Finite Group Databasebeta

← #35226332

Subgroups — #35226332

Order 20864.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#23782469C212yesmaximal, complemented
2#23782468C212yesmaximal, complemented
3#23782041C211yesmaximal, characteristic, complemented
4#23781934C212yesmaximal, complemented
5#23781933C212yesmaximal, complemented
6#23781834C212yesmaximal, complemented
7#23781832C211yesmaximal, characteristic, complemented
8#23781779C212yesmaximal, complemented
9#453×C163C211yesmaximal, characteristic, complemented
10#19104781C2212yescomplemented
11#19104780C2211yescharacteristic, complemented
12#19104763C2212yescomplemented
13#19104762—22no—
14#19104761—22no—
15#19104761C2212yescomplemented
16#19104760—22no—
17#19104760C2212yescomplemented
18#19104759—22no—
19#19104756—22no—
20#19104755—22no—
21#19104755C2212yes—
22C2×#14579684—22no—
23C2×#14579684C2212yescomplemented
24C2×#14579684C2212yescomplemented
25D8×D163—22no—
26#19104743C2212yescomplemented
27#19104742C2212yescomplemented
28#19104732C2211yescharacteristic, complemented
29#19104670C2212yescomplemented
30#19104663C2212yescomplemented
31#19104652C2211yescharacteristic, complemented
32#19104651C2212yescomplemented
33C4×C163⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C163C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C163C2212yescomplemented
36#127×C163C2211yescharacteristic, complemented
37#104×C163C2212yescomplemented
38#100×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579699—22no—
41#14579699C2312yes—
42#14579698—22no—
43C2×D652—22no—
44C2×D652—41no—
45C2×D652C2312yes—
46C2×C22⋊D163—41no—
47D4×D163—22no—
48D4×D163—22no—
49D4×D163—42no—
50#14579689D412yescomplemented
51#14579688—22no—
52#14579687—22no—
53#14579687D412yescomplemented
54#14579686—22no—
55#14579686D412yescomplemented
56D1304—22no—
57#14579684—22no—
58#14579684—22no—
59#14579684—22no—
60#14579684—22no—
61#14579684—22no—
62#14579684—22no—
63#14579684D412yescomplemented
64#14579683—22no—
65#14579683C2312yes—
66#14579682—22no—
67C2×C163⋊C8—22no—
68C2×C163⋊C8C2312yes—
69C8×D163—22no—
70#41×C163—22no—
71#41×C163C2312yes—
72C2×D4×C163—22no—
73C2×D4×C163—41no—
74C2×D4×C163—41no—
75C2×D4×C163C2311yescharacteristic
76C2×D4×C163C2311yescharacteristic
77SD16×C163—22no—
78D8×C163—22no—
79M16×C163C2312yes—
80C42×C163C2311yescharacteristic
81C22×D163—22no—
82C22×D163—41no—
83C22×D163—42no—
84Dic326C2×D412yes—
85D652—22no—
86D652—22no—
87D652—42no—
88D652—82no—
89D652C2×D412yes—
90C22⋊D163—22no—
91C22⋊D163—42no—
92C22⋊D163—42no—
93C2×Dic163—41no—
94C4×D163—22no—
95C4×D163—42no—
96C163⋊C8—22no—
97C163⋊C8C2×D412yes—
98C163⋊C8C2×D412yes—
99C23×C163—21no—
100C23×C163—21no—
101C23×C163—41no—
102Q8×C163C2×D412yes—
103D4×C163—22no—
104D4×C163—22no—
105D4×C163—22no—
106D4×C163—42no—
107D4×C163—82no—
108D4×C163C2×D412yes—
109C2×C4×C163—21no—
110C2×C4×C163—22no—
111C2×C4×C163—41no—
112C2×C4×C163C2411yescharacteristic
113C8×C163—22no—
114C22×C163—22no—
115C22×C163—41no—
116C22×C163—41no—
117C22×C163—81no—
118C22×C163—81no—
119C22×C163—161no—
120C22×C163H(5,𝔽2)11yescharacteristic
121C2×D163—22no—
122C2×D163—42no—
123C2×D163—82no—
124C2×D163—161no—
125C2×D163—162no—
126C4×C163—22no—
127C4×C163—22no—
128C4×C163C22×D412yes—
129Dic163—22no—
130C2×C163—21no—
131C2×C163—42no—
132C2×C163—81no—
133C2×C163—81no—
134C2×C163D4211yescharacteristic
135D163—42no—
136D163—82no—
137C163#329211yescharacteristic, complemented, Sylow
138#3292—1631nomaximal, Sylow
139#576—1632no—
140#576—1632no—
141#493—1631no—
142#472—1632no—
143#472—1632no—
144#453—1631no—
145#453—1632no—
146#453D16311yescharacteristic, complemented
