Finite Group Databasebeta

← #35222791

Subgroups — #35222791

Order 20864.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23781607C211yesmaximal, characteristic, complemented
2#23781604C211yesmaximal, characteristic, complemented
3#23781601C211yesmaximal, characteristic
4#23781599C211yesmaximal, characteristic, complemented
5#23781598C211yesmaximal, characteristic, complemented
6#23781598C211yesmaximal, characteristic, complemented
7#23781597C211yesmaximal, characteristic, complemented
8#23781588C211yesmaximal, characteristic
9#23781588C211yesmaximal, characteristic
10#23781586C211yesmaximal, characteristic, complemented
11#23781585C211yesmaximal, characteristic, complemented
12#23781577C211yesmaximal, characteristic
13#23781528C211yesmaximal, characteristic
14#23781525C211yesmaximal, characteristic
15#380×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C2×C4⋊Dic163C2211yescharacteristic
18C2×C4⋊Dic163C2211yescharacteristic
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2211yescharacteristic
21C2×C4⋊Dic163C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×C22⋊Dic163C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C163⋊#95C2211yescharacteristic
33C163⋊#95C2211yescharacteristic
34C163⋊#95C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C23×C4×C163C2211yescharacteristic
45C2×C4⋊C4×C163C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C22⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49#95×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51C2×Dic326—41no—
52C2×Dic326—41no—
53C2×Dic326—41no—
54C22×Dic163C2311yescharacteristic
55C22×Dic163C2311yescharacteristic
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C4⋊Dic163—41no—
63C4⋊Dic163—41no—
64C4⋊Dic163—41no—
65C4⋊Dic163—41no—
66C4⋊Dic163—41no—
67Dic163⋊C4—41no—
68Dic163⋊C4—41no—
69Dic163⋊C4—41no—
70Dic163⋊C4—41no—
71Dic163⋊C4—41no—
72Dic163⋊C4—41no—
73C22⋊Dic163—41no—
74C22⋊Dic163—41no—
75C22⋊Dic163—41no—
76C22⋊Dic163—41no—
77C24×C163C2311yescharacteristic
78C22×C4×C163—41no—
79C22×C4×C163—41no—
80C22×C4×C163Q814yescomplemented
81C22×C4×C163C2311yescharacteristic
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2311yescharacteristic
84C22×C4×C163C2311yescharacteristic
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C4⋊C4×C163—41no—
88C4⋊C4×C163—41no—
89C22⋊C4×C163—41no—
90C22⋊C4×C163—41no—
91Dic326—44no—
92C2×Dic163—21no—
93C2×Dic163—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—
115C2×Dic163—21no—
116C23×C163—21no—
117C23×C163—21no—
118C23×C163—21no—
119C23×C163—21no—
120C23×C163—21no—
121C23×C163—21no—
122C23×C163C2×Q812yes—
123C23×C163C2411yescharacteristic
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—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—44no—
141C2×C4×C163C2×Q812yes—
142C2×C4×C163C2×Q812yes—
143C22×C163—22no—
144C22×C163—22no—
145C22×C163—22no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163C22×Q811yescharacteristic
151C22×C163H(5,𝔽2)11yescharacteristic
152C22×C163H(5,𝔽2)11yescharacteristic
153C22×C163H(5,𝔽2)11yescharacteristic
154C22×C163H(5,𝔽2)11yescharacteristic
155C22×C1632−1+411yescharacteristic
156C22×C1632−1+411yescharacteristic
157C4×C163—22no—
158C4×C163—22no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163—41no—
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171C2×C163—42no—
172C2×C163#54311yescharacteristic
173C2×C163#54411yescharacteristic
174C2×C163#54411yescharacteristic
175C2×C163#56011yescharacteristic
176C2×C163#56211yescharacteristic
177C2×C163#56211yescharacteristic
178C2×C163#56311yescharacteristic
179C163#284111yescharacteristic, complemented, Sylow
