⋊C4C411yescharacteristic
27(C22⋊D163)⋊C4C411yescharacteristic
28#19104693C411yescharacteristic
29#19104691C411yescharacteristic
30#19104691C411yescharacteristic
31C2×C4×Dic163C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C42×D163C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35Dic163⋊C8C2211yescharacteristic
36Dic163⋊C8C2211yescharacteristic
37Dic163⋊C8C2211yescharacteristic
38Dic163⋊C8C2211yescharacteristic
39Dic163⋊C8C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43C22⋊(C163⋊C8)C2211yescharacteristic
44C22⋊(C163⋊C8)C2211yescharacteristic
45(C2×D163)⋊C8C2211yescharacteristic
46(C2×D163)⋊C8C2211yescharacteristic
47#19104646C2211yescharacteristic
48#19104645C2211yescharacteristic
49#19104645C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic
56C22⋊C8×C163C2211yescharacteristic
57C22⋊C8×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59#14579698—24no—
60C2×Dic326C2×C411yescharacteristic
61C2×C22⋊D163C2×C411yescharacteristic
62C2×C22⋊D163C2×C411yescharacteristic
63C22×Dic163C2×C411yescharacteristic
64C22×Dic163C2×C411yescharacteristic
65C22×Dic163C2×C411yescharacteristic
66C22×Dic163C2×C411yescharacteristic
67C2×C4×D163C2×C411yescharacteristic
68C2×C4×D163C2×C411yescharacteristic
69C2×C4×D163C2311yescharacteristic
70#14579683—41no—
71C2×C163⋊C8—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C8×D163—41no—
77C8×D163—41no—
78C4⋊Dic163C2×C411yescharacteristic
79Dic163⋊C4C2×C411yescharacteristic
80Dic163⋊C4C2×C411yescharacteristic
81Dic163⋊C4C2×C411yescharacteristic
82Dic163⋊C4C2×C411yescharacteristic
83C22⋊Dic163C2×C411yescharacteristic
84C22⋊Dic163C2×C411yescharacteristic
85(C2×D163)⋊C4C2×C411yescharacteristic
86(C2×D163)⋊C4C2×C411yescharacteristic
87C4×Dic163C2×C411yescharacteristic
88C4×Dic163C2×C411yescharacteristic
89C4×Dic163C2×C411yescharacteristic
90C4×Dic163C2×C411yescharacteristic
91C4×Dic163C2×C411yescharacteristic
92C4×Dic163C2×C411yescharacteristic
93C4×Dic163C2311yescharacteristic
94C4×Dic163C2311yescharacteristic
95C4×Dic163C2311yescharacteristic
96C2×D4×C163C2×C411yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99M16×C163—41no—
100C2×C8×C163—41no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163C2×C411yescharacteristic
106C22⋊C4×C163C2×C411yescharacteristic
107C22⋊C4×C163C2×C411yescharacteristic
108C42×C163C2311yescharacteristic
109C22×D163C22×C411yescharacteristic
110Dic326—22no—
111C22⋊D163—22no—
112C22⋊D163M1614yescomplemented
113C2×Dic163—22no—
114C2×Dic163—22no—
115C2×Dic163—22no—
116C2×Dic163—22no—
117C2×Dic163C22×C411yescharacteristic
118C2×Dic163C22×C411yescharacteristic
119C2×Dic163C22×C411yescharacteristic
120C2×Dic163C22×C411yescharacteristic
121C2×Dic163C22×C411yescharacteristic
122C2×Dic163C22×C411yescharacteristic
123C2×Dic163C22×C411yescharacteristic
124C4×D163—22no—
125C4×D163—21no—
126C4×D163—21no—
127C4×D163—22no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163C22×C411yescharacteristic
133C23×C163C22×C411yescharacteristic
134D4×C163—22no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163C22×C411yescharacteristic
140C2×C4×C163C22×C411yescharacteristic
141C2×C4×C163C22×C411yescharacteristic
142C2×C4×C163C22×C411yescharacteristic
143C2×C4×C163C2411yescharacteristic
144C8×C163—21no—
145C8×C163—21no—
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163C2×M1612yes—
154C22×C163C23×C411yescharacteristic
155C2×D163—21no—
156C2×D163—21no—
157C2×D163C2×M1612yes—
158C4×C163—21no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163H(5,𝔽2)11yescharacteristic
163C4×C1632−1+411yescharacteristic
164Dic163—21no—
165Dic163—21no—
166Dic163—21no—
167Dic163—21no—
168Dic163—21no—
169Dic163C2×M1612yes—
170Dic163#13112yes—
171C2×C163—22no—
172C2×C163—41no—
173C2×C163#52011yescharacteristic
174C2×C163C22×M1611yescharacteristic
175C2×C163C2×#13111yescharacteristic
176D163—22no—
177C163#299011yescharacteristic, complemented, Sylow
178#2990—1631nomaximal, Sylow
179C4×#41—1631no—
180#446—1631no—
181#446—1631no—
182#435—1631no—
183#435—1631no—
184#435—1631no—
185#435D16311yescharacteristic, complemented
186C8×D4—1631no—
187C8×D4—1631no—
