Finite Group Databasebeta

← #35226528

Subgroups — #35226528

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782322C211yesmaximal, characteristic, complemented
2#23782316C211yesmaximal, characteristic, complemented
3#23782023C211yesmaximal, characteristic, complemented
4#23782000C211yesmaximal, characteristic, complemented
5#23781977C211yesmaximal, characteristic, complemented
6#23781958C211yesmaximal, characteristic, complemented
7#23781911C211yesmaximal, characteristic, complemented
8#23781900C211yesmaximal, characteristic, complemented
9#23781879C211yesmaximal, characteristic, complemented
10#23781810C211yesmaximal, characteristic, complemented
11#23781805C211yesmaximal, characteristic, complemented
12#23781768C211yesmaximal, characteristic, complemented
13#23781734C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15#417×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×Q8×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C2×C8×D163C2211yescharacteristic
22#19104734C2211yescharacteristic
23#19104722—21no—
24#19104722C2211yescharacteristic
25#19104720—21no—
26#19104718—21no—
27#19104714—21no—
28#19104714C2211yescharacteristic
29#19104713—21no—
30#19104710C2211yescharacteristic
31#19104709C2211yescharacteristic
32#19104706—21no—
33#19104706C2211yescharacteristic
34Q8×Dic163C2211yescharacteristic
35#19104697C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104668C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C2×#41×C163C2211yescharacteristic
53C2×M16×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—41no—
65C2×D652—21no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163D411yescharacteristic
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579688—41no—
75#14579686—22no—
76#14579684—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C8×D163—41no—
80C4⋊Dic163—21no—
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163D411yescharacteristic
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98#41×C163—22no—
99#41×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×D4×C163—21no—
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163—21no—
105C22×C4×C163C2311yescharacteristic
106M16×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163—21no—
110C4⋊C4×C163C2311yescharacteristic
111C4⋊C4×C163C2311yescharacteristic
112C22×D163—21no—
113C22×D163C2×D411yescharacteristic
114Dic326—21no—
115Dic326—21no—
116Dic326—42no—
117Dic326—42no—
118D652—21no—
119D652—21no—
120D652—42no—
121C22⋊D163—22no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—42no—
133C4×D163—42no—
134C4×D163—42no—
135C163⋊C8—21no—
136C163⋊C8—21no—
137C23×C163—21no—
138C23×C163C2×D411yescharacteristic
139Q8×C163—21no—
140Q8×C163#4112yescomplemented
141D4×C163—21no—
142D4×C163—21no—
143D4×C163—42no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—42no—
150C2×C4×C163—42no—
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163C22×D411yescharacteristic
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163AGL(1,ℤ/8)12yes—
168C4×C163—21no—
169C4×C163—22no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—21no—
175Dic163—21no—
176Dic163—21no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—41no—
181C2×C163—81no—
182C2×C163#54611yescharacteristic
183C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C163#57811yescharacteristic
185D163—22no—
186D163—81no—
187C163#330811yescharacteristic, complemented, Sylow
188#3308—1631nomaximal, Sylow
189#548—1631no—
190#547—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#477—1631no—
195#468—1631no—
196#466—1631no—
197#463—1631no—
198#449—1631no—
199#448—1631no—
200#440—1631no—
201#435—1631no—
202#417D16311yescharacteristic, complemented
203C2×#103—1631no—
204C2×#41—3261no—
205C2×#41C2×D16311yescharacteristic
206C22×Q8—1631no—
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×D8—1631no—
210C2×M16C2×D16311yescharacteristic
211C22×C8—1631no—
212C4⋊Q8—1631no—
213#124—3261no—
214#124—1631no—
215#123—3261no—
216#123—3261no—
217C22⋊Q8—3261no—
218C22⋊Q8—3261no—
219C22⋊Q8—1631no—
220#121—3261no—
221#121—1631no—
222#121—1631no—
223#121—1631no—
224C4×Q8—1631no—
225C4×D4—3261no—
226C4×D4—1631no—
227C4×D4—1631no—
228C2×C4⋊C4—1631no—
229C2×C4⋊C4C2×D16311yescharacteristic
230#107—1631no—
231#107—1631no—
232#107—1631no—
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103—1631no—
238#103C2×D16311yescharacteristic
239#103C2×D16311yescharacteristic
240#102—1631no—
241#102—1631no—
242#102C2×D16311yescharacteristic
243#102C2×D16311yescharacteristic
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247#41—22no—
248#41—41no—
249#41—6524no—
250C2×Q8—3261no—
251C2×Q8—3261no—
252C2×Q8—3262no—
253C2×Q8—6521no—
254C2×Q8—1631no—
255C2×Q8C22×D16311yescharacteristic
256C2×D4—21no—
257C2×D4—3261no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—1631no—
261C2×D4—1631no—
262C2×D4C22×D16311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3261no—
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—1631no—
269C22×C4C22×D16311yescharacteristic
270SD16—3262no—
271SD16—6521no—
272D8—6521no—
273M16—41no—
274C2×C8—6521no—
275C2×C8—1631no—
276C2×C8—1631no—
277C2×C8C22×D16311yescharacteristic
278C2×C8C22×D16311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4C22×D16311yescharacteristic
288C4⋊C4C22×D16311yescharacteristic
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—1631no—
297C22⋊C4—1631no—
298C42—3261no—
299C42—1631no—
300C23—21no—
301C23—3261no—
302C23—1631no—
303C23D4×D16311yescharacteristic
304Q8—21no—
305Q8—3261no—
306Q8—3261no—
307Q8—6522no—
308Q8—6522no—
309Q8#1457969912yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—3262no—
314D4—3261no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4—6522no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—6522no—
332C2×C4—6522no—
333C2×C4—6522no—
334C2×C4—1631no—
335C2×C4—1631no—
336C2×C4—1631no—
337C2×C4D4×D16311yescharacteristic
338C2×C4C23×D16311yescharacteristic
339C8—21no—
340C8—21no—
341C8—3261no—
342C8—3261no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—3261no—
350C22—3261no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22—1632no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3261no—
361C4—3261no—
362C4—3261no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4C2×#1457969911yescharacteristic
367C4#1910478011yescharacteristic
368C2—41no—
369C2—81no—
370C2—13041no—
371C2—3262no—
372C2#2378230511yescharacteristic
373C2AGL(1,ℤ/8)×D16311yescharacteristic
374C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#417×C163C2
socleC22×C163C22×D4
hypercenter#417D163
hypocenterC163#3308
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #23782322C2semidirect
2. #23782316C2semidirect
3. #23782023C2semidirect
4. #23782000C2semidirect
5. #23781977C2semidirect
6. #23781958C2semidirect
7. #23781911C2semidirect
8. #23781900C2semidirect
9. #23781879C2semidirect
10. #23781810C2semidirect
11. #23781805C2semidirect
12. #23781768C2semidirect
13. #23781734C2semidirect
14. #103×D163C2semidirect
15. #417×C163C2semidirect
140. Q8×C163#41semidirect
187. C163#3308semidirect
202. #417D163semidirect
309. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823222
#237823162
#237820232
#237820002
#237819772
#237819582
#237819112
#237819002
#237818792
#237818102
#237818052
#237817682
#237817342
#103×D1632
#417×C1632
#3308163