Finite Group Databasebeta

← #35223891

Subgroups — #35223891

Order 20864.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104781C211yesmaximal, characteristic, complemented
2#23782463C211yesmaximal, characteristic, complemented
3#23782229C211yesmaximal, characteristic, complemented
4#23781841C214yesmaximal, complemented
5#23781831C214yesmaximal, complemented
6#23781682C211yesmaximal, characteristic, complemented
7#23781681C211yesmaximal, characteristic, complemented
8C2×#19104651C211yesmaximal, characteristic, complemented
9#575×C163C211yesmaximal, characteristic, complemented
10#19104781—22no—
11#19104781—22no—
12#19104781—22no—
13#19104781C2214yes—
14C2×#14579700—21no—
15C2×#14579699—21no—
16C2×#14579699C2211yescharacteristic
17C22×D652—21no—
18C22×D652C2211yescharacteristic
19C2×D4×D163—22no—
20C2×D4×D163—21no—
21#19104766C2212yescomplemented
22#19104765—21no—
23#19104761C2212yescomplemented
24#19104760—21no—
25#19104755—22no—
26C2×#14579683C2211yescharacteristic, complemented
27#19104731C2212yescomplemented
28#19104724C2212yescomplemented
29#19104709—22no—
30#19104703—22no—
31#19104694C2211yescharacteristic, complemented
32#19104662C2214yescomplemented
33#19104661C2214yescomplemented
34#19104651C2214yescomplemented
35C2×#41×C163C2211yescharacteristic
36C2×#41×C163C2211yescharacteristic, complemented
37#137×C163—21no—
38#137×C163C2212yescomplemented
39AGL(1,ℤ/8)×C163—21no—
40AGL(1,ℤ/8)×C163C2212yescomplemented
41#135×C163—22no—
42C2×M16×C163C2211yescharacteristic, complemented
43C23×D163—22no—
44C23×D163D412yes—
45#14579700—24no—
46#14579700—24no—
47#14579699—24no—
48#14579699—24no—
49#14579699—21no—
50#14579699—21no—
51#14579699D412yescomplemented
52#14579699C2312yes—
53C2×Dic326D411yescharacteristic, complemented
54C2×D652—22no—
55C2×D652—24no—
56C2×D652—24no—
57C2×D652—21no—
58C2×D652—21no—
59C2×D652—21no—
60C2×D652—21no—
61C2×D652D412yes—
62C2×D652D411yescharacteristic, complemented
63C2×D652C2312yes—
64C2×C22⋊D163—22no—
65C2×C22⋊D163—24no—
66C2×C22⋊D163—21no—
67D4×D163—28no—
68D4×D163—24no—
69D4×D163—28no—
70D4×D163—24no—
71C2×C4×D163—22no—
72C2×C4×D163—21no—
73#14579689—22no—
74#14579687—22no—
75#14579686—22no—
76#14579684—22no—
77#14579683—21no—
78#14579683C2312yescomplemented
79C2×C163⋊C8—21no—
80C4⋊Dic163—21no—
81C22⋊Dic163—21no—
82(C2×D163)⋊C4—24no—
83C4×Dic163C2312yescomplemented
84#41×C163—21no—
85#41×C163—21no—
86#41×C163—21no—
87#41×C163—21no—
88#41×C163—21no—
89#41×C163—21no—
90#41×C163D412yescomplemented
91#41×C163C2312yes—
92C2×Q8×C163D411yescharacteristic, complemented
93C2×Q8×C163C2311yescharacteristic, complemented
94C2×D4×C163—21no—
95C2×D4×C163—21no—
96C2×D4×C163D411yescharacteristic, complemented
97C2×D4×C163C2311yescharacteristic, complemented
98C22×C4×C163—21no—
99C22×C4×C163—21no—
100C22×C4×C163C2311yescharacteristic
101Q16×C163—22no—
102SD16×C163—22no—
103SD16×C163—22no—
104D8×C163—22no—
105M16×C163—21no—
106M16×C163C2312yescomplemented
107C2×C8×C163—21no—
108C22×D163—22no—
109C22×D163—21no—
110C22×D163—28no—
111C22×D163—22no—
112C22×D163—82no—
113C22×D163—84no—
114C22×D163C2×D414yes—
115Dic326—21no—
116Dic326C2×D412yes—
117D652—28no—
118D652—22no—
119D652—22no—
120D652—21no—
121D652—22no—
122D652—22no—
123D652C2×D412yes—
124C22⋊D163—28no—
125C22⋊D163—22no—
126C22⋊D163—22no—
