Finite Group Databasebeta

← #35223496

Subgroups — #35223496

Order 20864.

412 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#23781741C211yesmaximal, characteristic, complemented
5#23781740C211yesmaximal, characteristic, complemented
6#23781737C211yesmaximal, characteristic, complemented
7#23781736C211yesmaximal, characteristic, complemented
8#23781733C211yesmaximal, characteristic, complemented
9#23781717C211yesmaximal, characteristic, complemented
10#23781715C211yesmaximal, characteristic, complemented
11#23781715C211yesmaximal, characteristic, complemented
12#23781617C211yesmaximal, characteristic, complemented
13M16×Dic163C211yesmaximal, characteristic, complemented
14C2×C8×Dic163C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C411yescharacteristic
17C2×#14579683C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×#14579682C2211yescharacteristic
20C22×C163⋊C8C2211yescharacteristic
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
39C22⋊(C163⋊C8)C2211yescharacteristic
40C22⋊(C163⋊C8)C2211yescharacteristic
41(C2×D163)⋊C8C2211yescharacteristic
42(C2×D163)⋊C8C2211yescharacteristic
43#19104646C2211yescharacteristic
44#19104645C2211yescharacteristic
45#19104645C2211yescharacteristic
46C8×Dic163C2211yescharacteristic
47C8×Dic163C2211yescharacteristic
48C8×Dic163C2211yescharacteristic
49C8×Dic163C2211yescharacteristic
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—22no—
71#14579682—22no—
72#14579682—22no—
73C2×C163⋊C8—22no—
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
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—22no—
100C2×C8×C163—22no—
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⋊D163—22no—
113C2×Dic163—22no—
114C2×Dic163—22no—
115C2×Dic163—22no—
116C2×Dic163M1614yescomplemented
117C2×Dic163C22×C411yescharacteristic
118C2×Dic163C22×C411yescharacteristic
119C2×Dic163C22×C411yescharacteristic
120C2×Dic163C22×C411yescharacteristic
121C2×Dic163C22×C411yescharacteristic
122C2×Dic163C22×C411yescharacteristic
123C2×Dic163C22×C411yescharacteristic
124C4×D163—21no—
125C4×D163—21no—
126C4×D163—22no—
127C4×D163—22no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C163⋊C8—21no—
131C163⋊C8#4112yescomplemented
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#4112yescomplemented
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×D163—21no—
158C4×C163—21no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163C2×#4111yescharacteristic
163C4×C163C2×#4111yescharacteristic
164Dic163—21no—
165Dic163—21no—
166Dic163—21no—
167Dic163—21no—
168Dic163C2×M1612yes—
169Dic163C2×M1612yes—
170Dic163#13112yes—
171C2×C163—22no—
172C2×C163—41no—
173C2×C163C4×#4111yescharacteristic
174C2×C163C22×M1611yescharacteristic
175C2×C163C2×#13111yescharacteristic
176D163—41no—
177C163#296511yescharacteristic, complemented, Sylow
178#2965—1631nomaximal, Sylow
179C4×#41—1631no—
180#446—1631no—
181#446—1631no—
182#436—1631no—
183#436—1631no—
184#435—1631no—
185#435—1631no—
186#435—1631no—
187#435D16311yescharacteristic, complemented
188#431—1631no—
189#431—1631no—
190#431—1631no—
191C4×M16—1631no—
192C4×M16—1631no—
193C2×C4×C8—1631no—
194C2×#41—1631no—
195C2×M16—1631no—
196C2×M16—1631no—
197C2×M16—1631no—
198C2×M16C2×D16311yescharacteristic
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⋊C8C2×D16311yescharacteristic
221C22⋊C8—1631no—
222C22⋊C8—1631no—
223C22⋊C8—1631no—
224C22⋊C8—1631no—
225C22⋊C8C2×D16311yescharacteristic
226C22⋊C8C2×D16311yescharacteristic
227#97—1631no—
228#97—1631no—
229#97—1631no—
230#97C2×D16311yescharacteristic
231C4×C8—1631no—
232C4×C8—1631no—
233C4×C8—1631no—
234C4×C8—1631no—
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—22no—
252M16—3262no—
253M16—3262no—
254M16—3262no—
255C2×C8—22no—
256C2×C8—3262no—
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—3262no—
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—3261no—
304C2×C4—3261no—
305C2×C4—1631no—
306C2×C4—1631no—
307C2×C4—1631no—
308C2×C4—1631no—
309C2×C4—1631no—
310C2×C4—1631no—
311C2×C4—1631no—
312C2×C4—1634no—
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—3261no—
322C8—3261no—
323C8—3261no—
324C8—1632no—
325C8#1457969812yescomplemented
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—3261no—
332C22—3261no—
333C22—3261no—
334C22M16×D16312yes—
335C22C22×C4×D16311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—3261no—
341C4—3261no—
342C4—3261no—
343C4—3261no—
344C4—1632no—
345C4—1632no—
346C4—1632no—
347C4#41×D16311yescharacteristic
348C4C2×#1457969811yescharacteristic
349C2—22no—
350C2—41no—
351C2—6521no—
352C2C4×#1457969811yescharacteristic
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#2965
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. #23781741C2semidirect
5. #23781740C2semidirect
6. #23781737C2semidirect
7. #23781736C2semidirect
8. #23781733C2semidirect
9. #23781717C2semidirect
10. #23781715C2semidirect
11. #23781715C2semidirect
12. #23781617C2semidirect
13. M16×Dic163C2semidirect
14. C2×C8×Dic163C2semidirect
15. #435×C163C2semidirect
116. C2×Dic163M16semidirect
131. C163⋊C8#41semidirect
147. C8×C163#41semidirect
177. C163#2965semidirect
187. #435D163semidirect
325. C8#14579698semidirect

Maximal subgroups

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

subgroupindex
C4×#145796982
#237817942
#237817932
#237817412
#237817402
#237817372
#237817362
#237817332
#237817172
#237817152
#237816172
M16×Dic1632
C2×C8×Dic1632
#435×C1632
#2965163