Finite Group Databasebeta

← #35217394

Subgroups — #35217394

Order 20864.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104691C211yesmaximal, characteristic
2#23781552C211yesmaximal, characteristic, complemented
3#23781552C211yesmaximal, characteristic, complemented
4#23781546C211yesmaximal, characteristic, complemented
5#23781546C211yesmaximal, characteristic, complemented
6#23781540C211yesmaximal, characteristic
7#23781532C211yesmaximal, characteristic, complemented
8#23781530C211yesmaximal, characteristic, complemented
9#23781526C211yesmaximal, characteristic
10C4⋊C4×Dic163C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic163C211yesmaximal, characteristic
12C4×C22⋊Dic163C211yesmaximal, characteristic
13#23781510C211yesmaximal, characteristic, complemented
14C2×C163⋊#95C211yesmaximal, characteristic
15#385×C163C211yesmaximal, characteristic
16C23×Dic163C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104691C418yescomplemented
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×C22⋊Dic163C2211yescharacteristic
24C2×C22⋊Dic163C2211yescharacteristic
25C2×C4×Dic163C2211yescharacteristic
26C2×C4×Dic163C2211yescharacteristic
27C2×C4×Dic163C2211yescharacteristic
28C2×C4×Dic163C2211yescharacteristic
29C2×C4×Dic163C2211yescharacteristic
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
44C163⋊#95C2211yescharacteristic
45C23×C4×C163C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52C22×Dic163—22no—
53C22×Dic163—22no—
54C22×Dic163—22no—
55C22×Dic163—22no—
56C22×Dic163—41no—
57C22×Dic163C2×C414yes—
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C4⋊Dic163—22no—
67C4⋊Dic163—41no—
68Dic163⋊C4C2×C414yescomplemented
69Dic163⋊C4C2×C414yescomplemented
70C22⋊Dic163—22no—
71C22⋊Dic163C2×C414yes—
72C4×Dic163—22no—
73C4×Dic163—22no—
74C4×Dic163—41no—
75C4×Dic163—41no—
76C4×Dic163C2×C414yescomplemented
77C4×Dic163C2×C414yescomplemented
78C24×C163C2311yescharacteristic
79C22×C4×C163—22no—
80C22×C4×C163—22no—
81C22×C4×C163—41no—
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2311yescharacteristic
84C22×C4×C163C2311yescharacteristic
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C4⋊C4×C163—22no—
89C4⋊C4×C163—41no—
90C22⋊C4×C163—22no—
91C22⋊C4×C163C2×C414yes—
92C2×Dic163—28no—
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—44no—
113C2×Dic163C22×C412yes—
114C2×Dic163C22×C412yes—
115C2×Dic163C22×C412yes—
116C2×Dic163C22×C412yes—
117C2×Dic163#4112yes—
118C23×C163—21no—
119C23×C163—21no—
120C23×C163—21no—
121C23×C163—21no—
122C23×C163C22×C412yes—
123C23×C163#4112yes—
124C23×C163#4112yes—
125C23×C163C2411yescharacteristic
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—21no—
141C2×C4×C163—44no—
142C2×C4×C163C22×C412yes—
143C2×C4×C163C22×C412yes—
144C2×C4×C163#4112yes—
145C22×C163—22no—
146C22×C163—22no—
147C22×C163—22no—
148C22×C163—22no—
149C22×C163—22no—
150C22×C163—22no—
151C22×C163#11714yes—
152C22×C163C23×C411yescharacteristic
153C22×C163C2×#4111yescharacteristic
154C22×C163C2×#4111yescharacteristic
155C22×C163C2×#4111yescharacteristic
156C22×C163C2×#4111yescharacteristic
157C22×C163H(5,𝔽2)11yescharacteristic
158C22×C1632−1+411yescharacteristic
159C4×C163—22no—
160C4×C163—22no—
161C4×C163—22no—
162C4×C163—41no—
163C4×C163—41no—
164C4×C163—41no—
165Dic163—22no—
166Dic163—22no—
167Dic163—22no—
168Dic163—22no—
169Dic163—22no—
170Dic163—41no—
171Dic163—41no—
172Dic163—41no—
173C2×C163—24no—
174C2×C163C2×#11711yescharacteristic
175C2×C163C4×#4111yescharacteristic
176C2×C163#52011yescharacteristic
177C2×C163#55111yescharacteristic
178C2×C163#55111yescharacteristic
179C2×C163#55211yescharacteristic
180C2×C163#55211yescharacteristic
181C163#234311yescharacteristic, complemented, Sylow
182#2343—1631nomaximal, Sylow
