Finite Group Databasebeta

← #35226311

Subgroups — #35226311

Order 20864.

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

normal subgroups only

This group has 448 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#23782478C212yesmaximal, complemented
2#23782471C212yesmaximal, complemented
3#23781937C212yesmaximal, complemented
4#23781925C212yesmaximal, complemented
5#23781847C212yesmaximal, complemented
6#23781842C211yesmaximal, characteristic, complemented
7#23781837C211yesmaximal, characteristic, complemented
8#23781788C212yesmaximal, complemented
9#497×C163C211yesmaximal, characteristic, complemented
10#19104783C2212yes—
11#19104783C2211yescharacteristic, complemented
12#19104780C2211yescharacteristic
13#19104767—22no—
14#19104767—22no—
15#19104766C2212yes—
16#19104766C2212yes—
17#19104766C2212yescomplemented
18#19104764—22no—
19#19104762—22no—
20#19104761C2212yes—
21#19104758—22no—
22#19104757C2212yes—
23#19104756—22no—
24Q16×D163—22no—
25SD16×D163—22no—
26#19104741C2212yes—
27#19104741C2212yescomplemented
28#19104671C2212yescomplemented
29#19104663C2212yes—
30#19104663C2212yescomplemented
31#19104654C2212yescomplemented
32#19104654C2211yescharacteristic, complemented
33#19104652C2211yescharacteristic
34C2×Q16×C163C2212yescomplemented
35C2×SD16×C163C2212yes—
36C4⋊Q8×C163C2211yescharacteristic, complemented
37#124×C163C2211yescharacteristic
38#97×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579700—22no—
41#14579700—22no—
42#14579699—42no—
43#14579699—41no—
44#14579699C2312yes—
45#14579699C2312yes—
46#14579698—22no—
47Q8×D163—22no—
48Q8×D163—22no—
49Q8×D163—22no—
50C2×C22⋊D163—41no—
51D4×D163—22no—
52Dic652D412yes—
53#14579689—22no—
54#14579689—22no—
55#14579689—22no—
56#14579688D412yes—
57#14579688D412yes—
58#14579687—22no—
59#14579686—22no—
60#14579686—22no—
61#14579686—22no—
62D1304D412yes—
63#14579684—22no—
64#14579683C2312yes—
65#14579683C2312yes—
66#14579682—22no—
67#14579682—22no—
68C8×D163—22no—
69C8×D163—22no—
70C2×Q8×C163C2312yes—
71C2×Q8×C163C2311yescharacteristic
72C2×D4×C163C2311yescharacteristic
73Q16×C163—22no—
74Q16×C163—22no—
75SD16×C163—22no—
76SD16×C163—22no—
77C2×C8×C163C2312yes—
78C4⋊C4×C163—41no—
79C22⋊C4×C163—41no—
80C42×C163C2311yescharacteristic
81C22×D163—41no—
82Dic326—41no—
83Dic326—42no—
84Dic326C2×D412yes—
85Dic326C2×D412yes—
86D652—41no—
87D652—42no—
88D652C2×D412yes—
89D652C2×D412yes—
90C22⋊D163—22no—
91C22⋊D163—22no—
92C22⋊D163—42no—
93C2×Dic163—41no—
94C4×D163—22no—
95C4×D163—22no—
96C4×D163—42no—
97C4×D163—42no—
98C4×D163—42no—
99C163⋊C8—22no—
100C163⋊C8—22no—
101C23×C163—21no—
102Q8×C163—22no—
103Q8×C163—22no—
104Q8×C163—22no—
105D4×C163—22no—
106C2×C4×C163—21no—
107C2×C4×C163—21no—
108C2×C4×C163—22no—
109C2×C4×C163C2411yescharacteristic
110C8×C163C2×D412yes—
111C8×C163C2×D412yes—
112C22×C163—41no—
113C22×C163—81no—
114C22×C163H(5,𝔽2)11yescharacteristic
115C2×D163—22no—
116C2×D163—22no—
117C2×D163—161no—
118C4×C163—41no—
119C4×C163—41no—
120C4×C163—42no—
121C4×C163C22×D412yes—
122Dic163—22no—
123Dic163—22no—
124C2×C163—21no—
125C2×C163—81no—
126C2×C163D4211yescharacteristic
127D163—42no—
128D163—42no—
129C163#328911yescharacteristic, complemented, Sylow
130#3289—1631nomaximal, Sylow
131#578—1632no—
