Finite Group Databasebeta

← #35225956

Subgroups — #35225956

Order 20864.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782286C211yesmaximal, characteristic
2#23781969C211yesmaximal, characteristic, complemented
3#23781962C211yesmaximal, characteristic, complemented
4#23781960C211yesmaximal, characteristic, complemented
5#23781950C211yesmaximal, characteristic, complemented
6#23781921C211yesmaximal, characteristic, complemented
7#23781913C211yesmaximal, characteristic
8#23781904C211yesmaximal, characteristic, complemented
9#23781896C211yesmaximal, characteristic
10#23781887C211yesmaximal, characteristic, complemented
11#23781887C211yesmaximal, characteristic, complemented
12#23781872C211yesmaximal, characteristic, complemented
13#23781867C211yesmaximal, characteristic, complemented
14#23781692C211yesmaximal, characteristic, complemented
15#541×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579683C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22#19104735C2211yescharacteristic
23#19104730—21no—
24#19104723C2211yescharacteristic
25#19104721—21no—
26#19104717—21no—
27#19104714C2211yescharacteristic
28#19104713—21no—
29#19104709C2211yescharacteristic
30C4×Dic326C2211yescharacteristic
31C4×D652C2211yescharacteristic
32#19104668C2211yescharacteristic
33#19104668C2211yescharacteristic
34#19104666C2211yescharacteristic
35#19104666C2211yescharacteristic
36C4⋊(C163⋊C8)C2211yescharacteristic
37C4⋊(C163⋊C8)C2211yescharacteristic
38C4⋊(C163⋊C8)C2211yescharacteristic
39C4⋊(C163⋊C8)C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104659C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48C4⋊Q8×C163C2211yescharacteristic
49#126×C163—21no—
50#126×C163—21no—
51#125×C163—21no—
52#125×C163—21no—
53#125×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×Q8×C163C2211yescharacteristic
58#117×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C4×D163—21no—
64#14579689—41no—
65#14579689—41no—
66#14579686—41no—
67#14579686—41no—
68#14579683—41no—
69C2×C163⋊C8—41no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84C4×Dic163—21no—
85C2×Q8×C163C2311yescharacteristic
86C2×Q8×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C4⋊C4×C163—21no—
89C4⋊C4×C163—21no—
90C4⋊C4×C163—21no—
91C4⋊C4×C163—21no—
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163D411yescharacteristic
97C4⋊C4×C163D411yescharacteristic
98C4⋊C4×C163C2311yescharacteristic
99C4⋊C4×C163C2311yescharacteristic
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163D411yescharacteristic
103C22⋊C4×C163D411yescharacteristic
104C42×C163—21no—
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109Dic326—21no—
110Dic326—21no—
111Dic326—81no—
112D652—21no—
113D652—21no—
114C22⋊D163—81no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C4×D163—81no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163C2×D411yescharacteristic
124Q8×C163—21no—
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2411yescharacteristic
140C22×C163—21no—
141C22×C163—21no—
142C22×C163—21no—
143C22×C163C22×D411yescharacteristic
144C2×D163—41no—
145C2×D163—41no—
146C2×D163—41no—
147C4×C163—21no—
148C4×C163—21no—
149C4×C163—21no—
150C4×C163—21no—
151C4×C163—21no—
152C4×C163—41no—
153C4×C163—41no—
154C4×C163—41no—
155C4×C163—41no—
156C4×C163H(5,𝔽2)11yescharacteristic
157C4×C1632−1+411yescharacteristic
158Dic163—41no—
159Dic163—41no—
160Dic163—41no—
161C2×C163—41no—
162C2×C163#53711yescharacteristic
163C2×C163#57711yescharacteristic
164C2×C163#57811yescharacteristic
165D163—81no—
166C163#325811yescharacteristic, complemented, Sylow
167#3258—1631nomaximal, Sylow
168#542—1631no—
