Finite Group Databasebeta

← #35226393

Subgroups — #35226393

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782365C211yesmaximal, characteristic, complemented
2#23782308C211yesmaximal, characteristic, complemented
3#23782019C211yesmaximal, characteristic, complemented
4#23781999C211yesmaximal, characteristic, complemented
5#23781988C211yesmaximal, characteristic, complemented
6#23781966C211yesmaximal, characteristic, complemented
7#23781898C211yesmaximal, characteristic, complemented
8#23781891C211yesmaximal, characteristic, complemented
9#23781809C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic, complemented
11C4×#14579684C211yesmaximal, characteristic, complemented
12#23781730C211yesmaximal, characteristic, complemented
13#23781660C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#463×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic, complemented
21#19104728—21no—
22#19104727C2211yescharacteristic, complemented
23#19104718—21no—
24#19104717—21no—
25#19104711C2211yescharacteristic
26#19104708—21no—
27#19104708—21no—
28#19104706C2211yescharacteristic
29#19104706C2211yescharacteristic
30#19104706C2211yescharacteristic, complemented
31#19104705—21no—
32C4×D652C2211yescharacteristic
33C4×D652C2211yescharacteristic
34C4×C22⋊D163—21no—
35#19104693C2211yescharacteristic, complemented
36C4⋊C4×D163C2211yescharacteristic
37C22⋊C4×D163—21no—
38#19104669C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic, complemented
48#19104655C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C4×C163⋊C8C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic, complemented
53C2×D8×C163C2211yescharacteristic
54#124×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60#14579700—22no—
61#14579700—41no—
62C2×D652—21no—
63C2×D652D411yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69D4×D163—22no—
70D4×D163—41no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75C2×C4×D163C2311yescharacteristic
76#14579684—22no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C8×D163—41no—
80C8×D163—41no—
81C4⋊Dic163D411yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163—21no—
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163C2311yescharacteristic
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163—21no—
102SD16×C163—41no—
103D8×C163—41no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22⋊C4×C163—21no—
109C22⋊C4×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116C22×D163C2×D411yescharacteristic
117D652—22no—
118D652—21no—
119D652—42no—
120D652#4112yes—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C22⋊D163—81no—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—21no—
134C4×D163—42no—
135C4×D163—81no—
136C4×D163—81no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C23×C163—21no—
140C23×C163—21no—
141Q8×C163—21no—
142Q8×C163—21no—
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—21no—
146D4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—81no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163C22×D411yescharacteristic
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—21no—
165C2×D163—22no—
166C2×D163—21no—
167C2×D163—21no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—84no—
171C2×D163#13512yes—
172C4×C163—21no—
173C4×C163—21no—
174C4×C163—41no—
175C4×C163—41no—
176C4×C163C2×#4111yescharacteristic
177C4×C163H(5,𝔽2)11yescharacteristic
178Dic163—21no—
179Dic163—21no—
180Dic163—41no—
181Dic163—41no—
182Dic163—41no—
183C2×C163—81no—
184C2×C163—81no—
185C2×C163#54611yescharacteristic
186C2×C163C2×#13511yescharacteristic
187C2×C163#57611yescharacteristic
188D163—22no—
189D163—41no—
190D163—42no—
191C163#329711yescharacteristic, complemented, Sylow
192#3297—1631nomaximal, Sylow
193#553—1631no—
194#546—1631no—
195#488—1631no—
196#484—1631no—
197#482—1631no—
198#478—1631no—
199#466—1631no—
200#465—1631no—
201#463D16311yescharacteristic, complemented
202#449—1631no—
203#447—1631no—
204C4×D8—1631no—
205C8×D4—1631no—
206#416—1631no—
207C2×#102—1631no—
208C2×#41—1631no—
209C22×D4—1631no—
