Finite Group Databasebeta

← #35213511

Subgroups — #35213511

Order 20864.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#23782294C211yesmaximal, characteristic
2#23781823C211yesmaximal, characteristic
3#23781820C211yesmaximal, characteristic
4#23781586C211yesmaximal, characteristic
5#23781401C211yesmaximal, characteristic
6#23781390C211yesmaximal, characteristic
7#328×C163C211yesmaximal, characteristic
8C22×Dic326C2211yescharacteristic
9C2×Dic652—21no—
10C2×#14579689—21no—
11#19104734—21no—
12#19104733—21no—
13#19104725—21no—
14#19104725—21no—
15#19104714C2211yescharacteristic
16#19104712—21no—
17#19104691—21no—
18C2×C4⋊Dic163—21no—
19C2×Dic163⋊C4C2211yescharacteristic
20#19104660—21no—
21#19104658—21no—
22C22⋊(C163⋊C8)C2211yescharacteristic
23C163⋊#95—21no—
24C22⋊Q8×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C2×Dic326—21no—
28C2×Dic326—21no—
29C2×Dic326—21no—
30C2×Dic326—42no—
31C2×Dic326D411yescharacteristic
32C22×Dic163—21no—
33C22×Dic163D411yescharacteristic
34C22×Dic163D411yescharacteristic
35Dic652—42no—
36#14579689—42no—
37C2×C163⋊C8—21no—
38C4⋊Dic163—21no—
39C4⋊Dic163—42no—
40C4⋊Dic163D411yescharacteristic
41Dic163⋊C4—21no—
42Dic163⋊C4—21no—
43Dic163⋊C4—21no—
44Dic163⋊C4—21no—
45Dic163⋊C4—21no—
46Dic163⋊C4—21no—
47Dic163⋊C4—21no—
48C22⋊Dic163—21no—
49C4×Dic163—21no—
50C4×Dic163—21no—
51C2×Q8×C163D411yescharacteristic
52C22×C4×C163—21no—
53C22×C4×C163C2311yescharacteristic
54C2×C8×C163—21no—
55C4⋊C4×C163—21no—
56C4⋊C4×C163D411yescharacteristic
57C22⋊C4×C163—21no—
58Dic326—22no—
59Dic326—22no—
60Dic326—41no—
61Dic326—41no—
62Dic326—41no—
63Dic326—81no—
64C2×Dic163—21no—
65C2×Dic163—21no—
66C2×Dic163—22no—
67C2×Dic163—21no—
68C2×Dic163—21no—
69C2×Dic163—21no—
70C2×Dic163—41no—
71C2×Dic163—41no—
72C2×Dic163—41no—
73C2×Dic163Q1612yescomplemented
74C163⋊C8—41no—
75C23×C163C2×D411yescharacteristic
76Q8×C163—41no—
77C2×C4×C163—22no—
78C2×C4×C163—21no—
79C2×C4×C163—21no—
80C2×C4×C163—41no—
81C2×C4×C163—41no—
82C2×C4×C163—41no—
83C2×C4×C163C2×D411yescharacteristic
84C2×C4×C163C2×D411yescharacteristic
85C8×C163—41no—
86C22×C163—21no—
87C22×C163—21no—
88C22×C163#12011yescharacteristic
89C22×C163C2×Q1611yescharacteristic
90C22×C163#13711yescharacteristic
91C4×C163—21no—
92C4×C163—21no—
93C4×C163—42no—
94C4×C163—41no—
95C4×C163—41no—
96Dic163—22no—
97Dic163—41no—
98Dic163—42no—
99Dic163—41no—
100Dic163—41no—
101C2×C163—21no—
102C2×C163—21no—
103C2×C163#45111yescharacteristic
104C2×C163#45611yescharacteristic
105C2×C163B(4,𝔽2)11yescharacteristic
106C163#160511yescharacteristic, complemented, Sylow
107#1605—1631nomaximal, Sylow
108#544—1631no—
109#451—1631no—
110#451—1631no—
111#395—1631no—
112#328—1631no—
113#328D16311yescharacteristic
114#324—1631no—
115C22×Q8—1631no—
116C2×Q16—3261no—
117C2×Q16—3261no—
118C4⋊Q8—3261no—
119C4⋊Q8—3261no—
120#125—3261no—
121#125—3261no—
122C22⋊Q8—3261no—
123C22⋊Q8—1631no—
124C22⋊Q8C2×D16311yescharacteristic
125#117—3261no—
126C2×C4⋊C4—3261no—
127C2×C4⋊C4—1631no—
128#103—3261no—
129#103—3261no—
130C22⋊C8—1631no—
131C22⋊C8C2×D16311yescharacteristic
132#95—3261no—
133#95C2×D16311yescharacteristic
134C2×Q8—3261no—
135C2×Q8—3261no—
136C2×Q8—3261no—
137C2×Q8—6522no—
138C2×Q8—1631no—
139C2×Q8D65211yescharacteristic
140C22×C4—21no—
141C22×C4—3261no—
142C22×C4—1631no—
143C22×C4—1631no—
144C22×C4C22×D16311yescharacteristic
145Q16—6522no—
146Q16—6522no—
147C2×C8—21no—
148C2×C8—3261no—
149C4⋊C4—21no—
150C4⋊C4—3261no—
151C4⋊C4—3261no—
152C4⋊C4—3261no—
153C4⋊C4—3261no—
154C4⋊C4—3261no—
155C4⋊C4—3261no—
156C4⋊C4—3261no—
157C4⋊C4—3261no—
158C4⋊C4—6522no—
159C4⋊C4—1631no—
160C4⋊C4D65211yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3261no—
163C42—3261no—
164C42—3261no—
165C23D4×D16311yescharacteristic
166Q8—41no—
167Q8—13041no—
168Q8—3262no—
169Q8—3262no—
170Q8—6521no—
171Q8—6521no—
172Q8—6521no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—3261no—
180C2×C4—3261no—
181C2×C4—3262no—
182C2×C4—3261no—
183C2×C4—3261no—
184C2×C4—3261no—
185C2×C4—6521no—
186C2×C4—6521no—
187C2×C4—6521no—
188C2×C4—1632no—
189C2×C4D4×D16311yescharacteristic
190C2×C4C2×D65211yescharacteristic
191C8—41no—
192C8—6521no—
193C22—21no—
194C22—21no—
195C22#1910470511yescharacteristic
196C22Q16×D16311yescharacteristic
197C22#1910476411yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—3262no—
204C4—6521no—
205C4—6522no—
206C4—6521no—
207C4—6521no—
208C2—21no—
209C2—21no—
210C2#2378182211yescharacteristic
211C2#2378184911yescharacteristic
212C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#328×C163C2
socleC22×C163#120
prime radicalC23×C163C2×D4
hypercenter#328D163
hypocenterC163#1605
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
73. C2×Dic163Q16semidirect
106. C163#1605semidirect

Maximal subgroups

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

subgroupindex
#237822942
#237818232
#237818202
#237815862
#237814012
#237813902
#328×C1632
#1605163