Finite Group Databasebeta

← #35218224

Subgroups — #35218224

Order 20864.

508 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104733C211yesmaximal, characteristic, complemented
2C2×#19104692C211yesmaximal, characteristic
3#23781589C211yesmaximal, characteristic
4#23781587C211yesmaximal, characteristic
5#23781578C211yesmaximal, characteristic
6#23781576C211yesmaximal, characteristic
7#23781569C211yesmaximal, characteristic, complemented
8#23781562C211yesmaximal, characteristic, complemented
9#23781560C211yesmaximal, characteristic, complemented
10#23781549C211yesmaximal, characteristic, complemented
11#23781548C211yesmaximal, characteristic, complemented
12#23781529C211yesmaximal, characteristic, complemented
13#23781524C211yesmaximal, characteristic
14#23781512C211yesmaximal, characteristic
15#391×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18#19104733—24no—
19#19104692—24no—
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×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C2×C4×Dic163C2211yescharacteristic
37C2×C4×Dic163C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C22×Q8×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C42×C163C2211yescharacteristic
49#95×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C23×D163C2311yescharacteristic
54C2×Dic326—22no—
55C2×Dic326—22no—
56C2×Dic326—41no—
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C2×C4×D163—22no—
65C2×C4×D163—22no—
66C2×C4×D163Q814yescomplemented
67C4⋊Dic163—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—
85C2×Q8×C163—22no—
86C2×Q8×C163—22no—
87C2×Q8×C163—41no—
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C22×C4×C163C2311yescharacteristic
94C22×C4×C163C2311yescharacteristic
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—
104Dic326—44no—
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×Dic163—21no—
121C2×Dic163C2×D412yes—
122C2×Dic163C2×D412yes—
123C2×Dic163C2×Q812yes—
124C2×Dic163#4112yes—
125C2×Dic163#4112yes—
126C4×D163—28no—
127C23×C163C2411yescharacteristic
128Q8×C163—44no—
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×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163C2×D412yes—
148C2×C4×C163C2×Q812yes—
149C2×C4×C163#4112yes—
150C22×C163C22×D411yescharacteristic
151C22×C163C22×Q811yescharacteristic
152C22×C163C2×#4111yescharacteristic
153C22×C163C2×#4111yescharacteristic
154C22×C163C2×#4111yescharacteristic
155C22×C163H(5,𝔽2)11yescharacteristic
156C22×C1632−1+411yescharacteristic
157C2×D163—22no—
158C2×D163—22no—
159C2×D163—22no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C4×C163—22no—
165C4×C163—22no—
166C4×C163—22no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163—41no—
171Dic163—22no—
172Dic163—22no—
173Dic163—41no—
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163C22⋊Q814yes—
178C2×C163C2×C22⋊Q811yescharacteristic
179C2×C163#53311yescharacteristic
180C2×C163#54411yescharacteristic
181C2×C163#54611yescharacteristic
182C2×C163#54811yescharacteristic
183C2×C163#55511yescharacteristic
184C2×C163#55611yescharacteristic
185D163—42no—
186C163#243911yescharacteristic, complemented, Sylow
187#2439—1631nomaximal, Sylow
188C2×C4⋊Q8—1631no—
189C2×#117—1631no—
190#396—1631no—
191#396—1631no—
192#393—1631no—
193#393—1631no—
194#391—1631no—
195#391D16311yescharacteristic, complemented
196#389—1631no—
197#389—1631no—
198#387—1631no—
199#387—1631no—
200#382—1631no—
201#380—1631no—
202C4×C22⋊C4—1631no—
203C22×Q8—1631no—