183C2×#117—1631no—
184#387—1631no—
185#387—1631no—
186#387—1631no—
187#387—1631no—
188#385—1631no—
189#385D16311yescharacteristic
190#383—1631no—
191#382—1631no—
192#381—1631no—
193C4×C4⋊C4—1631no—
194C4×C22⋊C4—1631no—
195C4×C22⋊C4—1631no—
196#376—1631no—
197C2×#95—1631no—
198C23×C4—1631no—
199C23×C4—1631no—
200C23×C4C2×D16311yescharacteristic
201#117—1638no—
202C2×C4⋊C4—1631no—
203C2×C4⋊C4—1631no—
204C2×C4⋊C4—1631no—
205C2×C4⋊C4—1631no—
206C2×C4⋊C4C2×D16311yescharacteristic
207C2×C4⋊C4C2×D16311yescharacteristic
208C2×C22⋊C4—1631no—
209C2×C22⋊C4—1631no—
210C2×C22⋊C4C2×D16311yescharacteristic
211C2×C22⋊C4C2×D16311yescharacteristic
212C2×C42—1631no—
213C2×C42—1631no—
214C2×C42—1631no—
215C2×C42—1631no—
216C2×C42—1631no—
217C2×C42—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#95—1631no—
230#95—1631no—
231#95—1631no—
232#95C2×D16311yescharacteristic
233#95C2×D16311yescharacteristic
234C24C22×D16311yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—3262no—
239C22×C4—3262no—
240C22×C4—3262no—
241C22×C4—3262no—
242C22×C4—6521no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4—1631no—
246C22×C4—1634no—
247C22×C4—1631no—
248C22×C4—1631no—
249C22×C4—1631no—
250C22×C4—1631no—
251C22×C4—1631no—
252C22×C4C22×D16311yescharacteristic
253C22×C4C22×D16311yescharacteristic
254C22×C4C22×D16311yescharacteristic
255C22×C4C22×D16311yescharacteristic
256C22×C4C22×D16311yescharacteristic
257C22×C4C22×D16311yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—3262no—
261C4⋊C4—6521no—
262C4⋊C4—1634no—
263C4⋊C4—1634no—
264C22⋊C4—22no—
265C22⋊C4—3262no—
266C22⋊C4—1634no—
267C22⋊C4C4×D16314yes—
268C42—3262no—
269C42—3262no—
270C42—6521no—
271C42—6521no—
272C42—1634no—
273C42—1634no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D16312yes—
279C23#1457969812yes—
280C23#1457970012yes—
281C23C23×D16311yescharacteristic
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—21no—
297C2×C4—44no—
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—3268no—
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—6524no—
319C2×C4—1632no—
320C2×C4—1632no—
321C2×C4—1632no—
322C2×C4—1632no—
323C2×C4—1632no—
324C2×C4C2×C4×D16312yes—
325C2×C4C2×C4×D16312yes—
326C2×C4#1457969812yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#1910469314yes—
334C22C22×C4×D16311yescharacteristic
335C22#41×D16311yescharacteristic
336C22C2×#1457969811yescharacteristic
337C22C2×#1457969811yescharacteristic
338C22C2×#1457970011yescharacteristic
339C22#1910478011yescharacteristic
340C22#1910478211yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3262no—
348C4—3262no—
349C4—3262no—
350C4—3262no—
351C4—3262no—
352C4—6521no—
353C4—6521no—
354C4—6521no—
355C2—24no—
356C2C2×#1910469311yescharacteristic
357C2C4×#1457969811yescharacteristic
358C2#2378214411yescharacteristic
359C2#2378234211yescharacteristic
360C2#2378234611yescharacteristic
361C2#2378235211yescharacteristic
362C2#2378235711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#385×C163C2
socle, cosocleC23×C163C24
prime radicalC24×C163C23
hypercenter#385D163
hypocenterC163#2343
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #23781552C2semidirect
3. #23781552C2semidirect
4. #23781546C2semidirect
5. #23781546C2semidirect
7. #23781532C2semidirect
8. #23781530C2semidirect
10. C4⋊C4×Dic163C2semidirect
13. #23781510C2semidirect
18. #19104691C4semidirect
68. Dic163⋊C4C2×C4semidirect
69. Dic163⋊C4C2×C4semidirect
76. C4×Dic163C2×C4semidirect
77. C4×Dic163C2×C4semidirect
181. C163#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#191046912
#237815522
#237815462
#237815402
#237815322
#237815302
#237815262
C4⋊C4×Dic1632
C22⋊C4×Dic1632
C4×C22⋊Dic1632
#237815102
C2×C163⋊#952
#385×C1632
#2343163