Finite Group Databasebeta

← #35213528

Subgroups — #35213528

Order 20864.

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

normal subgroups only

This group has 228 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#23782290C211yesmaximal, characteristic
2#23781996C211yesmaximal, characteristic, complemented
3#23781993C211yesmaximal, characteristic, complemented
4#23781592C211yesmaximal, characteristic
5#23781401C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic
7#328×C163C211yesmaximal, characteristic, complemented
8#19104715—21no—
9#19104714C2211yescharacteristic
10#19104713—21no—
11#19104713—21no—
12#19104712—21no—
13#19104711—21no—
14C2×C4⋊Dic163C2211yescharacteristic
15C2×(C2×D163)⋊C4—21no—
16C2×(C2×D163)⋊C4C2211yescharacteristic
17C22⋊C4×D163—21no—
18#19104667—21no—
19#19104666—21no—
20#19104660—21no—
21#19104659—21no—
22C22⋊(C163⋊C8)C2211yescharacteristic
23C163⋊#95—21no—
24C22⋊Q8×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×Dic326—21no—
29C2×Dic326D411yescharacteristic
30C22×Dic163—21no—
31C22×Dic163D411yescharacteristic
32C2×C4×D163—21no—
33C2×C4×D163—21no—
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163—21no—
37C4⋊Dic163—21no—
38C4⋊Dic163D411yescharacteristic
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41Dic163⋊C4—21no—
42C22⋊Dic163—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—21no—
47(C2×D163)⋊C4—42no—
48C2×Q8×C163D411yescharacteristic
49C22×C4×C163—21no—
50C22×C4×C163C2311yescharacteristic
51C2×C8×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163D411yescharacteristic
54C22⋊C4×C163—21no—
55C22×D163—21no—
56C22×D163—21no—
57C22×D163—41no—
58C22×D163—41no—
59C22×D163SD1612yes—
60Dic326—41no—
61Dic326—81no—
62C2×Dic163—21no—
63C2×Dic163—21no—
64C2×Dic163—21no—
65C2×Dic163—41no—
66C2×Dic163—41no—
67C2×Dic163—41no—
68C2×Dic163—41no—
69C4×D163—81no—
70C4×D163—81no—
71C163⋊C8—41no—
72C23×C163C2×D411yescharacteristic
73Q8×C163—41no—
74C2×C4×C163—21no—
75C2×C4×C163—21no—
76C2×C4×C163—41no—
77C2×C4×C163—41no—
78C2×C4×C163—41no—
79C2×C4×C163—41no—
80C2×C4×C163C2×D411yescharacteristic
81C2×C4×C163C2×D411yescharacteristic
82C8×C163—41no—
83C22×C163—21no—
84C22×C163—21no—
85C22×C163#12011yescharacteristic
86C22×C163C2×SD1611yescharacteristic
87C22×C163#13711yescharacteristic
88C2×D163—22no—
89C2×D163—41no—
90C2×D163—41no—
91C2×D163—42no—
92C2×D163—81no—
93C4×C163—21no—
94C4×C163—21no—
95C4×C163—41no—
96C4×C163—41no—
97C4×C163—81no—
98Dic163—41no—
99Dic163—41no—
100Dic163—41no—
101Dic163—81no—
102C2×C163—21no—
103C2×C163—21no—
104C2×C163#44811yescharacteristic
105C2×C163#45511yescharacteristic
106C2×C163#45811yescharacteristic
107D163—42no—
108C163#160811yescharacteristic, complemented, Sylow
109#1608—1631nomaximal, Sylow
110#543—1631no—
111#483—1631no—
112#483—1631no—
113#397—1631no—
114#328—1631no—
115#328D16311yescharacteristic, complemented
116#323—1631no—
117C22⋊Q8—3261no—
118C22⋊Q8—3261no—
119C22⋊Q8—3261no—
120C22⋊Q8—3261no—
121C22⋊Q8—3261no—
122C22⋊Q8—1631no—
123C22⋊Q8C2×D16311yescharacteristic
124C2×C4⋊C4—1631no—
125C2×C22⋊C4—3261no—
126C2×C22⋊C4—3261no—
127C2×C22⋊C4—1631no—
128#106—3261no—
129#106—3261no—
130#103—3261no—
131#103—3261no—
132C22⋊C8—1631no—
133C22⋊C8C2×D16311yescharacteristic
134#95—3261no—
135#95C2×D16311yescharacteristic
136C24—1631no—
137C2×Q8—3261no—
138C2×Q8—1631no—
139C2×Q8D65211yescharacteristic
140C22×C4—21no—
141C22×C4—3261no—
142C22×C4—3261no—
143C22×C4—3261no—
144C22×C4—1631no—
145C22×C4C22×D16311yescharacteristic
146C2×C8—21no—
147C2×C8—3261no—
148C4⋊C4—21no—
149C4⋊C4—3261no—
150C4⋊C4—3261no—
151C4⋊C4—3261no—
152C4⋊C4—3261no—
153C4⋊C4—3261no—
154C4⋊C4—3261no—
155C4⋊C4—1631no—
156C4⋊C4D65211yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3261no—
159C22⋊C4—3261no—
160C22⋊C4—3261no—
161C22⋊C4—3261no—
162C22⋊C4—3261no—
163C22⋊C4—6522no—
164C23—3261no—
165C23—3261no—
166C23—6521no—
167C23—6521no—
168C23—1632no—
169C23D4×D16311yescharacteristic
170Q8—41no—
171Q8—13041no—
172Q8—6521no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—13041no—
180C2×C4—13041no—
181C2×C4—3261no—
182C2×C4—3261no—
183C2×C4—3261no—
184C2×C4—6521no—
185C2×C4—6521no—
186C2×C4—6521no—
187C2×C4—6521no—
188C2×C4D4×D16311yescharacteristic
189C2×C4C2×D65211yescharacteristic
190C8—41no—
191C8—6521no—
192C22—21no—
193C22—21no—
194C22—13041no—
195C22—3262no—
196C22—6522no—
197C22—6521no—
198C22—6521no—
199C22#1910470511yescharacteristic
200C22SD16×D16311yescharacteristic
201C22#1910476711yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—13041no—
208C4—6521no—
209C4—6521no—
210C4—6521no—
211C2—21no—
212C2—21no—
213C2—6522no—
214C2#2378180511yescharacteristic
215C2#2378184411yescharacteristic
216C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#328×C163C2
socleC22×C163#120
prime radicalC23×D163D4
hypercenter#328D163
hypocenterC163#1608
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
2. #23781996C2semidirect
3. #23781993C2semidirect
5. #23781401C2semidirect
7. #328×C163C2semidirect
108. C163#1608semidirect
115. #328D163semidirect

Maximal subgroups

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

subgroupindex
#237822902
#237819962
#237819932
#237815922
#237814012
#237813882
#328×C1632
#1608163