188#432—1631no—
189#432—1631no—
190#431—1631no—
191#423—1631no—
192C2×C4⋊C8—1631no—
193C2×C4⋊C8—1631no—
194C2×#41—1631no—
195C2×M16—1631no—
196C2×M16C2×D16311yescharacteristic
197C22×C8—1631no—
198C22×C8—1631no—
199C22×C8—1631no—
200C22×C8C2×D16311yescharacteristic
201C4×Q8—1631no—
202C4×Q8—1631no—
203C4×D4—1631no—
204C4×D4—1631no—
205C4×D4—1631no—
206C4×D4—1631no—
207C4×D4—1631no—
208C4×D4C2×D16311yescharacteristic
209#117—1631no—
210#117—1631no—
211#117—1631no—
212C2×C42—1631no—
213C2×C42—1631no—
214C2×C42—1631no—
215C4⋊C8—1631no—
216C4⋊C8—1631no—
217C4⋊C8—1631no—
218C4⋊C8—1631no—
219C4⋊C8—1631no—
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222C4⋊C8—1631no—
223C4⋊C8—1631no—
224C4⋊C8C2×D16311yescharacteristic
225C22⋊C8—1631no—
226C22⋊C8—1631no—
227C22⋊C8—1631no—
228C22⋊C8—1631no—
229C22⋊C8C2×D16311yescharacteristic
230C22⋊C8C2×D16311yescharacteristic
231#97—1631no—
232#97—1631no—
233#97—1631no—
234#97C2×D16311yescharacteristic
235C4×C8—1631no—
236C4×C8—1631no—
237#41—3264no—
238C2×Q8—1631no—
239C2×D4—1631no—
240C2×D4—1631no—
241C2×D4C4×D16311yescharacteristic
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1631no—
248C22×C4—1631no—
249C22×C4C22×D16311yescharacteristic
250C22×C4C22×D16311yescharacteristic
251M16—41no—
252M16—6521no—
253C2×C8—41no—
254C2×C8—6521no—
255C2×C8—6521no—
256C2×C8—6521no—
257C2×C8—1631no—
258C2×C8—1631no—
259C2×C8—1631no—
260C2×C8—1631no—
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C2×C8C22×D16311yescharacteristic
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4C4×D16311yescharacteristic
271C22⋊C4—1631no—
272C22⋊C4—1631no—
273C22⋊C4—1631no—
274C22⋊C4—1631no—
275C22⋊C4C4×D16311yescharacteristic
276C22⋊C4C4×D16311yescharacteristic
277C42—1631no—
278C42—1631no—
279C42—1631no—
280C42—1631no—
281C42—1631no—
282C42—1631no—
283C42—1631no—
284C42—1631no—
285C42—1631no—
286C42C22×D16311yescharacteristic
287C23—1631no—
288C23C2×C4×D16311yescharacteristic
289C23C2×C4×D16311yescharacteristic
290Q8—3262no—
291D4—22no—
292D4—3262no—
293D4—1634no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—3262no—
299C2×C4—3262no—
300C2×C4—3262no—
301C2×C4—3262no—
302C2×C4—3262no—
303C2×C4—3262no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4—1631no—
307C2×C4—1631no—
308C2×C4—1631no—
309C2×C4—1631no—
310C2×C4—1631no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4C2×C4×D16311yescharacteristic
314C2×C4C2×C4×D16311yescharacteristic
315C2×C4C2×C4×D16311yescharacteristic
316C2×C4C2×C4×D16311yescharacteristic
317C2×C4C23×D16311yescharacteristic
318C8—21no—
319C8—21no—
320C8—21no—
321C8—21no—
322C8—3261no—
323C8—3261no—
324C8—3261no—
325C8—3261no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—3261no—
332C22—3261no—
333C22—1632no—
334C22M16×D16312yes—
335C22C22×C4×D16311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—3261no—
341C4—3261no—
342C4—3261no—
343C4—3261no—
344C4—3261no—
345C4—1632no—
346C4—1632no—
347C4#1910478011yescharacteristic
348C4#1910478211yescharacteristic
349C2—22no—
350C2—41no—
351C2—3262no—
352C2#2378214411yescharacteristic
353C2C2×M16×D16311yescharacteristic
354C2#131×D16311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D163
commutator, socleC22×C163C23×C4
Fitting#435×C163C2
prime radicalC2×#14579698C4
cosocleC2×C4×C163C24
hypercenter#435D163
hypocenterC163#2990
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #23781794C2semidirect
3. #23781793C2semidirect
4. #23781739C2semidirect
5. #23781737C2semidirect
6. #23781734C2semidirect
7. C8×C22⋊D163C2semidirect
8. (C22⋊D163)⋊C8C2semidirect
9. #23781720C2semidirect
10. #23781719C2semidirect
11. #23781715C2semidirect
12. #23781688C2semidirect
13. C2×Dic163⋊C8C2semidirect
14. C4⋊C8×D163C2semidirect
15. #435×C163C2semidirect
112. C22⋊D163M16semidirect
177. C163#2990semidirect
185. #435D163semidirect

Maximal subgroups

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

subgroupindex
C4×#145796982
#237817942
#237817932
#237817392
#237817372
#237817342
C8×C22⋊D1632
(C22⋊D163)⋊C82
#237817202
#237817192
#237817152
#237816882
C2×Dic163⋊C82
C4⋊C8×D1632
#435×C1632
#2990163