180#2841—1631nomaximal, Sylow
181#400—1631no—
182#400—1631no—
183#399—1631no—
184#398—1631no—
185#398—1631no—
186#398—1631no—
187#398—1631no—
188#396—1631no—
189#396—1631no—
190#395—1631no—
191#395—1631no—
192#393—1631no—
193#381—1631no—
194#380—1631no—
195#380D16311yescharacteristic
196C22×Q8—1631no—
197C23×C4C2×D16311yescharacteristic
198C2×C4⋊C4—1631no—
199C2×C4⋊C4—1631no—
200C2×C4⋊C4—1631no—
201C2×C4⋊C4—1631no—
202C2×C4⋊C4—1631no—
203C2×C4⋊C4—1631no—
204C2×C4⋊C4—1631no—
205C2×C4⋊C4—1631no—
206C2×C4⋊C4—1631no—
207C2×C4⋊C4—1631no—
208C2×C4⋊C4—1631no—
209C2×C4⋊C4C2×D16311yescharacteristic
210C2×C4⋊C4C2×D16311yescharacteristic
211C2×C22⋊C4—1631no—
212C2×C22⋊C4—1631no—
213C2×C22⋊C4—1631no—
214C2×C22⋊C4—1631no—
215C2×C22⋊C4C2×D16311yescharacteristic
216C2×C22⋊C4C2×D16311yescharacteristic
217#95—1631no—
218#95—1631no—
219#95—1631no—
220#95—1631no—
221#95—1631no—
222#95—1631no—
223#95—1631no—
224#95—1631no—
225#95—1631no—
226#95—1631no—
227#95—1631no—
228#95—1631no—
229#95C2×D16311yescharacteristic
230#95C2×D16311yescharacteristic
231C24C22×D16311yescharacteristic
232C2×Q8—6521no—
233C2×Q8—6521no—
234C2×Q8—6521no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—1631no—
238C22×C4—1631no—
239C22×C4—1631no—
240C22×C4—1631no—
241C22×C4—1631no—
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4Dic32614yescomplemented
246C22×C4C22×D16311yescharacteristic
247C22×C4C22×D16311yescharacteristic
248C22×C4C22×D16311yescharacteristic
249C22×C4C22×D16311yescharacteristic
250C22×C4C22×D16311yescharacteristic
251C22×C4C22×D16311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—6521no—
255C4⋊C4—6521no—
256C4⋊C4—6521no—
257C4⋊C4—6521no—
258C4⋊C4—6521no—
259C4⋊C4—6521no—
260C4⋊C4—6521no—
261C4⋊C4—6521no—
262C4⋊C4—6521no—
263C4⋊C4—6521no—
264C4⋊C4—6521no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—6521no—
268C22⋊C4—6521no—
269C22⋊C4—6521no—
270C22⋊C4—6521no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23C2×Dic32612yes—
278C23C23×D16311yescharacteristic
279Q8—6524no—
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—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—44no—
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—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4C2×Dic32612yes—
322C2×C4C2×Dic32612yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×Dic32611yescharacteristic
331C22#1910478011yescharacteristic
332C22#1910478011yescharacteristic
333C22#1910478011yescharacteristic
334C22#1910478111yescharacteristic
335C22#1910478211yescharacteristic
336C22#1910478311yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—6521no—
344C4—6521no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4—6521no—
350C4—6521no—
351C2—42no—
352C2#2378229111yescharacteristic
353C2#2378229311yescharacteristic
354C2#2378229611yescharacteristic
355C2#2378240311yescharacteristic
356C2#2378241011yescharacteristic
357C2#2378241111yescharacteristic
358C2#2378241611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#380×C163C2
prime radicalC24×C163C23
hypercenter#380D163
hypocenterC163#2841
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. #23781607C2semidirect
2. #23781604C2semidirect
4. #23781599C2semidirect
5. #23781598C2semidirect
6. #23781598C2semidirect
7. #23781597C2semidirect
10. #23781586C2semidirect
11. #23781585C2semidirect
80. C22×C4×C163Q8semidirect
179. C163#2841semidirect
245. C22×C4Dic326semidirect

Maximal subgroups

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

subgroupindex
#237816072
#237816042
#237816012
#237815992
#237815982
#237815972
#237815882
#237815862
#237815852
#237815772
#237815282
#237815252
#380×C1632
#2841163