127C22⋊D163—84no—
128C2×Dic163—21no—
129C2×Dic163—22no—
130C4×D163—28no—
131C4×D163—22no—
132C4×D163—22no—
133C163⋊C8—22no—
134C23×C163—21no—
135C23×C163—21no—
136C23×C163C2×D411yescharacteristic
137Q8×C163—21no—
138Q8×C163—21no—
139Q8×C163—21no—
140Q8×C163C2×D412yes—
141D4×C163—22no—
142D4×C163—22no—
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—21no—
146D4×C163—41no—
147D4×C163—41no—
148D4×C163C2×D412yes—
149C2×C4×C163—21no—
150C2×C4×C163—22no—
151C2×C4×C163—22no—
152C2×C4×C163—21no—
153C2×C4×C163—41no—
154C2×C4×C163—41no—
155C2×C4×C163C2×D411yescharacteristic
156C2×C4×C163C2×D411yescharacteristic
157C2×C4×C163C2×D411yescharacteristic
158C2×C4×C163C2×D411yescharacteristic
159C2×C4×C163C2×D411yescharacteristic
160C2×C4×C163C2411yescharacteristic
161C8×C163—22no—
162C22×C163—22no—
163C22×C163—41no—
164C22×C163—42no—
165C22×C163—42no—
166C22×C163—41no—
167C22×C163#12011yescharacteristic
168C22×C163#12011yescharacteristic
169C22×C163C22×D411yescharacteristic
170C2×D163—24no—
171C2×D163—22no—
172C2×D163—44no—
173C2×D163—42no—
174C2×D163—44no—
175C2×D163—82no—
176C2×D163—88no—
177C4×C163—22no—
178C4×C163—41no—
179C4×C163#12011yescharacteristic
180C4×C163#12011yescharacteristic
181C4×C163C22×D411yescharacteristic
182C4×C163C22×D411yescharacteristic
183Dic163—22no—
184Dic163—42no—
185C2×C163—21no—
186C2×C163—21no—
187C2×C163—21no—
188C2×C163—42no—
189C2×C163—81no—
190C2×C163C2×#12011yescharacteristic
191D163—44no—
192D163—42no—
193C163#301911yescharacteristic, complemented, Sylow
194#3019—1631nomaximal, Sylow
195C2×H(5,𝔽2)—1631no—
196#575—1631no—
197#575D16311yescharacteristic, complemented
198#535—1631no—
199#455—1634no—
200#453—1634no—
201#421—1631no—
202#421—1631no—
203C2×#100—1631no—
204H(5,𝔽2)—3262no—
205H(5,𝔽2)—3262no—
206H(5,𝔽2)—3262no—
207H(5,𝔽2)—1634no—
208C2×#41—3261no—
209C2×#41—3261no—
210C2×#41—1631no—
211C2×#41C2×D16311yescharacteristic
212C2×#41C2×D16311yescharacteristic, complemented
213C22×D4—3261no—
214C22×D4—3262no—
215C22×D4—3261no—
216C22×D4—1631no—
217#137—21no—
218#137—3261no—
219#137—1632no—
220#137C2×D16312yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—3261no—
223AGL(1,ℤ/8)—1632no—
224AGL(1,ℤ/8)C2×D16312yescomplemented
225#135—22no—
226#135—3262no—
227C2×M16—1631no—
228C2×M16C2×D16311yescharacteristic, complemented
229#127—1632no—
230#124—1632no—
231#121—3262no—
232#120—3262no—
233#117—1631no—
234#104—1634no—
235#104—1634no—
236#100—1634no—
237C24—3262no—
238C24—1632no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—3264no—
246#41—3264no—
247#41—3264no—
248#41—3264no—
249#41—3261no—
250#41—3261no—
251#41—1632no—
252#41—1632no—
253#41C22⋊D16312yescomplemented
254#41C22×D16312yes—
255C2×Q8—1631no—
256C2×Q8C22⋊D16311yescharacteristic, complemented
257C2×Q8C22×D16311yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—3262no—
261C2×D4—3264no—
262C2×D4—3264no—
263C2×D4—3268no—
264C2×D4—3264no—
265C2×D4—3268no—
266C2×D4—3264no—
267C2×D4—3262no—
268C2×D4—3261no—
269C2×D4—3264no—
270C2×D4—3261no—
271C2×D4—3261no—
272C2×D4—3261no—
273C2×D4—3261no—
274C2×D4—1632no—
275C2×D4—1631no—
276C2×D4—1632no—
277C2×D4C22⋊D16311yescharacteristic, complemented
278C2×D4C22×D16311yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3262no—
282C22×C4—3261no—
283C22×C4C22×D16311yescharacteristic
