Finite Group Databasebeta

← #35222849

Subgroups — #35222849

Order 20864.

582 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×#19104728C211yesmaximal, characteristic, complemented
2C22×C4×Dic163C211yesmaximal, characteristic, complemented
3#23781559C211yesmaximal, characteristic, complemented
4#23781558C211yesmaximal, characteristic, complemented
5#23781553C211yesmaximal, characteristic, complemented
6#23781552C211yesmaximal, characteristic, complemented
7#23781550C211yesmaximal, characteristic, complemented
8#23781547C211yesmaximal, characteristic, complemented
9#23781545C211yesmaximal, characteristic
10#23781545C211yesmaximal, characteristic
11#23781540C211yesmaximal, characteristic, complemented
12#23781538C211yesmaximal, characteristic, complemented
13#23781537C211yesmaximal, characteristic, complemented
14#23781532C211yesmaximal, characteristic, complemented
15#386×C163C211yesmaximal, characteristic
16C23×Dic163C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104728—24no—
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×C22⋊Dic163C2211yescharacteristic
26C2×C22⋊Dic163C2211yescharacteristic
27C2×C22⋊Dic163C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×C22⋊Dic163C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C4×Dic163—24no—
32C2×C4×Dic163C2211yescharacteristic
33C2×C4×Dic163C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C22×D4×C163C2211yescharacteristic
47C23×C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C22×Dic163—22no—
54C22×Dic163—22no—
55C22×Dic163—22no—
56C22×Dic163—22no—
57C22×Dic163—22no—
58C22×Dic163—22no—
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C4⋊Dic163—22no—
68C4⋊Dic163—22no—
69Dic163⋊C4—22no—
70Dic163⋊C4—22no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73C22⋊Dic163—22no—
74C22⋊Dic163—22no—
75C22⋊Dic163—22no—
76C22⋊Dic163—22no—
77C22⋊Dic163—22no—
78C22⋊Dic163—22no—
79C4×Dic163—22no—
80C4×Dic163—22no—
81C4×Dic163—22no—
82C4×Dic163D414yescomplemented
83C24×C163C2311yescharacteristic
84C24×C163C2311yescharacteristic
85C2×D4×C163—22no—
86C2×D4×C163—22no—
87C2×D4×C163—22no—
88C22×C4×C163—22no—
89C22×C4×C163—22no—
90C22×C4×C163—22no—
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C22×C4×C163C2311yescharacteristic
94C22×C4×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96C22⋊C4×C163—22no—
97C22⋊C4×C163—22no—
98C22⋊C4×C163—22no—
99C2×Dic163—28no—
100C2×Dic163—28no—
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—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163C2×D412yes—
122C2×Dic163C2×D412yes—
123C2×Dic163#4112yes—
124C2×Dic163#4112yes—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163—21no—
128C23×C163—21no—
129C23×C163—21no—
130C23×C163—21no—
131C23×C163—21no—
132C23×C163—21no—
133C23×C163—21no—
134C23×C163—21no—
135C23×C163—21no—
136C23×C163#4112yes—
137C23×C163#4112yes—
138C23×C163#4112yes—
139C23×C163C2411yescharacteristic
140D4×C163—44no—
141C2×C4×C163—28no—
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×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163C2×D412yes—
156C2×C4×C163#4112yes—
157C22×C163—22no—
158C22×C163—22no—
159C22×C163—22no—
160C22×C163—22no—
161C22×C163—22no—
162C22×C163—22no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163#12614yescomplemented
171C22×C163C22×D411yescharacteristic
172C22×C163C2×#4111yescharacteristic
173C22×C163C2×#4111yescharacteristic
174C22×C163C2×#4111yescharacteristic
175C22×C163C2×#4111yescharacteristic
176C22×C163C2×#4111yescharacteristic
177C22×C163C2×#4111yescharacteristic
178C4×C163—22no—
179C4×C163—22no—
180C4×C163—41no—
181C4×C163—41no—
182C4×C163—41no—
183Dic163—22no—
184Dic163—22no—
185Dic163—22no—
186Dic163—22no—
187Dic163—41no—
188Dic163—41no—
189Dic163—41no—
190Dic163—41no—
191C2×C163—24no—
192C2×C163—81no—
193C2×C163#52511yescharacteristic
194C2×C163#52511yescharacteristic
195C2×C163#52511yescharacteristic
196C2×C163C2×#12611yescharacteristic
197C2×C163#52911yescharacteristic
198C2×C163#52911yescharacteristic