169#541D16311yescharacteristic, complemented
170#479—1631no—
171#478—1631no—
172#477—1631no—
173#475—1631no—
174#470—1631no—
175#468—1631no—
176#467—1631no—
177#465—1631no—
178#464—1631no—
179#464—1631no—
180#462—1631no—
181#461—1631no—
182#424—1631no—
183C2×Q16—1631no—
184C2×Q16—1631no—
185C2×SD16—1631no—
186C2×SD16—1631no—
187C2×M16—1631no—
188C22×C8—1631no—
189C4⋊Q8—1631no—
190C4⋊Q8C2×D16311yescharacteristic
191#126—21no—
192#126—21no—
193#126—3261no—
194#125—21no—
195#125—21no—
196#125C2×D16311yescharacteristic
197#124—3261no—
198#124—1631no—
199#123—3261no—
200C22⋊Q8—3261no—
201C22⋊Q8—1631no—
202C22⋊Q8C2×D16311yescharacteristic
203C22⋊Q8C2×D16311yescharacteristic
204#121—1631no—
205C4×Q8—1631no—
206C4×Q8C2×D16311yescharacteristic
207C4×Q8C2×D16311yescharacteristic
208C4×D4—1631no—
209#117C2×D16311yescharacteristic
210#107—1631no—
211#107—1631no—
212#106—1631no—
213#106—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216C4⋊C8—1631no—
217C4⋊C8—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#103—1631no—
222#103—1631no—
223#103—1631no—
224#102—1631no—
225#102—1631no—
226C2×Q8—3261no—
227C2×Q8—1631no—
228C2×Q8C22×D16311yescharacteristic
229C2×Q8C22×D16311yescharacteristic
230C2×D4—3261no—
231C2×D4—1631no—
232C22×C4—3261no—
233C22×C4C22×D16311yescharacteristic
234Q16—6521no—
235Q16—6521no—
236SD16—6521no—
237SD16—6521no—
238M16—6521no—
239C2×C8—6521no—
240C2×C8—1631no—
241C2×C8—1631no—
242C2×C8—1631no—
243C2×C8—1631no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4C22⋊D16311yescharacteristic
258C4⋊C4C22⋊D16311yescharacteristic
259C4⋊C4C22×D16311yescharacteristic
260C4⋊C4C22×D16311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C22⋊C4—3261no—
266C22⋊C4—3261no—
267C22⋊C4—3261no—
268C22⋊C4C22⋊D16311yescharacteristic
269C22⋊C4C22⋊D16311yescharacteristic
270C42—21no—
271C42—21no—
272C42—3261no—
273C42C22×D16311yescharacteristic
274C42C22×D16311yescharacteristic
275C23—3261no—
276C23C2×C22⋊D16311yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—13041no—
282Q8—3261no—
283Q8—3261no—
284D4—13041no—
285D4—3261no—
286D4—3261no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—13041no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3261no—
297C2×C4C2×C22⋊D16311yescharacteristic
298C2×C4C2×C22⋊D16311yescharacteristic
299C2×C4C2×C22⋊D16311yescharacteristic
300C2×C4C2×C22⋊D16311yescharacteristic
301C2×C4C2×C22⋊D16311yescharacteristic
302C2×C4C23×D16311yescharacteristic
303C8—3261no—
304C8—3261no—
305C8—3261no—
306C8—3261no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—6521no—
311C22—6521no—
312C22—6521no—
313C22C22×C22⋊D16311yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—6521no—
324C4—6521no—
325C4—6521no—
326C4#1910478111yescharacteristic
327C4#1910478211yescharacteristic
328C2—41no—
329C2—13041no—
330C2#2378224311yescharacteristic
331C2#2378247311yescharacteristic
332C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#541×C163C2
socleC22×C163C22×D4
prime radical#19104709C22
hypercenter#541D163
hypocenterC163#3258
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #23781969C2semidirect
3. #23781962C2semidirect
4. #23781960C2semidirect
5. #23781950C2semidirect
6. #23781921C2semidirect
8. #23781904C2semidirect
10. #23781887C2semidirect
11. #23781887C2semidirect
12. #23781872C2semidirect
13. #23781867C2semidirect
14. #23781692C2semidirect
15. #541×C163C2semidirect
166. C163#3258semidirect
169. #541D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237819692
#237819622
#237819602
#237819502
#237819212
#237819132
#237819042
#237818962
#237818872
#237818722
#237818672
#237816922
#541×C1632
#3258163