Finite Group Databasebeta

← #35213535

Subgroups — #35213535

Order 20864.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782262C211yesmaximal, characteristic
2#23781996C211yesmaximal, characteristic, complemented
3#23781979C211yesmaximal, characteristic
4#23781595C211yesmaximal, characteristic, complemented
5#23781403C211yesmaximal, characteristic, complemented
6#23781400C211yesmaximal, characteristic
7#324×C163C211yesmaximal, characteristic, complemented
8#19104730—21no—
9#19104725—21no—
10#19104720—21no—
11#19104718—21no—
12#19104714C2211yescharacteristic
13#19104713—21no—
14#19104709C2211yescharacteristic
15C4×C22⋊D163—21no—
16C2×C4⋊Dic163C2211yescharacteristic
17C2×Dic163⋊C4—21no—
18#19104669—21no—
19#19104667—21no—
20#19104660—21no—
21#19104657—21no—
22C163⋊#95—21no—
23C163⋊#95C2211yescharacteristic
24C2×C4⋊C4×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26C22⋊C8×C163C2211yescharacteristic
27C2×Dic326D411yescharacteristic
28C2×D652D411yescharacteristic
29C2×C22⋊D163—21no—
30C22×Dic163—21no—
31C22×Dic163—21no—
32C2×C4×D163—21no—
33C4⋊Dic163—21no—
34C4⋊Dic163—21no—
35C4⋊Dic163D411yescharacteristic
36C4⋊Dic163D411yescharacteristic
37Dic163⋊C4—21no—
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41Dic163⋊C4—42no—
42C22⋊Dic163—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—21no—
47C4×Dic163—21no—
48C22×C4×C163D411yescharacteristic
49C22×C4×C163D411yescharacteristic
50C22×C4×C163C2311yescharacteristic
51C2×C8×C163—21no—
52C2×C8×C163—21no—
53C4⋊C4×C163—21no—
54C4⋊C4×C163—21no—
55C22×D163—21no—
56Dic326—41no—
57D652—41no—
58C22⋊D163—41no—
59C22⋊D163—41no—
60C2×Dic163—22no—
61C2×Dic163—21no—
62C2×Dic163—21no—
63C2×Dic163—21no—
64C2×Dic163—41no—
65C2×Dic163—41no—
66C2×Dic163—41no—
67C2×Dic163—41no—
68C2×Dic163—41no—
69C4×D163—81no—
70C23×C163C2×D411yescharacteristic
71C2×C4×C163—21no—
72C2×C4×C163—21no—
73C2×C4×C163—21no—
74C2×C4×C163—21no—
75C2×C4×C163—41no—
76C2×C4×C163—41no—
77C2×C4×C163C2×D411yescharacteristic
78C2×C4×C163C2×D411yescharacteristic
79C8×C163—41no—
80C8×C163—41no—
81C22×C163—21no—
82C22×C163—21no—
83C22×C163#12011yescharacteristic
84C22×C163#13511yescharacteristic
85C22×C163AGL(1,ℤ/8)11yescharacteristic
86C2×D163—41no—
87C2×D163—41no—
88C2×D163—41no—
89C4×C163—21no—
90C4×C163—21no—
91C4×C163—41no—
92C4×C163—41no—
93Dic163—42no—
94Dic163—41no—
95Dic163—41no—
96Dic163—41no—
97Dic163—81no—
98C2×C163—21no—
99C2×C163—21no—
100C2×C163#44911yescharacteristic
101C2×C163#45611yescharacteristic
102C2×C163#45811yescharacteristic
103D163—81no—
104C163#160911yescharacteristic, complemented, Sylow
105#1609—1631nomaximal, Sylow
106#539—1631no—
107#483—1631no—
108#480—1631no—
109#398—1631no—
110#328—1631no—
111#327—1631no—
112#324D16311yescharacteristic, complemented
113#126—3261no—
114#125—3261no—
115#123—3261no—
116#123—3261no—
117C22⋊Q8—3261no—
118C22⋊Q8—1631no—
119#121—1631no—
120C4×D4—3261no—
121C2×C4⋊C4—3261no—
122C2×C4⋊C4—1631no—
123C2×C4⋊C4C2×D16311yescharacteristic
124#107—3261no—
125#106—3261no—
126#103—3261no—
127#102—3261no—
128C22⋊C8C2×D16311yescharacteristic
129C22⋊C8C2×D16311yescharacteristic
130#95—3261no—
131#95—1631no—
132C2×Q8—1631no—
133C2×D4—3261no—
134C2×D4—1631no—
135C22×C4—3261no—
136C22×C4—3261no—
137C22×C4—3261no—
138C22×C4D65211yescharacteristic
139C22×C4D65211yescharacteristic
140C22×C4C22×D16311yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—3261no—
146C4⋊C4—3261no—
147C4⋊C4—3261no—
148C4⋊C4—3261no—
149C4⋊C4—3261no—
150C4⋊C4—3261no—
151C4⋊C4—6522no—
152C4⋊C4—1631no—
153C4⋊C4—1631no—
154C22⋊C4—3261no—
155C22⋊C4—3261no—
156C22⋊C4—3261no—
157C22⋊C4—3261no—
158C22⋊C4—3261no—
159C42—3261no—
160C23—3261no—
161C23C2×D65211yescharacteristic
162Q8—6521no—
163D4—6521no—
164D4—6521no—
165D4—6521no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—13041no—
173C2×C4—3262no—
174C2×C4—3261no—
175C2×C4—3261no—
176C2×C4—3261no—
177C2×C4—6521no—
178C2×C4—6521no—
179C2×C4—6521no—
180C2×C4—6521no—
181C2×C4—6521no—
182C2×C4D4×D16311yescharacteristic
183C2×C4D4×D16311yescharacteristic
184C8—41no—
185C8—41no—
186C22—21no—
187C22—21no—
188C22—6521no—
189C22—6521no—
190C22—6521no—
191C22#1910470511yescharacteristic
192C22#1910475711yescharacteristic
193C22#1910476311yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—13041no—
199C4—6522no—
200C4—6521no—
201C4—6521no—
202C4—6521no—
203C2—21no—
204C2—21no—
205C2—13041no—
206C2#2378181211yescharacteristic
207C2#2378185011yescharacteristic
208C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#324×C163C2
socleC22×C163#120
prime radical#19104709C22
hypercenter#324D163
hypocenterC163#1609
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #23781996C2semidirect
4. #23781595C2semidirect
5. #23781403C2semidirect
7. #324×C163C2semidirect
104. C163#1609semidirect
112. #324D163semidirect

Maximal subgroups

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

subgroupindex
#237822622
#237819962
#237819792
#237815952
#237814032
#237814002
#324×C1632
#1609163