Finite Group Databasebeta

← #35222771

Subgroups — #35222771

Order 20864.

366 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C163⋊#401C211yesmaximal, characteristic, complemented
2#23781605C211yesmaximal, characteristic, complemented
3#23781604C211yesmaximal, characteristic, complemented
4#23781600C211yesmaximal, characteristic
5#23781598C211yesmaximal, characteristic, complemented
6#23781594C211yesmaximal, characteristic
7#23781592C211yesmaximal, characteristic
8#23781591C211yesmaximal, characteristic
9#23781587C211yesmaximal, characteristic
10#23781584C211yesmaximal, characteristic, complemented
11#23781576C211yesmaximal, characteristic
12#23781561C211yesmaximal, characteristic, complemented
13#23781554C211yesmaximal, characteristic
14#23781547C211yesmaximal, characteristic, complemented
15#395×C163C211yesmaximal, characteristic, complemented
16C2×C4⋊Dic163C2211yescharacteristic
17C2×C4⋊Dic163C2211yescharacteristic
18C2×Dic163⋊C4C2211yescharacteristic
19C2×Dic163⋊C4C2211yescharacteristic
20C2×Dic163⋊C4C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×(C2×D163)⋊C4C2211yescharacteristic
24C2×(C2×D163)⋊C4C2211yescharacteristic
25C2×(C2×D163)⋊C4C2211yescharacteristic
26C2×(C2×D163)⋊C4C2211yescharacteristic
27C2×(C2×D163)⋊C4C2211yescharacteristic
28C2×(C2×D163)⋊C4C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×C4×Dic163C2211yescharacteristic
31C163⋊#95C2211yescharacteristic
32C163⋊#95C2211yescharacteristic
33C163⋊#95C2211yescharacteristic
34C163⋊#95C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C22×Q8×C163C2211yescharacteristic
45C2×C4⋊C4×C163C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48#95×C163C2211yescharacteristic
49#95×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51C23×D163C2311yescharacteristic
52C22×Dic163C2311yescharacteristic
53C22×Dic163C2311yescharacteristic
54C22×Dic163C2311yescharacteristic
55C22×Dic163C2311yescharacteristic
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C4⋊Dic163—22no—
60C4⋊Dic163—41no—
61Dic163⋊C4—22no—
62Dic163⋊C4—41no—
63Dic163⋊C4—41no—
64Dic163⋊C4—41no—
65Dic163⋊C4—41no—
66(C2×D163)⋊C4—22no—
67(C2×D163)⋊C4—41no—
68(C2×D163)⋊C4—41no—
69(C2×D163)⋊C4—41no—
70(C2×D163)⋊C4—41no—
71(C2×D163)⋊C4—41no—
72(C2×D163)⋊C4—41no—
73C4×Dic163—41no—
74C2×Q8×C163—41no—
75C2×Q8×C163—41no—
76C2×Q8×C163—41no—
77C22×C4×C163C2311yescharacteristic
78C22×C4×C163C2311yescharacteristic
79C22×C4×C163C2311yescharacteristic
80C22×C4×C163C2311yescharacteristic
81C22×C4×C163C2311yescharacteristic
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2311yescharacteristic
84C4⋊C4×C163—41no—
85C4⋊C4×C163—41no—
86C4⋊C4×C163—41no—
87C22×D163—21no—
88C22×D163—21no—
89C22×D163—21no—
90C22×D163—21no—
91C22×D163—21no—
92C22×D163—21no—
93C22×D163—21no—
94C2×Dic163—21no—
95C2×Dic163—21no—
96C2×Dic163—21no—
97C2×Dic163—21no—
98C2×Dic163—21no—
99C2×Dic163—21no—
100C2×Dic163—21no—
101C2×Dic163—21no—
102C2×Dic163—21no—
103C2×Dic163—21no—
104C2×Dic163—21no—
105C2×Dic163—21no—
106C2×Dic163—21no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C23×C163C2411yescharacteristic
116Q8×C163—44no—
117C2×C4×C163—21no—
118C2×C4×C163—21no—
119C2×C4×C163—21no—
120C2×C4×C163—21no—
121C2×C4×C163—21no—
122C2×C4×C163—21no—
123C2×C4×C163—21no—
124C2×C4×C163—21no—
125C2×C4×C163—21no—
126C2×C4×C163—21no—
127C2×C4×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×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163#4112yes—
138C22×C163C2×#4111yescharacteristic
139C22×C163H(5,𝔽2)11yescharacteristic
140C22×C163H(5,𝔽2)11yescharacteristic
141C22×C163H(5,𝔽2)11yescharacteristic
142C22×C1632−1+411yescharacteristic
143C22×C1632−1+411yescharacteristic
144C22×C1632−1+411yescharacteristic
145C2×D163—41no—
146C2×D163—41no—
147C2×D163—41no—
148C2×D163—41no—
149C2×D163—41no—
150C2×D163—41no—
151C2×D163—41no—
152C4×C163—22no—
153C4×C163—41no—
154C4×C163—41no—
155C4×C163—41no—
156C4×C163—41no—
157C4×C163—41no—
158C4×C163—41no—
159Dic163—41no—
160Dic163—41no—
161Dic163—41no—