284Q16—22no—
285Q16—3262no—
286SD16—22no—
287SD16—22no—
288SD16—3262no—
289SD16—3262no—
290D8—22no—
291D8—3262no—
292M16—21no—
293M16—3261no—
294M16—1632no—
295M16C22×D16312yescomplemented
296C2×C8—21no—
297C2×C8—3261no—
298C4⋊C4—3261no—
299C22⋊C4—3264no—
300C22⋊C4—3261no—
301C42—1632no—
302C23—21no—
303C23—21no—
304C23—13044no—
305C23—13042no—
306C23—3268no—
307C23—3262no—
308C23—3262no—
309C23—3261no—
310C23—1634no—
311C23D4×D16311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—3261no—
316Q8—1632no—
317Q8C2×C22⋊D16312yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—13044no—
326D4—3268no—
327D4—3268no—
328D4—3262no—
329D4—3262no—
330D4—3262no—
331D4—3262no—
332D4—3262no—
333D4—3261no—
334D4—3262no—
335D4—1632no—
336D4C2×C22⋊D16312yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—3268no—
344C2×C4—3261no—
345C2×C4—3262no—
346C2×C4—3262no—
347C2×C4—3262no—
348C2×C4D4×D16311yescharacteristic
349C2×C4D4×D16311yescharacteristic
350C2×C4D4×D16311yescharacteristic
351C2×C4C2×C22⋊D16311yescharacteristic
352C2×C4C2×C22⋊D16311yescharacteristic
353C2×C4C23×D16311yescharacteristic
354C8—22no—
355C8—3262no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—13042no—
362C22—13048no—
363C22—3264no—
364C22—3262no—
365C22—6524no—
366C22—6522no—
367C22—6524no—
368C22#1910470311yescharacteristic
369C22#1910470311yescharacteristic
370C22C2×D4×D16311yescharacteristic
371C4—22no—
372C4—41no—
373C4—3262no—
374C4—6522no—
375C4#1910470311yescharacteristic
376C4#1910470311yescharacteristic
377C4C2×D4×D16311yescharacteristic
378C4C22×C22⋊D16311yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—6522no—
385C2—6524no—
386C2C2×#1910470311yescharacteristic
subgroupG/H
centerC2C2×#19104703
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#575×C163C2
socleC2×C163C2×#120
hypercenter#575D163
hypocenterC163#3019
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#19104781C2semidirect
2. #23782463C2semidirect
3. #23782229C2semidirect
4. #23781841C2semidirect
5. #23781831C2semidirect
6. #23781682C2semidirect
7. #23781681C2semidirect
8. C2×#19104651C2semidirect
9. #575×C163C2semidirect
21. #19104766C22semidirect
23. #19104761C22semidirect
26. C2×#14579683C22semidirect
27. #19104731C22semidirect
28. #19104724C22semidirect
31. #19104694C22semidirect
32. #19104662C22semidirect
33. #19104661C22semidirect
34. #19104651C22semidirect
36. C2×#41×C163C22semidirect
38. #137×C163C22semidirect
40. AGL(1,ℤ/8)×C163C22semidirect
42. C2×M16×C163C22semidirect
51. #14579699D4semidirect
53. C2×Dic326D4semidirect
62. C2×D652D4semidirect
78. #14579683C23semidirect
83. C4×Dic163C23semidirect
90. #41×C163D4semidirect
92. C2×Q8×C163D4semidirect
93. C2×Q8×C163C23semidirect
96. C2×D4×C163D4semidirect
97. C2×D4×C163C23semidirect
106. M16×C163C23semidirect
193. C163#3019semidirect
197. #575D163semidirect
212. C2×#41C2×D163semidirect
220. #137C2×D163semidirect
224. AGL(1,ℤ/8)C2×D163semidirect
228. C2×M16C2×D163semidirect
253. #41C22⋊D163semidirect
256. C2×Q8C22⋊D163semidirect
257. C2×Q8C22×D163semidirect
277. C2×D4C22⋊D163semidirect
278. C2×D4C22×D163semidirect
295. M16C22×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047812
#237824632
#237822292
#237818412
#237818312
#237816822
#237816812
C2×#191046512
#575×C1632
#3019163