Finite Group Databasebeta

← #35217475

Subgroups — #35217475

Order 20864.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104697C211yesmaximal, characteristic
2#23781557C211yesmaximal, characteristic, complemented
3#23781555C211yesmaximal, characteristic, complemented
4#23781551C211yesmaximal, characteristic, complemented
5#23781543C211yesmaximal, characteristic
6#23781543C211yesmaximal, characteristic
7#23781532C211yesmaximal, characteristic, complemented
8#23781527C211yesmaximal, characteristic, complemented
9#23781522C211yesmaximal, characteristic, complemented
10C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
11C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
12#23781512C211yesmaximal, characteristic, complemented
13#23781510C211yesmaximal, characteristic, complemented
14#95×D163C211yesmaximal, characteristic, complemented
15#387×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104697C418yescomplemented
22C2×C4⋊Dic163C2211yescharacteristic
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×(C2×D163)⋊C4C2211yescharacteristic
31C2×C4×Dic163C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C2×C4×Dic163C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C2×C4⋊C4×C163C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C42×C163C2211yescharacteristic
48#95×C163C2211yescharacteristic
49#95×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52C23×D163C2311yescharacteristic
53C23×D163C2311yescharacteristic
54C2×D652—22no—
55C2×D652—22no—
56C2×D652C2×C414yes—
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C2×C4×D163—22no—
64C2×C4×D163—22no—
65C2×C4×D163—22no—
66C2×C4×D163—22no—
67C2×C4×D163—41no—
68C2×C4×D163—41no—
69C2×C4×D163—41no—
70C2×C4×D163—41no—
71C2×C4×D163—41no—
72C2×C4×D163—41no—
73C2×C4×D163C2×C414yes—
74C2×C4×D163C2×C414yescomplemented
75C4⋊Dic163—41no—
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—
82(C2×D163)⋊C4C2×C414yes—
83(C2×D163)⋊C4C2×C414yescomplemented
84C4×Dic163—22no—
85C4×Dic163—22no—
86C4×Dic163—41no—
87C4×Dic163C2×C414yescomplemented
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—
96C4⋊C4×C163C2×C414yescomplemented
97C42×C163—41no—
98C22×D163—21no—
99C22×D163—21no—
100C22×D163—21no—
101C22×D163—21no—
102C22×D163—21no—
103C22×D163—21no—
104C22×D163—21no—
105C22×D163—21no—
106C22×D163—21no—
107C22×D163—21no—
108C22×D163C22×C412yes—
109C22×D163C22×C412yes—
110C22×D163#4112yes—
111C22×D163#4112yes—
112D652—28no—
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×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163C22×C412yes—
129C2×Dic163C22×C412yes—
130C2×Dic163#4112yes—
131C4×D163—44no—
132C4×D163—44no—
133C4×D163—44no—
134C4×D163—44no—
135C23×C163C2411yescharacteristic
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×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163C22×C412yes—
154C2×C4×C163C22×C412yes—
155C2×C4×C163C22×C412yes—
156C2×C4×C163#4112yes—
157C22×C163C23×C411yescharacteristic
158C22×C163C2×#4111yescharacteristic
159C22×C163C2×#4111yescharacteristic
160C22×C163C2×#4111yescharacteristic
161C22×C163C2×#4111yescharacteristic
162C22×C163H(5,𝔽2)11yescharacteristic
163C22×C163H(5,𝔽2)11yescharacteristic
164C2×D163—22no—
165C2×D163—22no—
166C2×D163—22no—
167C2×D163—22no—
168C2×D163—22no—
169C2×D163—22no—
170C2×D163—22no—
171C2×D163—22no—
172C2×D163—22no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—41no—
177C2×D163#11714yes—
178C4×C163—22no—
179C4×C163—22no—
180C4×C163—22no—
181C4×C163—22no—
182C4×C163—41no—
183C4×C163—41no—
184C4×C163—41no—
185Dic163—22no—
186Dic163—22no—
187Dic163—22no—
188Dic163—41no—
189Dic163—41no—
190Dic163—41no—
191C2×C163C2×#11711yescharacteristic
192C2×C163C4×#4111yescharacteristic
193C2×C163#51811yescharacteristic
194C2×C163#55111yescharacteristic
195C2×C163#55111yescharacteristic
196C2×C163#55311yescharacteristic
197C2×C163#55311yescharacteristic
198D163—24no—
199D163—42no—
200C163#235311yescharacteristic, complemented, Sylow
201#2353—1631nomaximal, Sylow
202C2×C4×D4—1631no—
203#388—1631no—
204#388—1631no—
205#387—1631no—
206#387D16311yescharacteristic, complemented
207#386—1631no—
208#386—1631no—
209#383—1631no—
210#381—1631no—
211#380—1631no—
212C4×C22⋊C4—1631no—
