Finite Group Databasebeta

← #35226002

Subgroups — #35226002

Order 20864.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#23782288C212yesmaximal, complemented
2#23781954C212yesmaximal, complemented
3#23781948C212yesmaximal, complemented
4#23781918C211yesmaximal, characteristic
5#23781896C212yesmaximal, complemented
6#23781889C212yesmaximal, complemented
7#23781883C212yesmaximal, complemented
8#23781693C211yesmaximal, characteristic, complemented
9#470×C163C211yesmaximal, characteristic, complemented
10C2×#14579686C2212yes—
11C2×D1304C2211yescharacteristic
12C22×C163⋊C8C2211yescharacteristic
13#19104734C2212yes—
14#19104730—22no—
15#19104724C2212yes—
16#19104721—22no—
17#19104717—22no—
18#19104715—22no—
19#19104709C2212yes—
20Q8×Dic163C2212yes—
21#19104697C2212yes—
22#19104694C2211yescharacteristic
23#19104667C2211yescharacteristic
24#19104666C2212yes—
25Dic163⋊C8C2212yes—
26Dic163⋊C8C2212yes—
27#19104659C2212yes—
28#19104657C2212yes—
29#19104655C2212yes—
30C2×Q16×C163C2211yescharacteristic
31C2×M16×C163C2211yescharacteristic
32C22⋊Q8×C163C2212yes—
33#106×C163C2211yescharacteristic
34#103×C163C2212yes—
35C2×Dic326—22no—
36C2×D652C2312yes—
37C2×C22⋊D163—22no—
38C2×C4×D163—22no—
39#14579686—42no—
40D1304—41no—
41C2×C163⋊C8—41no—
42C2×C163⋊C8C2312yes—
43C4⋊Dic163—22no—
44C4⋊Dic163D412yes—
45C4⋊Dic163C2312yes—
46Dic163⋊C4—22no—
47C22⋊Dic163D411yescharacteristic
48C22⋊Dic163D411yescharacteristic
49(C2×D163)⋊C4—22no—
50(C2×D163)⋊C4—22no—
51(C2×D163)⋊C4—22no—
52(C2×D163)⋊C4—22no—
53C4×Dic163—22no—
54C4×Dic163C2312yes—
55C2×Q8×C163C2312yes—
56C22×C4×C163C2311yescharacteristic
57Q16×C163—41no—
58M16×C163—41no—
59C2×C8×C163C2311yescharacteristic
60C2×C8×C163C2311yescharacteristic
61C4⋊C4×C163—22no—
62C4⋊C4×C163C2312yes—
63C22⋊C4×C163—22no—
64C22×D163—22no—
65Dic326—82no—
66D652—22no—
67D652—22no—
68C22⋊D163—82no—
69C2×Dic163—22no—
70C2×Dic163C2×D412yes—
71C2×Dic163C2×D412yes—
72C4×D163—82no—
73C163⋊C8—22no—
74C23×C163C2×D411yescharacteristic
75Q8×C163—22no—
76Q8×C163—22no—
77C2×C4×C163—21no—
78C2×C4×C163—21no—
79C2×C4×C163—22no—
80C2×C4×C163—22no—
81C2×C4×C163C2×D411yescharacteristic
82C2×C4×C163C2411yescharacteristic
83C8×C163—21no—
84C8×C163—21no—
85C22×C163—21no—
86C22×C163—22no—
87C22×C163C22×D411yescharacteristic
88C2×D163—42no—
89C2×D163—42no—
90C2×D163—42no—
91C4×C163—21no—
92C4×C163—42no—
93C4×C163—42no—
94C4×C163H(5,𝔽2)11yescharacteristic
95C4×C1632−1+411yescharacteristic
96Dic163—22no—
97Dic163—22no—
98Dic163—42no—
99C2×C163—41no—
100C2×C163#53711yescharacteristic
101C2×C163#57712yes—
102D163—82no—
103C163#326311yescharacteristic, complemented, Sylow
104#3263—1631nomaximal, Sylow
105#542—1632no—
106#476—1632no—
107#475—1632no—
108#470D16311yescharacteristic, complemented
109#469—1631no—
110#465—1632no—
111#464—1632no—
112#463—1632no—
113#424—1631no—
114C2×Q16C2×D16311yescharacteristic
115C2×SD16—1632no—
116C2×D8—1631no—
117C2×M16C2×D16311yescharacteristic
118C22×C8—1631no—
119C4⋊Q8—1632no—
120#126—3262no—
121#124—3262no—
122#124—1632no—
123#123—3262no—
124C22⋊Q8—3262no—
125C22⋊Q8C2×D16312yes—
126#121—1632no—
127C4×Q8—1632no—
128C4×D4—1632no—
129#117—1631no—
130#106—1631no—
131#106—1632no—
132#106C2×D16311yescharacteristic
133C4⋊C8—1632no—
134C4⋊C8—1632no—
135#103—1632no—
136#103C2×D16312yes—
137#102—1632no—
138#102—1632no—
139C2×Q8—3262no—
140C2×Q8C22×D16312yes—
141C2×D4—3262no—
142C2×D4—1632no—
143C22×C4—3262no—
144C22×C4C22×D16311yescharacteristic
145Q16—41no—
146SD16—6522no—
147D8—6521no—
148M16—41no—
149C2×C8—6521no—
150C2×C8—1632no—
151C2×C8C22×D16311yescharacteristic
152C2×C8C22×D16311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—3262no—
155C4⋊C4—3262no—
156C4⋊C4—1632no—
157C4⋊C4—1632no—
158C4⋊C4C22×D16312yes—
159C22⋊C4—22no—
160C22⋊C4—3262no—
161C22⋊C4—3262no—
162C22⋊C4—3262no—
163C22⋊C4—3262no—
164C22⋊C4—1631no—
165C22⋊C4—1631no—
166C42—3262no—
167C42—1632no—
168C23—3262no—
169C23D4×D16311yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—13042no—
173D4—13042no—
174D4—3262no—
175D4—3262no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—13042no—
181C2×C4—3262no—
182C2×C4—1632no—
183C2×C4—1632no—
184C2×C4D4×D16311yescharacteristic
185C2×C4C23×D16311yescharacteristic
186C8—21no—
187C8—21no—
188C8—3262no—
189C22—22no—
190C22—21no—
191C22—6522no—
192C22—6522no—
193C22—6522no—
194C22C2×D4×D16311yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—3262no—
199C4—3262no—
200C4—6522no—
201C4#1910478011yescharacteristic
202C4#1910478211yescharacteristic
203C2—41no—
204C2—13042no—
205C2#2378224211yescharacteristic
206C2#2378247412yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#470×C163C2
socleC22×C163C22×D4
prime radical#23781918C2
hypercenter#470D163
hypocenterC163#3263
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #23782288C2semidirect
2. #23781954C2semidirect
3. #23781948C2semidirect
5. #23781896C2semidirect
6. #23781889C2semidirect
7. #23781883C2semidirect
8. #23781693C2semidirect
9. #470×C163C2semidirect
103. C163#3263semidirect
108. #470D163semidirect

Maximal subgroups

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

subgroupindex
#237822882
#237819542
#237819482
#237819182
#237818962
#237818892
#237818832
#237816932
#470×C1632
#3263163