204C22×Q8C2×D16311yescharacteristic
205C23×C4—1631no—
206C4⋊Q8—3264no—
207#117—3264no—
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⋊C4—1631no—
217C2×C4⋊C4C2×D16311yescharacteristic
218C2×C22⋊C4—1631no—
219C2×C22⋊C4—1631no—
220C2×C22⋊C4—1631no—
221C2×C22⋊C4—1631no—
222C2×C22⋊C4—1631no—
223C2×C22⋊C4—1631no—
224C2×C42—1631no—
225C2×C42—1631no—
226C2×C42—1631no—
227C2×C42C2×D16311yescharacteristic
228#95—1631no—
229#95—1631no—
230#95—1631no—
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95C2×D16311yescharacteristic
237#95C2×D16311yescharacteristic
238#95C2×D16311yescharacteristic
239#95C2×D16311yescharacteristic
240C24—1631no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—3262no—
245C2×Q8—3262no—
246C2×Q8—6521no—
247C22×C4—3262no—
248C22×C4—3262no—
249C22×C4—1631no—
250C22×C4—1631no—
251C22×C4—1631no—
252C22×C4—1631no—
253C22×C4—1631no—
254C22×C4—1631no—
255C22×C4—1634no—
256C22×C4—1631no—
257C22×C4C22×D16311yescharacteristic
258C22×C4C22×D16311yescharacteristic
259C22×C4C22×D16311yescharacteristic
260C22×C4C22×D16311yescharacteristic
261C22×C4C22×D16311yescharacteristic
262C22×C4C22×D16311yescharacteristic
263C22×C4C22×D16311yescharacteristic
264C4⋊C4—41no—
265C4⋊C4—3262no—
266C4⋊C4—3262no—
267C4⋊C4—3262no—
268C4⋊C4—3262no—
269C4⋊C4—3262no—
270C4⋊C4—3262no—
271C4⋊C4—3262no—
272C4⋊C4—6521no—
273C4⋊C4—1634no—
274C22⋊C4—3262no—
275C22⋊C4—3262no—
276C22⋊C4—3262no—
277C22⋊C4—3262no—
278C22⋊C4—6521no—
279C22⋊C4—6521no—
280C42—22no—
281C42—3262no—
282C42—3262no—
283C42—3262no—
284C23—3261no—
285C23—3261no—
286C23—3261no—
287C23—3261no—
288C23—3261no—
289C23—3261no—
290C23—1632no—
291C23C23×D16311yescharacteristic
292Q8—44no—
293Q8—6524no—
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—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3268no—
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—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—1632no—
330C2×C4—1632no—
331C2×C4—1632no—
332C2×C4—1632no—
333C2×C4—1632no—
334C2×C4D4×D16312yes—
335C2×C4Q8×D16312yes—
336C2×C4#1457969912yes—
337C22—3262no—
338C22—3262no—
339C22—3262no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22—6521no—
344C22C2×D4×D16311yescharacteristic
345C22C2×Q8×D16311yescharacteristic
346C22#41×D16311yescharacteristic
347C22#41×D16311yescharacteristic
348C22C2×#1457969911yescharacteristic
349C22#1910478011yescharacteristic
350C22#1910478311yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3262no—
359C4—3262no—
360C4—6521no—
361C4—6521no—
362C4—6521no—
363C4—6521no—
364C4—1634no—
365C2—6522no—
366C2C22⋊Q8×D16311yescharacteristic
367C2#2378221911yescharacteristic
368C2#2378229211yescharacteristic
369C2#2378230511yescharacteristic
370C2#2378232511yescharacteristic
371C2#2378237711yescharacteristic
372C2#2378238711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#391×C163C2
prime radicalC23×D163C23
hypercenter#391D163
hypocenterC163#2439
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#19104733C2semidirect
7. #23781569C2semidirect
8. #23781562C2semidirect
9. #23781560C2semidirect
10. #23781549C2semidirect
11. #23781548C2semidirect
12. #23781529C2semidirect
15. #391×C163C2semidirect
66. C2×C4×D163Q8semidirect
186. C163#2439semidirect
195. #391D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047332
C2×#191046922
#237815892
#237815872
#237815782
#237815762
#237815692
#237815622
#237815602
#237815492
#237815482
#237815292
#237815242
#237815122
#391×C1632
#2439163