162Dic163—41no—
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166C2×C163#53911yescharacteristic
167C2×C163#54211yescharacteristic
168C2×C163#54211yescharacteristic
169C2×C163#56011yescharacteristic
170C2×C163#56311yescharacteristic
171C2×C163#56311yescharacteristic
172C2×C163#56311yescharacteristic
173D163—81no—
174C163#283811yescharacteristic, complemented, Sylow
175#2838—1631nomaximal, Sylow
176#401—1631no—
177#400—1631no—
178#400—1631no—
179#399—1631no—
180#398—1631no—
181#397—1631no—
182#397—1631no—
183#397—1631no—
184#396—1631no—
185#395—1631no—
186#395D16311yescharacteristic, complemented
187#393—1631no—
188#389—1631no—
189#388—1631no—
190#387—1631no—
191C22×Q8C2×D16311yescharacteristic
192C2×C4⋊C4—1631no—
193C2×C4⋊C4—1631no—
194C2×C4⋊C4—1631no—
195C2×C4⋊C4—1631no—
196C2×C4⋊C4—1631no—
197C2×C4⋊C4—1631no—
198C2×C4⋊C4—1631no—
199C2×C4⋊C4C2×D16311yescharacteristic
200C2×C4⋊C4C2×D16311yescharacteristic
201C2×C4⋊C4C2×D16311yescharacteristic
202C2×C22⋊C4—1631no—
203C2×C22⋊C4—1631no—
204C2×C22⋊C4—1631no—
205C2×C22⋊C4—1631no—
206C2×C22⋊C4—1631no—
207C2×C22⋊C4—1631no—
208C2×C22⋊C4—1631no—
209C2×C42—1631no—
210#95—1631no—
211#95—1631no—
212#95—1631no—
213#95—1631no—
214#95—1631no—
215#95—1631no—
216#95—1631no—
217#95—1631no—
218#95—1631no—
219#95—1631no—
220#95—1631no—
221#95—1631no—
222#95—1631no—
223#95C2×D16311yescharacteristic
224#95C2×D16311yescharacteristic
225#95C2×D16311yescharacteristic
226C24—1631no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C22×C4—1631no—
231C22×C4—1631no—
232C22×C4—1631no—
233C22×C4—1631no—
234C22×C4—1631no—
235C22×C4—1631no—
236C22×C4—1631no—
237C22×C4C22×D16311yescharacteristic
238C22×C4C22×D16311yescharacteristic
239C22×C4C22×D16311yescharacteristic
240C22×C4C22×D16311yescharacteristic
241C22×C4C22×D16311yescharacteristic
242C22×C4C22×D16311yescharacteristic
243C22×C4C22×D16311yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—3262no—
248C4⋊C4—3262no—
249C4⋊C4—6521no—
250C4⋊C4—6521no—
251C4⋊C4—6521no—
252C4⋊C4—6521no—
253C4⋊C4—6521no—
254C22⋊C4—3262no—
255C22⋊C4—6521no—
256C22⋊C4—6521no—
257C22⋊C4—6521no—
258C22⋊C4—6521no—
259C22⋊C4—6521no—
260C22⋊C4—6521no—
261C42—6521no—
262C23—3261no—
263C23—3261no—
264C23—3261no—
265C23—3261no—
266C23—3261no—
267C23—3261no—
268C23—3261no—
269C23C23×D16311yescharacteristic
270Q8—44no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3261no—
292C2×C4—3261no—
293C2×C4—3261no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—3261no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4#1457970012yes—
313C22—6521no—
314C22—6521no—
315C22—6521no—
316C22—6521no—
317C22—6521no—
318C22—6521no—
319C22—6521no—
320C22C2×#1457970011yescharacteristic
321C22#1910478011yescharacteristic
322C22#1910478011yescharacteristic
323C22#1910478011yescharacteristic
324C22#1910478211yescharacteristic
325C22#1910478311yescharacteristic
326C22#1910478311yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—6521no—
335C4—6521no—
336C4—6521no—
337C4—6521no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C2—13041no—
342C2#2378226411yescharacteristic
343C2#2378228911yescharacteristic
344C2#2378228911yescharacteristic
345C2#2378240411yescharacteristic
346C2#2378241411yescharacteristic
347C2#2378241611yescharacteristic
348C2#2378241611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#395×C163C2
prime radicalC23×D163C23
hypercenter#395D163
hypocenterC163#2838
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C163⋊#401C2semidirect
2. #23781605C2semidirect
3. #23781604C2semidirect
5. #23781598C2semidirect
10. #23781584C2semidirect
12. #23781561C2semidirect
14. #23781547C2semidirect
15. #395×C163C2semidirect
174. C163#2838semidirect
186. #395D163semidirect

Maximal subgroups

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

subgroupindex
C163⋊#4012
#237816052
#237816042
#237816002
#237815982
#237815942
#237815922
#237815912
#237815872
#237815842
#237815762
#237815612
#237815542
#237815472
#395×C1632
#2838163