213C4×C22⋊C4—1631no—
214C4×C22⋊C4—1631no—
215#376—1631no—
216C2×#95—1631no—
217C22×D4—1631no—
218C23×C4—1631no—
219C23×C4—1631no—
220C23×C4—1631no—
221C23×C4—1631no—
222C4×D4—1638no—
223C2×C4⋊C4—1631no—
224C2×C4⋊C4C2×D16311yescharacteristic
225C2×C4⋊C4C2×D16311yescharacteristic
226C2×C22⋊C4—1631no—
227C2×C22⋊C4—1631no—
228C2×C22⋊C4—1631no—
229C2×C22⋊C4—1631no—
230C2×C22⋊C4—1631no—
231C2×C22⋊C4—1631no—
232C2×C22⋊C4—1631no—
233C2×C22⋊C4—1631no—
234C2×C42—1631no—
235C2×C42—1631no—
236C2×C42—1631no—
237C2×C42—1631no—
238C2×C42C2×D16311yescharacteristic
239#95—1631no—
240#95—1631no—
241#95—1631no—
242#95—1631no—
243#95—1631no—
244#95—1631no—
245#95—1631no—
246#95—1631no—
247#95—1631no—
248#95—1631no—
249#95C2×D16311yescharacteristic
250#95C2×D16311yescharacteristic
251#95C2×D16311yescharacteristic
252#95C2×D16311yescharacteristic
253C24—1631no—
254C24—1631no—
255C2×D4—3262no—
256C2×D4—3262no—
257C2×D4—1634no—
258C22×C4—3262no—
259C22×C4—3262no—
260C22×C4—3262no—
261C22×C4—3262no—
262C22×C4—6521no—
263C22×C4—6521no—
264C22×C4—6521no—
265C22×C4—6521no—
266C22×C4—6521no—
267C22×C4—6521no—
268C22×C4—1631no—
269C22×C4—1631no—
270C22×C4—1631no—
271C22×C4—1631no—
272C22×C4—1631no—
273C22×C4—1631no—
274C22×C4—1634no—
275C22×C4—1634no—
276C22×C4C22×D16311yescharacteristic
277C22×C4C22×D16311yescharacteristic
278C22×C4C22×D16311yescharacteristic
279C22×C4C22×D16311yescharacteristic
280C22×C4C22×D16311yescharacteristic
281C22×C4C22×D16311yescharacteristic
282C22×C4C22×D16311yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—6521no—
285C4⋊C4C4×D16314yescomplemented
286C22⋊C4—3262no—
287C22⋊C4—3262no—
288C22⋊C4—3262no—
289C22⋊C4—3262no—
290C22⋊C4—6521no—
291C22⋊C4—6521no—
292C22⋊C4—1634no—
293C22⋊C4—1634no—
294C42—41no—
295C42—3262no—
296C42—3262no—
297C42—6521no—
298C42—1634no—
299C23—3261no—
300C23—3261no—
301C23—3261no—
302C23—3261no—
303C23—3261no—
304C23—3261no—
305C23—3261no—
306C23—3261no—
307C23—3261no—
308C23—3261no—
309C23—1632no—
310C23—1632no—
311C23—1632no—
312C23—1632no—
313C23C23×D16311yescharacteristic
314D4—3268no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
341C2×C4—3261no—
342C2×C4—3261no—
343C2×C4—3261no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—6524no—
348C2×C4—6524no—
349C2×C4—6524no—
350C2×C4—6524no—
351C2×C4—1632no—
352C2×C4—1632no—
353C2×C4—1632no—
354C2×C4C2×C4×D16312yes—
355C2×C4C2×C4×D16312yes—
356C2×C4C2×C4×D16312yes—
357C2×C4#1457970012yes—
358C22—3262no—
359C22—3262no—
360C22—3262no—
361C22—3262no—
362C22—3262no—
363C22—3262no—
364C22—3262no—
365C22—3262no—
366C22—3262no—
367C22—6521no—
368C22—6521no—
369C22—6521no—
370C22—6521no—
371C22—1634no—
372C22C22×C4×D16311yescharacteristic
373C22#41×D16311yescharacteristic
374C22#41×D16311yescharacteristic
375C22#41×D16311yescharacteristic
376C22C2×#1457970011yescharacteristic
377C22#1910478011yescharacteristic
378C22#1910478111yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—3262no—
387C4—3262no—
388C4—3262no—
389C4—6521no—
390C4—6521no—
391C4—6521no—
392C2—3264no—
393C2—6522no—
394C2#117×D16311yescharacteristic
395C2C4×#1457970011yescharacteristic
396C2#2378213711yescharacteristic
397C2#2378234011yescharacteristic
398C2#2378234211yescharacteristic
399C2#2378236011yescharacteristic
400C2#2378236811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#387×C163C2
socle, cosocleC23×C163C24
prime radicalC22×D652C22
hypercenter#387D163
hypocenterC163#2353
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #23781557C2semidirect
3. #23781555C2semidirect
4. #23781551C2semidirect
7. #23781532C2semidirect
8. #23781527C2semidirect
9. #23781522C2semidirect
10. C4×(C2×D163)⋊C4C2semidirect
11. C4×(C2×D163)⋊C4C2semidirect
12. #23781512C2semidirect
13. #23781510C2semidirect
14. #95×D163C2semidirect
15. #387×C163C2semidirect
21. #19104697C4semidirect
74. C2×C4×D163C2×C4semidirect
83. (C2×D163)⋊C4C2×C4semidirect
87. C4×Dic163C2×C4semidirect
96. C4⋊C4×C163C2×C4semidirect
200. C163#2353semidirect
206. #387D163semidirect
285. C4⋊C4C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191046972
#237815572
#237815552
#237815512
#237815432
#237815322
#237815272
#237815222
C4×(C2×D163)⋊C42
#237815122
#237815102
#95×D1632
#387×C1632
#2353163