Finite Group Databasebeta

← #35224113

Subgroups — #35224113

Order 20864.

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

normal subgroups only

This group has 520 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
1C22×C8×D163C211yesmaximal, characteristic, complemented
2#23782172C212yesmaximal, complemented
3#23781903C212yesmaximal, complemented
4#23781898C212yesmaximal, complemented
5#23781891C214yesmaximal, complemented
6#23781703C211yesmaximal, characteristic, complemented
7#23781661C212yesmaximal, complemented
8C2×#135×C163C211yesmaximal, characteristic, complemented
9C22×C4×D163C2211yescharacteristic
10C22×C163⋊C8C2211yescharacteristic
11C2×C8×D163—24no—
12C2×C8×D163C2212yes—
13C2×C8×D163C2212yes—
14#19104716—22no—
15#19104711C2214yes—
16#19104706C2214yes—
17#19104703—22no—
18C4×C22⋊D163—24no—
19#19104694C2212yes—
20#19104669C2212yescomplemented
21#19104667C2212yescomplemented
22#19104659C2214yescomplemented
23#19104656C2214yescomplemented
24C2×#41×C163C2212yes—
25#135×C163—24no—
26C2×Q16×C163C2211yescharacteristic, complemented
27C2×SD16×C163C2212yescomplemented
28C2×D8×C163C2211yescharacteristic, complemented
29C22×C8×C163C2211yescharacteristic
30C23×D163D411yescharacteristic
31C2×C22⋊D163—24no—
32C22×Dic163D411yescharacteristic
33C2×C4×D163—22no—
34C2×C4×D163—22no—
35C2×C4×D163D412yes—
36C2×C4×D163C2312yes—
37C2×C163⋊C8—22no—
38C2×C163⋊C8C2312yes—
39C8×D163—24no—
40C8×D163—24no—
41C4⋊Dic163C2314yes—
42Dic163⋊C4—24no—
43C22⋊Dic163—24no—
44C22⋊Dic163—22no—
45(C2×D163)⋊C4—24no—
46C4×Dic163—22no—
47#41×C163—24no—
48#41×C163—24no—
49C2×Q8×C163C2312yes—
50C2×D4×C163—22no—
51C2×D4×C163C2312yes—
52C22×C4×C163—22no—
53C22×C4×C163C2311yescharacteristic
54Q16×C163—22no—
55SD16×C163—24no—
56D8×C163—22no—
57C2×C8×C163D414yescomplemented
58C2×C8×C163C2311yescharacteristic
59C2×C8×C163C2311yescharacteristic
60C22×D163—22no—
61C22×D163—24no—
62C22×D163C2×D412yes—
63C22⋊D163—48no—
64C2×Dic163—22no—
65C2×Dic163—24no—
66C2×Dic163C2×D412yes—
67C4×D163—24no—
68C4×D163—22no—
69C4×D163—22no—
70C163⋊C8—22no—
71C23×C163—22no—
72C23×C163C2×D411yescharacteristic
73Q8×C163—22no—
74Q8×C163—22no—
75D4×C163—22no—
76D4×C163—22no—
77D4×C163—44no—
78C2×C4×C163—22no—
79C2×C4×C163—44no—
80C2×C4×C163C2×D411yescharacteristic
81C2×C4×C163C2×D412yes—
82C2×C4×C163#4112yes—
83C2×C4×C163C2411yescharacteristic
84C8×C163C2×D412yes—
85C8×C163C2×D412yes—
86C22×C163—22no—
87C22×C163—42no—
88C22×C163—44no—
89C22×C163#12112yes—
90C22×C163C22×D411yescharacteristic
91C2×D163—28no—
92C2×D163—22no—
93C2×D163—22no—
94C2×D163#13514yescomplemented
95C4×C163—42no—
96C4×C163#12112yes—
97C4×C163C22×D411yescharacteristic
98C4×C163C2×#4111yescharacteristic
99Dic163—22no—
100Dic163—44no—
101C2×C163—22no—
102C2×C163—82no—
103C2×C163C2×#12111yescharacteristic
104C2×C163C2×#13512yes—
105D163—24no—
106C163#305011yescharacteristic, complemented, Sylow
107#3050—1631nomaximal, Sylow
108C2×#135D16311yescharacteristic, complemented
