Finite Group Databasebeta

← #35216532

Subgroups — #35216532

Order 20864.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#23781854C212yesmaximal, complemented
2#23781832C211yesmaximal, characteristic
3#23781458C211yesmaximal, characteristic, complemented
4#23781455C212yesmaximal, complemented
5#559×C163C211yesmaximal, characteristic, complemented
6#19104780C2211yescharacteristic
7#19104761—21no—
8#19104718—22no—
9#19104663—21no—
10#19104652C2211yescharacteristic
11#19104650C2212yes—
12#19104649—22no—
13#127×C163C2211yescharacteristic
14#124×C163—22no—
15#123×C163—21no—
16#123×C163C2212yes—
17C4×Q8×C163—21no—
18C4×D4×C163—22no—
19#14579699—21no—
20#14579698—41no—
21C2×C22⋊D163—22no—
22D4×D163—41no—
23#14579687—41no—
24#14579684—41no—
25#14579683—21no—
26Dic163⋊C4—42no—
27C22⋊Dic163—22no—
28(C2×D163)⋊C4—42no—
29C2×Q8×C163—21no—
30C2×D4×C163—22no—
31C2×D4×C163D411yescharacteristic
32C2×D4×C163C2311yescharacteristic
33C22×C4×C163—21no—
34C22×C4×C163D412yes—
35C4⋊C4×C163—21no—
36C4⋊C4×C163—22no—
37C22⋊C4×C163—22no—
38C22⋊C4×C163—21no—
39C22⋊C4×C163—22no—
40C22⋊C4×C163D412yes—
41C42×C163—22no—
42C42×C163D411yescharacteristic
43C22×D163—22no—
44Dic326—21no—
45D652—21no—
46C22⋊D163—42no—
47C22⋊D163—41no—
48C2×Dic163—22no—
49C2×Dic163—42no—
50C4×D163—41no—
51C163⋊C8—41no—
52C23×C163—21no—
53C23×C163C2×D412yes—
54Q8×C163—42no—
55D4×C163—41no—
56D4×C163—41no—
57D4×C163—44no—
58C2×C4×C163—21no—
59C2×C4×C163—22no—
60C2×C4×C163—22no—
61C2×C4×C163—42no—
62C2×C4×C163—81no—
63C2×C4×C163C2×D411yescharacteristic
64C22×C163—22no—
65C22×C163—42no—
66C22×C163—42no—
67C22×C163—41no—
68C22×C163#12011yescharacteristic
69C2×D163—41no—
70C2×D163—82no—
71C4×C163—22no—
72C4×C163—21no—
73C4×C163—42no—
74C4×C163—42no—
75Dic163—41no—
76Dic163—82no—
77C2×C163—21no—
78C2×C163—42no—
79C2×C163—81no—
80C2×C163B(4,𝔽2)11yescharacteristic
81D163—81no—
82C163#219811yescharacteristic, complemented, Sylow
83#2198—1631nomaximal, Sylow
84#559D16311yescharacteristic, complemented
85#458—1632no—
86#453—1631no—
87#353—1631no—
88#352—1632no—
89H(5,𝔽2)—1631no—
90AGL(1,ℤ/8)—3261no—
91#127C2×D16311yescharacteristic
92#124—22no—
93#123—21no—
94#123—3262no—
95#123C2×D16312yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—3261no—
99#100—1631no—
100#99—3262no—
101#99—1632no—
102#41—3261no—
103#41—6521no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—3262no—
107C2×D4—6521no—
108C2×D4C22⋊D16311yescharacteristic
109C2×D4C22×D16311yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D16312yes—
112SD16—6521no—
113D8—6521no—
114M16—3261no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—6522no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—3262no—
122C22⋊C4—6522no—
123C22⋊C4C22⋊D16312yes—
124C42—22no—
125C42C22⋊D16311yescharacteristic
126C23—21no—
127C23—3262no—
128C23C2×C22⋊D16312yes—
129Q8—42no—
130Q8—3261no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3261no—
135D4—6522no—
136D4—6521no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—3262no—
143C2×C4—6521no—
144C2×C4—6522no—
145C2×C4C2×C22⋊D16311yescharacteristic
146C8—6521no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—13042no—
152C22—6521no—
153C22#1910470411yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—13042no—
159C4—6521no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—13041no—
164C2#2378185211yescharacteristic
subgroupG/H
centerC2#23781852
commutator, cosocleC2×D4×C163C23
FrattiniC2×D4C22×D163
Fitting#559×C163C2
socleC2×C163B(4,𝔽2)
prime radical#23781832C2
hypercenter#559D163
hypocenterC163#2198
LCSG ▷ C2×D4×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #23781854C2semidirect
3. #23781458C2semidirect
4. #23781455C2semidirect
5. #559×C163C2semidirect
82. C163#2198semidirect
84. #559D163semidirect

Maximal subgroups

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

subgroupindex
#237818542
#237818322
#237814582
#237814552
#559×C1632
#2198163