199C2×C163#52911yescharacteristic
200C163#285211yescharacteristic, complemented, Sylow
201#2852—1631nomaximal, Sylow
202C2×#126—1631no—
203C22×C42—1631no—
204#388—1631no—
205#388—1631no—
206#388—1631no—
207#387—1631no—
208#387—1631no—
209#387—1631no—
210#386—1631no—
211#386—1631no—
212#386D16311yescharacteristic
213#385—1631no—
214#385—1631no—
215#385—1631no—
216#383—1631no—
217C22×D4C2×D16311yescharacteristic
218C23×C4—1631no—
219C23×C4—1631no—
220C23×C4C2×D16311yescharacteristic
221#126—3264no—
222C2×C4⋊C4—1631no—
223C2×C4⋊C4—1631no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4—1631no—
226C2×C4⋊C4—1631no—
227C2×C4⋊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×C22⋊C4C2×D16311yescharacteristic
235C2×C22⋊C4C2×D16311yescharacteristic
236C2×C22⋊C4C2×D16311yescharacteristic
237C2×C42—3264no—
238C2×C42—1631no—
239C2×C42—1631no—
240C2×C42—1631no—
241C2×C42—1631no—
242#95—1631no—
243#95—1631no—
244#95—1631no—
245#95—1631no—
246#95—1631no—
247#95—1631no—
248#95—1631no—
249#95—1631no—
250#95—1631no—
251#95—1631no—
252#95C2×D16311yescharacteristic
253#95C2×D16311yescharacteristic
254C24C22×D16311yescharacteristic
255C24C22×D16311yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—3262no—
263C22×C4—3262no—
264C22×C4—3262no—
265C22×C4—3262no—
266C22×C4—3262no—
267C22×C4—3262no—
268C22×C4—1631no—
269C22×C4—1631no—
270C22×C4—1631no—
271C22×C4—1631no—
272C22×C4—1631no—
273C22×C4—1631no—
274C22×C4—1631no—
275C22×C4—1631no—
276C22×C4C22×D16311yescharacteristic
277C22×C4C22×D16311yescharacteristic
278C22×C4C22×D16311yescharacteristic
279C22×C4C22×D16311yescharacteristic
280C22×C4C22×D16311yescharacteristic
281C4⋊C4—3262no—
282C4⋊C4—3262no—
283C4⋊C4—3262no—
284C4⋊C4—3262no—
285C4⋊C4—3262no—
286C4⋊C4—3262no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—3262no—
291C22⋊C4—3262no—
292C22⋊C4—3262no—
293C22⋊C4—3262no—
294C22⋊C4—3262no—
295C22⋊C4—3262no—
296C42—3262no—
297C42—3262no—
298C42—3262no—
299C42—1634no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23#1457969812yes—
312C23#1457969812yes—
313C23#1457969912yes—
314C23C23×D16311yescharacteristic
315D4—44no—
316C2×C4—28no—
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—3268no—
331C2×C4—3268no—
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—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—1632no—
353C2×C4—1632no—
354C2×C4—1632no—
355C2×C4—1632no—
356C2×C4D4×D16312yes—
357C2×C4#1457969812yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#1910472814yescomplemented
372C22C2×D4×D16311yescharacteristic
373C22#41×D16311yescharacteristic
374C22#41×D16311yescharacteristic
375C22C2×#1457969811yescharacteristic
376C22C2×#1457969811yescharacteristic
377C22C2×#1457969811yescharacteristic
378C22C2×#1457969911yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3262no—
385C4—3262no—
386C4—3262no—
387C4—3262no—
388C4—6521no—
389C4—6521no—
390C4—6521no—
391C4—6521no—
392C2—24no—
393C2—81no—
394C2#2378217111yescharacteristic
395C2#2378217411yescharacteristic
396C2#2378217511yescharacteristic
397C2C2×#1910472811yescharacteristic
398C2#2378219211yescharacteristic
399C2#2378219511yescharacteristic
400C2#2378219611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#386×C163C2
prime radicalC22×D4×C163C22
hypercenter#386D163
hypocenterC163#2852
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#19104728C2semidirect
2. C22×C4×Dic163C2semidirect
3. #23781559C2semidirect
4. #23781558C2semidirect
5. #23781553C2semidirect
6. #23781552C2semidirect
7. #23781550C2semidirect
8. #23781547C2semidirect
11. #23781540C2semidirect
12. #23781538C2semidirect
13. #23781537C2semidirect
14. #23781532C2semidirect
82. C4×Dic163D4semidirect
170. C22×C163#126semidirect
200. C163#2852semidirect
371. C22#19104728semidirect

Maximal subgroups

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

subgroupindex
C2×#191047282
C22×C4×Dic1632
#237815592
#237815582
#237815532
#237815522
#237815502
#237815472
#237815452
#237815402
#237815382
#237815372
#237815322
#386×C1632
#2852163