Finite Group Databasebeta

← #35213158

Subgroups — #35213158

Order 20864.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D163C211yesmaximal, characteristic
2#23781720C212yesmaximal, complemented
3#23781405C211yesmaximal, characteristic, complemented
4#23781404C212yesmaximal, complemented
5#329×C163C211yesmaximal, characteristic, complemented
6C2×Q8×D163C411yescharacteristic
7C2×D4×D163C411yescharacteristic
8#19104724C411yescharacteristic
9#19104723C2211yescharacteristic
10#19104722C411yescharacteristic
11#19104721—22no—
12C22⋊C4×D163—22no—
13C42×D163C2211yescharacteristic
14Dic163⋊C8—22no—
15(C2×D163)⋊C8—22no—
16#19104646C2212yescomplemented
17#124×C163C2211yescharacteristic
18#97×C163C2212yescomplemented
19C23×D163—21no—
20C2×Dic326—21no—
21C2×Dic326C2×C411yescharacteristic
22Q8×D163—42no—
23C2×D652C2×C411yescharacteristic
24C2×C22⋊D163—21no—
25D4×D163—42no—
26C2×C4×D163—21no—
27C2×C4×D163C2×C411yescharacteristic
28C2×C4×D163D412yes—
29C2×C163⋊C8—22no—
30C22⋊Dic163—22no—
31(C2×D163)⋊C4—22no—
32(C2×D163)⋊C4—22no—
33C4×Dic163C2×C411yescharacteristic
34C4×Dic163D412yes—
35C2×Q8×C163C2×C411yescharacteristic
36C2×D4×C163C2×C411yescharacteristic
37C2×C8×C163—22no—
38C22⋊C4×C163—22no—
39C42×C163C2311yescharacteristic
40C22×D163—21no—
41C22×D163—44no—
42C22×D163—42no—
43C22×D163C22⋊C411yescharacteristic
44Dic326—42no—
45Dic326—41no—
46D652—41no—
47C22⋊D163—42no—
48C2×Dic163—21no—
49C2×Dic163C22⋊C412yes—
50C2×Dic163C22⋊C411yescharacteristic
51C4×D163—22no—
52C4×D163—42no—
53C4×D163—42no—
54C163⋊C8—42no—
55C23×C163—21no—
56Q8×C163—41no—
57D4×C163—41no—
58C2×C4×C163—21no—
59C2×C4×C163C22×C411yescharacteristic
60C2×C4×C163C2×D412yes—
61C8×C163—42no—
62C22×C163—41no—
63C22×C163—42no—
64C22×C163C2×C22⋊C411yescharacteristic
65C2×D163—22no—
66C2×D163—41no—
67C2×D163—42no—
68C2×D163—84no—
69C2×D163#10012yes—
70C4×C163—21no—
71C4×C163—22no—
72C4×C163—41no—
73Dic163—21no—
74Dic163—22no—
75Dic163—41no—
76C2×C163—81no—
77C2×C163C2×#10011yescharacteristic
78C2×C163#42112yes—
79D163—22no—
80D163—81no—
81C163#153311yescharacteristic, complemented, Sylow
82#1533—1631nomaximal, Sylow
83C2×#124—1631no—
84#432—1632no—
85#329—1631no—
86#329—1632no—
87#329D16311yescharacteristic, complemented
88C22×Q8—1631no—
89C22×D4—1631no—
90#124—3262no—
91#124—1631no—
92#124—1631no—
93#124—1631no—
94#124C2×D16311yescharacteristic
95C2×C22⋊C4—3262no—
96C2×C42—1631no—
97C4⋊C8—3262no—
98C22⋊C8—3262no—
99#97—1632no—
100#97C2×D16312yescomplemented
101C24—3261no—
102C2×Q8—3261no—
103C2×Q8—6522no—
104C2×Q8—1631no—
105C2×Q8C4×D16311yescharacteristic
106C2×D4—3261no—
107C2×D4—6522no—
108C2×D4—1631no—
109C2×D4C4×D16311yescharacteristic
110C22×C4—3261no—
111C22×C4—1631no—
112C22×C4—1632no—
113C2×C8—22no—
114C2×C8—3262no—
115C22⋊C4—22no—
116C22⋊C4—3262no—
117C22⋊C4—3262no—
118C22⋊C4—3262no—
119C42—1631no—
120C42—1632no—
121C42C22×D16311yescharacteristic
122C23—21no—
123C23—3261no—
124C23—6524no—
125C23—6522no—
126C23—1631no—
127Q8—41no—
128Q8—6521no—
129Q8—6522no—
130D4—41no—
131D4—6521no—
132D4—6522no—
133C2×C4—21no—
134C2×C4—3262no—
135C2×C4—3261no—
136C2×C4—6522no—
137C2×C4—6522no—
138C2×C4—1632no—
139C2×C4—1631no—
140C2×C4C2×C4×D16311yescharacteristic
141C2×C4D4×D16312yes—
142C8—42no—
143C8—6522no—
144C22—41no—
145C22—42no—
146C22—13044no—
147C22—3262no—
148C22—6521no—
149C22—6522no—
150C22—1632no—
151C22C22⋊C4×D16311yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—3261no—
156C4—3262no—
157C4—6521no—
158C2—81no—
159C2—13041no—
160C2—3262no—
161C2#100×D16311yescharacteristic
162C2#2378168012yes—
subgroupG/H
centerC22C22⋊C4×D163
commutatorC2×C4×C163C22×C4
FrattiniC42C22×D163
Fitting#329×C163C2
socleC22×C163C2×C22⋊C4
prime radicalC2×D4×D163C4
cosocleC42×C163C23
hypercenter#329D163
hypocenterC163#1533
LCSG ▷ C2×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #23781720C2semidirect
3. #23781405C2semidirect
4. #23781404C2semidirect
5. #329×C163C2semidirect
16. #19104646C22semidirect
18. #97×C163C22semidirect
81. C163#1533semidirect
87. #329D163semidirect
100. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#124×D1632
#237817202
#237814052
#237814042
#329×C1632
#1533163