Finite Group Databasebeta

← #35213846

Subgroups — #35213846

Order 20864.

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

normal subgroups only

This group has 196 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#23782419C211yesmaximal, characteristic
2#23782071C211yesmaximal, characteristic, complemented
3#23782057C211yesmaximal, characteristic
4#23781407C211yesmaximal, characteristic, complemented
5#23781406C211yesmaximal, characteristic, complemented
6#23781404C211yesmaximal, characteristic
7#501×C163C211yesmaximal, characteristic, complemented
8C2×#14579689—21no—
9C2×#14579686—21no—
10#19104734—21no—
11#19104730—21no—
12#19104730—21no—
13#19104727C2211yescharacteristic
14#19104723C2211yescharacteristic
15#19104720—21no—
16#19104715—21no—
17#19104713—21no—
18#19104668—21no—
19#19104666—21no—
20#19104646C2211yescharacteristic
21#19104646C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23#125×C163C2211yescharacteristic
24#107×C163—21no—
25#106×C163—21no—
26#97×C163C2211yescharacteristic
27C2×Dic326—21no—
28C2×Dic326D411yescharacteristic
29C2×D652D411yescharacteristic
30C2×C4×D163—21no—
31#14579689—42no—
32#14579686—42no—
33C2×C163⋊C8—21no—
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163D411yescharacteristic
37C4⋊Dic163D411yescharacteristic
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41(C2×D163)⋊C4—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46C4×Dic163—21no—
47C2×Q8×C163—21no—
48C2×C8×C163—21no—
49C4⋊C4×C163—21no—
50C4⋊C4×C163—21no—
51C4⋊C4×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163D411yescharacteristic
54C4⋊C4×C163D411yescharacteristic
55C42×C163C2311yescharacteristic
56C22×D163—21no—
57Dic326—41no—
58Dic326—81no—
59D652—41no—
60C2×Dic163—21no—
61C2×Dic163—21no—
62C2×Dic163—21no—
63C4×D163—81no—
64C163⋊C8—22no—
65C163⋊C8—22no—
66Q8×C163—41no—
67Q8×C163—41no—
68C2×C4×C163—21no—
69C2×C4×C163—21no—
70C2×C4×C163—41no—
71C2×C4×C163C2×D411yescharacteristic
72C2×C4×C163C2×D411yescharacteristic
73C2×C4×C163C2×D411yescharacteristic
74C8×C163—22no—
75C22×C163#12011yescharacteristic
76C2×D163—41no—
77C2×D163—41no—
78C2×D163—41no—
79C4×C163—21no—
80C4×C163—21no—
81C4×C163—21no—
82C4×C163—41no—
83C4×C163—41no—
84C4×C163—81no—
85Dic163—41no—
86Dic163—41no—
87Dic163—41no—
88C2×C163#45311yescharacteristic
89C2×C163#45611yescharacteristic
90C2×C163#45611yescharacteristic
91D163—81no—
92C163#166011yescharacteristic, complemented, Sylow
93#1660—1631nomaximal, Sylow
94#563—1631no—
95#501—1631no—
96#501D16311yescharacteristic, complemented
97#497—1631no—
98#330—1631no—
99#330—1631no—
100#329—1631no—
101C2×Q16—3261no—
102C2×SD16—3261no—
103C4⋊Q8—3261no—
104C4⋊Q8C2×D16311yescharacteristic
105#126—3261no—
106#126—3261no—
107#125—1631no—
108#125C2×D16311yescharacteristic
109#124—1631no—
110#123—3261no—
111C22⋊Q8—3261no—
112C22⋊Q8—3261no—
113#107—21no—
114#107—3261no—
115#106—21no—
116#106—3261no—
117#97—1631no—
118#97—1631no—
119#97C2×D16311yescharacteristic
120C2×Q8—21no—
121C2×Q8—3261no—
122C2×Q8—1631no—
123C2×D4—1631no—
124C22×C4—3261no—
125Q16—6522no—
126SD16—6522no—
127C2×C8—21no—
128C2×C8—3261no—
129C2×C8—3261no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3261no—
135C4⋊C4—3261no—
136C4⋊C4—3261no—
137C4⋊C4—3261no—
138C4⋊C4—1631no—
139C4⋊C4—1631no—
140C4⋊C4C22⋊D16311yescharacteristic
141C4⋊C4C22⋊D16311yescharacteristic
142C22⋊C4—3261no—
143C22⋊C4—3261no—
144C22⋊C4—3261no—
145C22⋊C4—3261no—
146C22⋊C4—3261no—
147C42—3261no—
148C42C22×D16311yescharacteristic
149C23—3261no—
150Q8—41no—
151Q8—41no—
152Q8—13041no—
153Q8—6521no—
154D4—6521no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—13041no—
159C2×C4—3261no—
160C2×C4—3261no—
161C2×C4—3261no—
162C2×C4D4×D16311yescharacteristic
163C2×C4D4×D16311yescharacteristic
164C2×C4C2×C22⋊D16311yescharacteristic
165C8—22no—
166C8—3262no—
167C8—3262no—
168C22—6521no—
169C22—6521no—
170C22—6521no—
171C22#1910470311yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—6521no—
179C4—6521no—
180C4—6521no—
181C2—13041no—
182C2#2378183111yescharacteristic
183C2#2378184611yescharacteristic
184C2#2378184711yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#501×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#501D163
hypocenterC163#1660
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #23782071C2semidirect
4. #23781407C2semidirect
5. #23781406C2semidirect
7. #501×C163C2semidirect
92. C163#1660semidirect
96. #501D163semidirect

Maximal subgroups

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

subgroupindex
#237824192
#237820712
#237820572
#237814072
#237814062
#237814042
#501×C1632
#1660163