210C2×SD16C2×D16311yescharacteristic, complemented
211C2×D8—1631no—
212C2×D8C2×D16311yescharacteristic
213C22×C8—1631no—
214C22×C8—1631no—
215#126—3261no—
216#125—1631no—
217#124C2×D16311yescharacteristic
218#123—3261no—
219#123—3261no—
220C22⋊Q8—1631no—
221#121—3261no—
222#121—3261no—
223#121—1631no—
224#121—1631no—
225#121—1631no—
226#120—3261no—
227C4×D4—3261no—
228C4×D4—1631no—
229C4×D4—1631no—
230C4×D4C2×D16311yescharacteristic, complemented
231#117—1631no—
232C2×C4⋊C4—1631no—
233C2×C22⋊C4—3261no—
234#107—1631no—
235#107—1631no—
236#106—1631no—
237C4⋊C8C2×D16311yescharacteristic, complemented
238#103—1631no—
239#103—1631no—
240#103C2×D16311yescharacteristic, complemented
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102C2×D16311yescharacteristic
248C22⋊C8—1631no—
249C22⋊C8—1631no—
250C4×C8—1631no—
251C24—3261no—
252#41—3262no—
253#41—6521no—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—3261no—
256C2×D4—3262no—
257C2×D4—3261no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—6521no—
262C2×D4—1631no—
263C2×D4—1631no—
264C2×D4C22×D16311yescharacteristic
265C2×D4C22×D16311yescharacteristic
266C22×C4—21no—
267C22×C4—3261no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4—1631no—
271C22×C4—1631no—
272SD16—41no—
273D8—41no—
274D8—3262no—
275C2×C8—6521no—
276C2×C8—6521no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4—1631no—
288C4⋊C4—1631no—
289C4⋊C4C22×D16311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—3261no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4—1631no—
300C22⋊C4—1631no—
301C42—3261no—
302C42C22×D16311yescharacteristic
303C23—21no—
304C23—21no—
305C23—13041no—
306C23—3261no—
307C23—6522no—
308C23—6522no—
309C23—1631no—
310C23—1631no—
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—13041no—
318D4—13041no—
319D4—3262no—
320D4—3261no—
321D4—6522no—
322D4—6522no—
323D4—6522no—
324D4—1632no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—13041no—
329C2×C4—13041no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—6522no—
338C2×C4—1631no—
339C2×C4—1631no—
340C2×C4D4×D16311yescharacteristic
341C2×C4D4×D16311yescharacteristic
342C2×C4C23×D16311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3261no—
346C8—3261no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—13044no—
354C22—3261no—
355C22—3261no—
356C22—3261no—
357C22—3261no—
358C22—3261no—
359C22—3262no—
360C22—6521no—
361C22—6521no—
362C22—1632no—
363C22C2×D4×D16311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3261no—
369C4—3261no—
370C4—6521no—
371C4—6521no—
372C4—6521no—
373C4#41×D16311yescharacteristic
374C4#1910478011yescharacteristic
375C2—81no—
376C2—81no—
377C2—3262no—
378C2—6522no—
379C2—6521no—
380C2#2378231211yescharacteristic
381C2#135×D16311yescharacteristic
382C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#463×C163C2
socleC22×C163C22×D4
hypercenter#463D163
hypocenterC163#3297
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #23782365C2semidirect
2. #23782308C2semidirect
3. #23782019C2semidirect
4. #23781999C2semidirect
5. #23781988C2semidirect
6. #23781966C2semidirect
7. #23781898C2semidirect
8. #23781891C2semidirect
9. #23781809C2semidirect
10. #23781797C2semidirect
11. C4×#14579684C2semidirect
12. #23781730C2semidirect
13. #23781660C2semidirect
14. #102×D163C2semidirect
15. #463×C163C2semidirect
20. C2×C8×D163C22semidirect
22. #19104727C22semidirect
30. #19104706C22semidirect
35. #19104693C22semidirect
38. #19104669C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
47. #19104656C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C4×C163⋊C8C22semidirect
52. C2×SD16×C163C22semidirect
55. C4×D4×C163C22semidirect
56. C4⋊C8×C163C22semidirect
57. #103×C163C22semidirect
191. C163#3297semidirect
201. #463D163semidirect
210. C2×SD16C2×D163semidirect
230. C4×D4C2×D163semidirect
237. C4⋊C8C2×D163semidirect
240. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823652
#237823082
#237820192
#237819992
#237819882
#237819662
#237818982
#237818912
#237818092
#237817972
C4×#145796842
#237817302
#237816602
#102×D1632
#463×C1632
#3297163