Finite Group Databasebeta

← #35213501

Subgroups — #35213501

Order 20864.

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

normal subgroups only

This group has 244 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#23781823C211yesmaximal, characteristic, complemented
3#23781806C211yesmaximal, characteristic
4#23781585C211yesmaximal, characteristic, complemented
5#23781403C211yesmaximal, characteristic, complemented
6#23781400C211yesmaximal, characteristic
7#324×C163C211yesmaximal, characteristic, complemented
8C22×Dic326C2211yescharacteristic
9C2×Dic652—21no—
10C2×#14579688—21no—
11#19104730—21no—
12#19104725—21no—
13#19104720—21no—
14#19104718—21no—
15#19104714C2211yescharacteristic
16#19104713—21no—
17#19104709C2211yescharacteristic
18C4×C22⋊D163—21no—
19C2×Dic163⋊C4—21no—
20#19104660—21no—
21#19104660—21no—
22C163⋊#95—21no—
23C163⋊#95C2211yescharacteristic
24C2×C4⋊C4×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26C22⋊C8×C163C2211yescharacteristic
27C2×Dic326—21no—
28C2×Dic326—21no—
29C2×Dic326—42no—
30C2×Dic326D411yescharacteristic
31C2×D652D411yescharacteristic
32C2×C22⋊D163—21no—
33C22×Dic163—21no—
34C22×Dic163—21no—
35C2×C4×D163—21no—
36Dic652—42no—
37#14579688—42no—
38C4⋊Dic163D411yescharacteristic
39C4⋊Dic163D411yescharacteristic
40Dic163⋊C4—21no—
41Dic163⋊C4—21no—
42Dic163⋊C4—21no—
43Dic163⋊C4—21no—
44Dic163⋊C4—42no—
45C22⋊Dic163—21no—
46(C2×D163)⋊C4—21no—
47(C2×D163)⋊C4—21no—
48(C2×D163)⋊C4—21no—
49(C2×D163)⋊C4—21no—
50C4×Dic163—21no—
51C22×C4×C163D411yescharacteristic
52C22×C4×C163D411yescharacteristic
53C22×C4×C163C2311yescharacteristic
54C2×C8×C163—21no—
55C2×C8×C163—21no—
56C4⋊C4×C163—21no—
57C4⋊C4×C163—21no—
58C22×D163—21no—
59Dic326—22no—
60Dic326—22no—
61Dic326—41no—
62Dic326—41no—
63Dic326—41no—
64D652—41no—
65C22⋊D163—41no—
66C22⋊D163—41no—
67C2×Dic163—22no—
68C2×Dic163—22no—
69C2×Dic163—21no—
70C2×Dic163—21no—
71C2×Dic163—21no—
72C2×Dic163—41no—
73C2×Dic163—41no—
74C2×Dic163—41no—
75C2×Dic163—41no—
76C4×D163—81no—
77C23×C163C2×D411yescharacteristic
78C2×C4×C163—21no—
79C2×C4×C163—21no—
80C2×C4×C163—21no—
81C2×C4×C163—21no—
82C2×C4×C163—41no—
83C2×C4×C163—41no—
84C2×C4×C163C2×D411yescharacteristic
85C2×C4×C163C2×D411yescharacteristic
86C8×C163—41no—
87C8×C163—41no—
88C22×C163—21no—
89C22×C163—21no—
90C22×C163#12011yescharacteristic
91C22×C163#13511yescharacteristic
92C22×C163#13711yescharacteristic
93C2×D163—41no—
94C2×D163—41no—
95C2×D163—41no—
96C4×C163—21no—
97C4×C163—21no—
98C4×C163—41no—
99C4×C163—41no—
100Dic163—42no—
101Dic163—41no—
102Dic163—41no—
103Dic163—41no—
104Dic163—42no—
105C2×C163—21no—
106C2×C163—21no—
107C2×C163#45211yescharacteristic
108C2×C163#45611yescharacteristic
109C2×C163B(4,𝔽2)11yescharacteristic
110D163—81no—
111C163#160311yescharacteristic, complemented, Sylow
112#1603—1631nomaximal, Sylow
113#539—1631no—
114#451—1631no—
115#448—1631no—
116#395—1631no—
117#328—1631no—
118#327—1631no—
119#324D16311yescharacteristic, complemented
120C22×Q8—1631no—
121C2×Q16—3261no—
122C2×SD16—3261no—
123#126—3261no—
124#125—3261no—
125#123—3261no—
126#123—3261no—
127C22⋊Q8—3261no—
128C22⋊Q8—1631no—
129#121—1631no—
130C4×D4—3261no—
131C2×C4⋊C4—3261no—
132C2×C4⋊C4C2×D16311yescharacteristic
133#103—3261no—
134#103—3261no—
135C22⋊C8C2×D16311yescharacteristic
136C22⋊C8C2×D16311yescharacteristic
137#95—3261no—
138#95—1631no—
139C2×Q8—3261no—
140C2×Q8—3261no—
141C2×Q8—6522no—
142C2×Q8—1631no—
143C2×D4—3261no—
144C2×D4—1631no—
145C22×C4—3261no—
146C22×C4—3261no—
147C22×C4—3261no—
148C22×C4D65211yescharacteristic
149C22×C4D65211yescharacteristic
150C22×C4C22×D16311yescharacteristic
151Q16—6522no—
152SD16—6522no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3261no—
158C4⋊C4—3261no—
159C4⋊C4—3261no—
160C4⋊C4—3261no—
161C4⋊C4—6522no—
162C4⋊C4—1631no—
163C4⋊C4—1631no—
164C22⋊C4—3261no—
165C22⋊C4—3261no—
166C22⋊C4—3261no—
167C22⋊C4—3261no—
168C22⋊C4—3261no—
169C42—3261no—
170C23—3261no—
171C23C2×D65211yescharacteristic
172Q8—3262no—
173Q8—3262no—
174Q8—6521no—
175Q8—6521no—
176Q8—6521no—
177D4—6521no—
178D4—6521no—
179D4—6521no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—13041no—
187C2×C4—3262no—
188C2×C4—3262no—
189C2×C4—3261no—
190C2×C4—3261no—
191C2×C4—3261no—
192C2×C4—6521no—
193C2×C4—6521no—
194C2×C4—6521no—
195C2×C4—6521no—
196C2×C4D4×D16311yescharacteristic
197C2×C4D4×D16311yescharacteristic
198C8—41no—
199C8—41no—
200C22—21no—
201C22—21no—
202C22—6521no—
203C22—6521no—
204C22—6521no—
205C22#1910470511yescharacteristic
206C22#1910475711yescharacteristic
207C22#1910476811yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—6522no—
213C4—6521no—
214C4—6521no—
215C4—6521no—
216C4—6522no—
217C2—21no—
218C2—21no—
219C2—13041no—
220C2#2378182911yescharacteristic
221C2#2378185011yescharacteristic
222C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#324×C163C2
socleC22×C163#120
prime radical#19104709C22
hypercenter#324D163
hypocenterC163#1603
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. #23781823C2semidirect
4. #23781585C2semidirect
5. #23781403C2semidirect
7. #324×C163C2semidirect
111. C163#1603semidirect
119. #324D163semidirect

Maximal subgroups

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

subgroupindex
#237822622
#237818232
#237818062
#237815852
#237814032
#237814002
#324×C1632
#1603163