Finite Group Databasebeta

← #35226007

Subgroups — #35226007

Order 20864.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782286C212yesmaximal, complemented
2#23781956C212yesmaximal, complemented
3#23781951C212yesmaximal, complemented
4#23781923C211yesmaximal, characteristic, complemented
5#23781918C211yesmaximal, characteristic
6#23781894C212yesmaximal, complemented
7#23781888C212yesmaximal, complemented
8#23781881C212yesmaximal, complemented
9#424×C163C211yesmaximal, characteristic, complemented
10C2×Dic652C2211yescharacteristic
11C2×#14579688C2212yes—
12C2×D1304C2211yescharacteristic
13#19104735C2212yes—
14#19104730—22no—
15#19104723C2212yes—
16#19104721—22no—
17#19104717—22no—
18#19104714C2212yes—
19#19104713—22no—
20#19104709C2212yes—
21C4×Dic326C2212yes—
22C4×D652C2212yes—
23#19104667C2212yes—
24#19104667C2211yescharacteristic
25#19104667C2211yescharacteristic
26#19104660C2212yes—
27#19104660C2212yes—
28#19104657C2212yes—
29#19104657C2212yes—
30C2×M16×C163C2211yescharacteristic
31C22×C8×C163C2211yescharacteristic
32#117×C163C2211yescharacteristic
33C4⋊C8×C163C2212yes—
34C4⋊C8×C163C2212yes—
35C2×Dic326—22no—
36C2×Dic326C2312yes—
37C2×D652C2312yes—
38C2×C22⋊D163—22no—
39C2×C4×D163—22no—
40Dic652—41no—
41#14579688—42no—
42D1304—41no—
43C4⋊Dic163—22no—
44C4⋊Dic163C2312yes—
45C4⋊Dic163C2312yes—
46Dic163⋊C4—22no—
47Dic163⋊C4—22no—
48C22⋊Dic163—22no—
49(C2×D163)⋊C4—22no—
50(C2×D163)⋊C4—22no—
51(C2×D163)⋊C4—22no—
52(C2×D163)⋊C4—22no—
53C4×Dic163—22no—
54C22×C4×C163C2311yescharacteristic
55M16×C163—41no—
56C2×C8×C163—41no—
57C2×C8×C163C2312yes—
58C2×C8×C163C2311yescharacteristic
59C2×C8×C163C2311yescharacteristic
60C4⋊C4×C163D412yes—
61C22⋊C4×C163D411yescharacteristic
62C22⋊C4×C163D411yescharacteristic
63C42×C163C2312yes—
64C22×D163—22no—
65Dic326—22no—
66Dic326—22no—
67Dic326—82no—
68D652—22no—
69D652—22no—
70C22⋊D163—82no—
71C2×Dic163—22no—
72C2×Dic163—22no—
73C2×Dic163—22no—
74C4×D163—82no—
75C23×C163C2×D411yescharacteristic
76C2×C4×C163—21no—
77C2×C4×C163—21no—
78C2×C4×C163C2×D412yes—
79C2×C4×C163C2×D412yes—
80C2×C4×C163C2×D411yescharacteristic
81C2×C4×C163C2411yescharacteristic
82C8×C163—22no—
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—22no—
93C4×C163—22no—
94C4×C163H(5,𝔽2)11yescharacteristic
95C4×C1632−1+411yescharacteristic
96Dic163—42no—
97Dic163—42no—
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#470—1631no—
109#469—1631no—
110#465—1632no—
111#464—1632no—
112#463—1632no—
113#424D16311yescharacteristic, complemented
114C2×Q16—1631no—
115C2×SD16—1632no—
116C2×D8—1631no—
117C2×M16C2×D16311yescharacteristic
118C22×C8C2×D16311yescharacteristic
119C4⋊Q8—1632no—
120#126—3262no—
121#124—3262no—
122#124—1632no—
123#123—3262no—
124C22⋊Q8—3262no—
125C22⋊Q8—1632no—
126#121—1632no—
127C4×Q8—1632no—
128C4×D4—1632no—
129#117C2×D16311yescharacteristic
130#106—1632no—
131#106—1631no—
132#106—1631no—
133C4⋊C8C2×D16312yes—
134C4⋊C8C2×D16312yes—
135#103—1632no—
136#103—1632no—
137#102—1632no—
138#102—1632no—
139C2×Q8—3262no—
140C2×Q8—1632no—
141C2×D4—3262no—
142C2×D4—1632no—
143C22×C4—3262no—
144C22×C4C22×D16311yescharacteristic
145Q16—6521no—
146SD16—6522no—
147D8—6521no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D16311yescharacteristic
151C2×C8C22×D16312yes—
152C2×C8C22×D16311yescharacteristic
153C4⋊C4—3262no—
154C4⋊C4—3262no—
155C4⋊C4—3262no—
156C4⋊C4—1632no—
157C4⋊C4—1632no—
158C4⋊C4D65212yes—
159C22⋊C4—3262no—
160C22⋊C4—3262no—
161C22⋊C4—3262no—
162C22⋊C4—3262no—
163C22⋊C4—3262no—
164C22⋊C4D65211yescharacteristic
165C22⋊C4D65211yescharacteristic
166C42—3262no—
167C42C22×D16312yes—
168C23—3262no—
169C23C2×D65211yescharacteristic
170Q8—13042no—
171Q8—3262no—
172Q8—3262no—
173D4—13042no—
174D4—3262no—
175D4—3262no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—13042no—
179C2×C4—3262no—
180C2×C4—3262no—
181C2×C4—3262no—
182C2×C4C2×D65212yes—
183C2×C4C2×D65212yes—
184C2×C4C2×D65211yescharacteristic
185C2×C4C23×D16311yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—6522no—
192C22—6522no—
193C22—6522no—
194C22C22×D65211yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—6522no—
199C4—6522no—
200C4—6522no—
201C4#1910478011yescharacteristic
202C4#1910478311yescharacteristic
203C2—41no—
204C2—13042no—
205C2#2378224611yescharacteristic
206C2#2378247712yes—
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#424×C163C2
socleC22×C163C22×D4
prime radical#23781918C2
hypercenter#424D163
hypocenterC163#3263
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #23782286C2semidirect
2. #23781956C2semidirect
3. #23781951C2semidirect
4. #23781923C2semidirect
6. #23781894C2semidirect
7. #23781888C2semidirect
8. #23781881C2semidirect
9. #424×C163C2semidirect
103. C163#3263semidirect
113. #424D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237819562
#237819512
#237819232
#237819182
#237818942
#237818882
#237818812
#424×C1632
#3263163