147#443—1632no—
148H(5,𝔽2)—1632no—
149H(5,𝔽2)—1631no—
150H(5,𝔽2)C2×D16311yescharacteristic, complemented
151AGL(1,ℤ/8)—3262no—
152AGL(1,ℤ/8)—3262no—
153AGL(1,ℤ/8)—3262no—
154AGL(1,ℤ/8)—3262no—
155AGL(1,ℤ/8)—1632no—
156AGL(1,ℤ/8)—1632no—
157AGL(1,ℤ/8)—1632no—
158AGL(1,ℤ/8)C2×D16312yescomplemented
159#135—3262no—
160#135—3262no—
161#135—1632no—
162C2×D8—3262no—
163C2×D8—3262no—
164C2×D8—1632no—
165C2×D8—1632no—
166#131—1632no—
167#131—1632no—
168#127—1631no—
169#127C2×D16311yescharacteristic, complemented
170#108—1632no—
171#104—1632no—
172#104C2×D16312yescomplemented
173#100—1631no—
174#100—1632no—
175#100C2×D16311yescharacteristic, complemented
176C4×C8—1631no—
177#41—22no—
178#41—3262no—
179#41—3262no—
180#41—3262no—
181#41—1632no—
182#41C22×D16312yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3262no—
187C2×D4—3262no—
188C2×D4—3262no—
189C2×D4—6521no—
190C2×D4—6522no—
191C2×D4—6521no—
192C2×D4—1632no—
193C2×D4C22×D16311yescharacteristic
194C2×D4C22×D16311yescharacteristic
195Q16—1632no—
196SD16—22no—
197SD16—3262no—
198SD16—3262no—
199SD16—3262no—
200SD16—1632no—
201SD16—1632no—
202D8—22no—
203D8—3262no—
204D8—3262no—
205D8—3262no—
206D8—3262no—
207D8—3262no—
208D8—3262no—
209D8—3262no—
210D8—1632no—
211M16—3262no—
212M16—3262no—
213M16—1632no—
214M16C22×D16312yes—
215C2×C8—3262no—
216C2×C8—3262no—
217C2×C8—1632no—
218C42C22×D16311yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—3262no—
223C23—6522no—
224C23—6521no—
225Q8—1632no—
226Q8D4×D16312yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—13042no—
233D4—3262no—
234D4—3262no—
235D4—3262no—
236D4—6522no—
237D4—6522no—
238D4—6522no—
239D4—1632no—
240D4D4×D16312yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—3262no—
245C2×C4—6522no—
246C2×C4—6521no—
247C2×C4C23×D16311yescharacteristic
248C8—22no—
249C8—3262no—
250C8—1632no—
251C8—1632no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—13042no—
259C22—26082no—
260C22—26081no—
261C22—3262no—
262C22—6522no—
263C22#1910478011yescharacteristic
264C4—22no—
265C4—22no—
266C4—3262no—
267C4C2×D4×D16312yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—13042no—
273C2—6522no—
274C2#2378229711yescharacteristic
subgroupG/H
centerC2#23782297
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#453×C163C2
socleC2×C163D42
hypercenter#453D163
hypocenterC163#3292
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23782469C2semidirect
2. #23782468C2semidirect
3. #23782041C2semidirect
4. #23781934C2semidirect
5. #23781933C2semidirect
6. #23781834C2semidirect
7. #23781832C2semidirect
8. #23781779C2semidirect
9. #453×C163C2semidirect
10. #19104781C22semidirect
11. #19104780C22semidirect
12. #19104763C22semidirect
15. #19104761C22semidirect
17. #19104760C22semidirect
23. C2×#14579684C22semidirect
24. C2×#14579684C22semidirect
26. #19104743C22semidirect
27. #19104742C22semidirect
28. #19104732C22semidirect
29. #19104670C22semidirect
30. #19104663C22semidirect
31. #19104652C22semidirect
32. #19104651C22semidirect
33. C4×C163⋊C8C22semidirect
34. H(5,𝔽2)×C163C22semidirect
35. AGL(1,ℤ/8)×C163C22semidirect
36. #127×C163C22semidirect
37. #104×C163C22semidirect
38. #100×C163C22semidirect
50. #14579689D4semidirect
53. #14579687D4semidirect
55. #14579686D4semidirect
63. #14579684D4semidirect
137. C163#3292semidirect
146. #453D163semidirect
150. H(5,𝔽2)C2×D163semidirect
158. AGL(1,ℤ/8)C2×D163semidirect
169. #127C2×D163semidirect
172. #104C2×D163semidirect
175. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824692
#237824682
#237820412
#237819342
#237819332
#237818342
#237818322
#237817792
#453×C1632
#3292163