132#577—1632no—
133#497D16311yescharacteristic, complemented
134#473—1632no—
135#471—1632no—
136#456—1632no—
137#455—1631no—
138#454—1631no—
139#444—1632no—
1402−1+4—1632no—
1412−1+4—1631no—
142H(5,𝔽2)—1631no—
143#137—3262no—
144#137—3262no—
145#137—3262no—
146#137—1632no—
147#137—1632no—
148#137—1632no—
149AGL(1,ℤ/8)—3262no—
150AGL(1,ℤ/8)—1632no—
151#135—3262no—
152#135—3262no—
153#135—1632no—
154C2×Q16—3262no—
155C2×Q16C2×D16312yescomplemented
156C2×SD16—3262no—
157C2×SD16C2×D16312yes—
158#131—1632no—
159#131—1632no—
160C4⋊Q8C2×D16311yescharacteristic, complemented
161#124C2×D16311yescharacteristic
162#108—1632no—
163#104—1632no—
164#104—1632no—
165#101—1631no—
166#101—1632no—
167#100—1631no—
168#97C2×D16311yescharacteristic, complemented
169#41—3262no—
170#41—3262no—
171#41—3262no—
172#41—3262no—
173#41—6521no—
174#41—6522no—
175#41—1632no—
176#41—1632no—
177C2×Q8—3262no—
178C2×Q8—3262no—
179C2×Q8—3262no—
180C2×Q8C22×D16311yescharacteristic
181C2×Q8C22×D16312yes—
182C2×D4—3262no—
183C2×D4—6521no—
184C2×D4C22×D16311yescharacteristic
185Q16—22no—
186Q16—22no—
187Q16—3262no—
188Q16—3262no—
189Q16—3262no—
190Q16—1632no—
191SD16—22no—
192SD16—22no—
193SD16—3262no—
194SD16—3262no—
195SD16—3262no—
196SD16—3262no—
197SD16—1632no—
198SD16—1632no—
199D8—3262no—
200D8—1632no—
201M16—3262no—
202M16—3262no—
203M16—1632no—
204M16—1632no—
205C2×C8—3262no—
206C2×C8—3262no—
207C2×C8C22×D16312yes—
208C4⋊C4—41no—
209C22⋊C4—41no—
210C42C22×D16311yescharacteristic
211C23—21no—
212C23—6521no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—6522no—
217Q8—6521no—
218Q8—1632no—
219Q8—1632no—
220D4—22no—
221D4—3262no—
222D4—3262no—
223D4—6521no—
224D4—6522no—
225D4—6522no—
226D4—1632no—
227D4—1632no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—3262no—
232C2×C4—3262no—
233C2×C4—6522no—
234C2×C4—6522no—
235C2×C4—6522no—
236C2×C4—6521no—
237C2×C4C23×D16311yescharacteristic
238C8—3262no—
239C8—3262no—
240C8D4×D16312yes—
241C8D4×D16312yes—
242C22—41no—
243C22—81no—
244C22—26081no—
245C22—3262no—
246C22—3262no—
247C22#1910478011yescharacteristic
248C4—41no—
249C4—41no—
250C4—42no—
251C4—3262no—
252C4—3262no—
253C4C2×D4×D16312yes—
254C2—21no—
255C2—81no—
256C2—6522no—
257C2—6522no—
258C2#2378230011yescharacteristic
subgroupG/H
centerC2#23782300
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#497×C163C2
socleC2×C163D42
hypercenter#497D163
hypocenterC163#3289
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #23782478C2semidirect
2. #23782471C2semidirect
3. #23781937C2semidirect
4. #23781925C2semidirect
5. #23781847C2semidirect
6. #23781842C2semidirect
7. #23781837C2semidirect
8. #23781788C2semidirect
9. #497×C163C2semidirect
11. #19104783C22semidirect
17. #19104766C22semidirect
27. #19104741C22semidirect
28. #19104671C22semidirect
30. #19104663C22semidirect
31. #19104654C22semidirect
32. #19104654C22semidirect
34. C2×Q16×C163C22semidirect
36. C4⋊Q8×C163C22semidirect
38. #97×C163C22semidirect
129. C163#3289semidirect
133. #497D163semidirect
155. C2×Q16C2×D163semidirect
160. C4⋊Q8C2×D163semidirect
168. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824782
#237824712
#237819372
#237819252
#237818472
#237818422
#237818372
#237817882
#497×C1632
#3289163