Finite Group Databasebeta

← #35218246

Subgroups — #35218246

Order 20864.

492 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104725C211yesmaximal, characteristic, complemented
2C2×#19104692C211yesmaximal, characteristic
3#23781600C211yesmaximal, characteristic
4#23781600C211yesmaximal, characteristic
5#23781587C211yesmaximal, characteristic
6#23781587C211yesmaximal, characteristic
7#23781562C211yesmaximal, characteristic, complemented
8#23781560C211yesmaximal, characteristic, complemented
9#23781560C211yesmaximal, characteristic, complemented
10#23781552C211yesmaximal, characteristic, complemented
11#23781549C211yesmaximal, characteristic, complemented
12#23781534C211yesmaximal, characteristic, complemented
13#23781524C211yesmaximal, characteristic
14#23781512C211yesmaximal, characteristic
15#389×C163C211yesmaximal, characteristic, complemented
16C22×C4×D163C2211yescharacteristic
17#19104725—24no—
18#19104692—24no—
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×C4×Dic163C2211yescharacteristic
38C2×C4×Dic163C2211yescharacteristic
39C2×C4×Dic163C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C4⋊C4×C163C2211yescharacteristic
50C2×C42×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C23×D163C2311yescharacteristic
54C22×Dic163C2311yescharacteristic
55C22×Dic163C2311yescharacteristic
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C2×C4×D163—22no—
62C2×C4×D163—22no—
63C2×C4×D163Q814yes—
64C4⋊Dic163—22no—
65C4⋊Dic163—41no—
66Dic163⋊C4—22no—
67Dic163⋊C4—22no—
68Dic163⋊C4—22no—
69Dic163⋊C4—22no—
70Dic163⋊C4—22no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—41no—
75Dic163⋊C4D414yes—
76(C2×D163)⋊C4—22no—
77(C2×D163)⋊C4—22no—
78(C2×D163)⋊C4—22no—
79(C2×D163)⋊C4—22no—
80(C2×D163)⋊C4—41no—
81(C2×D163)⋊C4—41no—
82C4×Dic163—22no—
83C4×Dic163—22no—
84C4×Dic163—22no—
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C4⋊C4×C163—22no—
93C4⋊C4×C163—22no—
94C4⋊C4×C163—41no—
95C4⋊C4×C163—41no—
96C42×C163—22no—
97C22×D163—21no—
98C22×D163—21no—
99C22×D163—21no—
100C22×D163—21no—
101C22×D163—21no—
102C22×D163—21no—
103C22×D163C2×Q812yes—
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—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163C2×D412yes—
121C2×Dic163C2×D412yes—
122C2×Dic163C2×Q812yes—
123C2×Dic163#4112yes—
124C2×Dic163#4112yes—
125C4×D163—28no—
126C23×C163C2411yescharacteristic
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—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D412yes—
146C2×C4×C163C2×Q812yes—
147C2×C4×C163#4112yes—
148C22×C163C22×D411yescharacteristic
149C22×C163C22×Q811yescharacteristic
150C22×C163C2×#4111yescharacteristic
151C22×C163C2×#4111yescharacteristic
152C22×C163C2×#4111yescharacteristic
153C22×C163H(5,𝔽2)11yescharacteristic
154C22×C1632−1+411yescharacteristic
155C2×D163—22no—
156C2×D163—22no—
157C2×D163—22no—
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C4×C163—22no—
163C4×C163—22no—
164C4×C163—22no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163—41no—
169Dic163—22no—
170Dic163—22no—
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174Dic163—41no—
175Dic163C22⋊Q814yes—
176C2×C163C2×C22⋊Q811yescharacteristic
177C2×C163#53311yescharacteristic
178C2×C163#54411yescharacteristic
179C2×C163#54711yescharacteristic
180C2×C163#55011yescharacteristic
181C2×C163#55211yescharacteristic
182C2×C163#55311yescharacteristic
183D163—42no—
184C163#244111yescharacteristic, complemented, Sylow
185#2441—1631nomaximal, Sylow
186C2×#125—1631no—
187C2×#117—1631no—
188#399—1631no—
189#399—1631no—
190#396—1631no—