109C23×C8—1631no—
110#525—1632no—
111#467—1632no—
112#466—1632no—
113#465—1634no—
114#427—1631no—
115#416—1632no—
116C2×#41C2×D16312yes—
117C23×C4—1631no—
118#135—24no—
119C2×Q16C2×D16311yescharacteristic, complemented
120C2×SD16C2×D16312yescomplemented
121C2×D8C2×D16311yescharacteristic, complemented
122C22×C8—3264no—
123C22×C8—1631no—
124C22×C8—1632no—
125C22×C8—1632no—
126C22×C8C2×D16311yescharacteristic
127#123—3262no—
128C22⋊Q8—1634no—
129#121—1634no—
130#120—3262no—
131C4×D4—3264no—
132#117—1632no—
133#107—1632no—
134#106—1632no—
135#103—1634no—
136#102—1634no—
137C24—1631no—
138#41—24no—
139#41—24no—
140C2×Q8C22×D16312yes—
141C2×D4—22no—
142C2×D4—3264no—
143C2×D4C22×D16312yes—
144C22×C4—22no—
145C22×C4—3262no—
146C22×C4—3262no—
147C22×C4—1631no—
148C22×C4—1632no—
149C22×C4—1632no—
150C22×C4C22×D16311yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154C2×C8—3262no—
155C2×C8—3264no—
156C2×C8—3264no—
157C2×C8—1632no—
158C2×C8C22⋊D16314yescomplemented
159C2×C8C22×D16311yescharacteristic
160C2×C8C22×D16311yescharacteristic
161C4⋊C4—3264no—
162C4⋊C4—1634no—
163C22⋊C4—3264no—
164C22⋊C4—3262no—
165C22⋊C4—3264no—
166C42—3262no—
167C23—22no—
168C23—3264no—
169C23—3262no—
170C23—1632no—
171C23D4×D16311yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—6528no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—3262no—
181C2×C4—3262no—
182C2×C4—3262no—
183C2×C4—3264no—
184C2×C4—3264no—
185C2×C4—1632no—
186C2×C4D4×D16311yescharacteristic
187C2×C4C2×C22⋊D16312yes—
188C2×C4#1457969812yes—
189C2×C4C23×D16311yescharacteristic
190C8—3262no—
191C8C2×C22⋊D16312yes—
192C8C2×C22⋊D16312yes—
193C22—22no—
194C22—42no—
195C22—44no—
196C22—3268no—
197C22—3262no—
198C22—3262no—
199C22—1634no—
200C22#1910470612yes—
201C22C2×D4×D16311yescharacteristic
202C4—42no—
203C4—3262no—
204C4—6524no—
205C4#1910470612yes—
206C4C22×C22⋊D16311yescharacteristic
207C4C2×#1457969811yescharacteristic
208C2—22no—
209C2—82no—
210C2—3264no—
211C2C2×#1910470611yescharacteristic
212C2#135×D16312yes—
subgroupG/H
centerC2×C4D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#135×C163C2
socleC22×C163C22×D4
hypercenterC2×#135D163
hypocenterC163#3050
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×C8×D163C2semidirect
2. #23782172C2semidirect
3. #23781903C2semidirect
4. #23781898C2semidirect
5. #23781891C2semidirect
6. #23781703C2semidirect
7. #23781661C2semidirect
8. C2×#135×C163C2semidirect
20. #19104669C22semidirect
21. #19104667C22semidirect
22. #19104659C22semidirect
23. #19104656C22semidirect
26. C2×Q16×C163C22semidirect
27. C2×SD16×C163C22semidirect
28. C2×D8×C163C22semidirect
57. C2×C8×C163D4semidirect
94. C2×D163#135semidirect
106. C163#3050semidirect
108. C2×#135D163semidirect
119. C2×Q16C2×D163semidirect
120. C2×SD16C2×D163semidirect
121. C2×D8C2×D163semidirect
158. C2×C8C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D1632
#237821722
#237819032
#237818982
#237818912
#237817032
#237816612
C2×#135×C1632
#3050163