191#396—1631no—
192#389—1631no—
193#389—1631no—
194#389—1631no—
195#389D16311yescharacteristic, complemented
196#387—1631no—
197#387—1631no—
198#384—1631no—
199#380—1631no—
200C4×C22⋊C4—1631no—
201C23×C4—1631no—
202#125—3264no—
203#117—3264no—
204C2×C4⋊C4—1631no—
205C2×C4⋊C4—1631no—
206C2×C4⋊C4—1631no—
207C2×C4⋊C4—1631no—
208C2×C4⋊C4—1631no—
209C2×C4⋊C4—1631no—
210C2×C4⋊C4—1631no—
211C2×C4⋊C4—1631no—
212C2×C4⋊C4—1631no—
213C2×C4⋊C4—1631no—
214C2×C4⋊C4—1631no—
215C2×C4⋊C4—1631no—
216C2×C4⋊C4C2×D16311yescharacteristic
217C2×C4⋊C4C2×D16311yescharacteristic
218C2×C4⋊C4C2×D16311yescharacteristic
219C2×C4⋊C4C2×D16311yescharacteristic
220C2×C22⋊C4—1631no—
221C2×C22⋊C4—1631no—
222C2×C22⋊C4—1631no—
223C2×C22⋊C4—1631no—
224C2×C22⋊C4—1631no—
225C2×C22⋊C4—1631no—
226C2×C42—1631no—
227C2×C42—1631no—
228C2×C42—1631no—
229C2×C42C2×D16311yescharacteristic
230#95—1631no—
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95C2×D16311yescharacteristic
237#95C2×D16311yescharacteristic
238C24—1631no—
239C22×C4—3262no—
240C22×C4—3262no—
241C22×C4—1631no—
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1634no—
248C22×C4—1631no—
249C22×C4C22×D16311yescharacteristic
250C22×C4C22×D16311yescharacteristic
251C22×C4C22×D16311yescharacteristic
252C22×C4C22×D16311yescharacteristic
253C22×C4C22×D16311yescharacteristic
254C22×C4C22×D16311yescharacteristic
255C22×C4C22×D16311yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—3262no—
261C4⋊C4—3262no—
262C4⋊C4—3262no—
263C4⋊C4—3262no—
264C4⋊C4—3262no—
265C4⋊C4—3262no—
266C4⋊C4—3262no—
267C4⋊C4—3262no—
268C4⋊C4—3262no—
269C4⋊C4—6521no—
270C4⋊C4—6521no—
271C4⋊C4—1634no—
272C22⋊C4—3262no—
273C22⋊C4—3262no—
274C22⋊C4—3262no—
275C22⋊C4—3262no—
276C22⋊C4—6521no—
277C22⋊C4—6521no—
278C42—22no—
279C42—3262no—
280C42—3262no—
281C42—3262no—
282C23—3261no—
283C23—3261no—
284C23—3261no—
285C23—3261no—
286C23—3261no—
287C23—3261no—
288C23—1632no—
289C23C23×D16311yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
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—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—1632no—
326C2×C4—1632no—
327C2×C4—1632no—
328C2×C4—1632no—
329C2×C4—1632no—
330C2×C4D4×D16312yes—
331C2×C4Q8×D16312yes—
332C2×C4#1457969912yes—
333C22—3262no—
334C22—3262no—
335C22—3262no—
336C22—6521no—
337C22—6521no—
338C22—6521no—
339C22—6521no—
340C22C2×D4×D16311yescharacteristic
341C22C2×Q8×D16311yescharacteristic
342C22#41×D16311yescharacteristic
343C22#41×D16311yescharacteristic
344C22C2×#1457969911yescharacteristic
345C22#1910478011yescharacteristic
346C22#1910478311yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3262no—
355C4—3262no—
356C4—6521no—
357C4—6521no—
358C4—6521no—
359C4—6521no—
360C4—1634no—
361C2—6522no—
362C2C22⋊Q8×D16311yescharacteristic
363C2#2378221911yescharacteristic
364C2#2378229211yescharacteristic
365C2#2378231611yescharacteristic
366C2#2378233311yescharacteristic
367C2#2378234711yescharacteristic
368C2#2378236411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#389×C163C2
prime radicalC23×D163C23
hypercenter#389D163
hypocenterC163#2441
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#19104725C2semidirect
7. #23781562C2semidirect
8. #23781560C2semidirect
9. #23781560C2semidirect
10. #23781552C2semidirect
11. #23781549C2semidirect
12. #23781534C2semidirect
15. #389×C163C2semidirect
184. C163#2441semidirect
195. #389D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047252
C2×#191046922
#237816002
#237815872
#237815622
#237815602
#237815522
#237815492
#237815342
#237815242
#237815122
#389×C1632
#2441163