Finite Group Databasebeta

← #36356139

Subgroups — #36356139

Order 22144.

1160 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×C22⋊D173C211yesmaximal, characteristic
2C2×C22⋊C4×D173C212yesmaximal, complemented
3#26011367C218yesmaximal, complemented
4#26011352C212yesmaximal, complemented
5#26011347C211yesmaximal, characteristic, complemented
6#379×C173C211yesmaximal, characteristic, complemented
7C24×D173C2211yescharacteristic
8C22×C22⋊D173—28no—
9C22×C22⋊D173C418yescomplemented
10C22×C22⋊D173C2214yes—
11C23×Dic173C2211yescharacteristic
12C22×C4×D173C2214yescomplemented
13C2×C22⋊Dic173C2212yes—
14C2×C22⋊Dic173C2214yescomplemented
15C2×#14583803C2214yescomplemented
16C22⋊C4×D173—28no—
17#19149586C2218yescomplemented
18C25×C173C2211yescharacteristic
19C2×C22⋊C4×C173C2212yes—
20C2×C22⋊C4×C173C2214yescomplemented
21C23×D173—44no—
22C23×D173C2×C414yes—
23C23×D173D418yes—
24C23×D173C2312yes—
25C2×C22⋊D173—232no—
26C2×C22⋊D173—28no—
27C2×C22⋊D173—28no—
28C2×C22⋊D173C2×C4116yescomplemented
29C22×Dic173—24no—
30C22×Dic173C2×C414yescomplemented
31C22×Dic173C2314yes—
32C22×Dic173C2312yes—
33C2×C4×D173—28no—
34C2×C4×D173—48no—
35C22⋊Dic173—24no—
36C22⋊Dic173—44no—
37#14583803—28no—
38C24×C173—24no—
39C24×C173—44no—
40C24×C173C2×C414yescomplemented
41C24×C173C2312yes—
42C24×C173C2311yescharacteristic
43C22×C4×C173C2314yes—
44C22⋊C4×C173—24no—
45C22⋊C4×C173—44no—
46C22×D173—28no—
47C22×D173—48no—
48C22×D173—416no—
49C22×D173C22⋊C4116yescomplemented
50C22×D173C22×C414yes—
51C22×D173C2×D414yes—
52C22×D173C2×D414yes—
53C22⋊D173—232no—
54C2×Dic173—28no—
55C2×Dic173—24no—
56C2×Dic173—24no—
57C2×Dic173—28no—
58C2×Dic173C22×C414yes—
59C4×D173—416no—
60C23×C173—24no—
61C23×C173—28no—
62C23×C173—24no—
63C23×C173—24no—
64C23×C173—24no—
65C23×C173—48no—
66C23×C173—416no—
67C23×C173C22×C414yes—
68C23×C173C22×C412yes—
69C23×C173C2×D414yes—
70C23×C173C2411yescharacteristic
71C2×C4×C173—24no—
72C2×C4×C173—28no—
73C22×C173—24no—
74C22×C173—28no—
75C22×C173—24no—
76C22×C173—24no—
77C22×C173—24no—
78C22×C173—416no—
79C22×C173—48no—
80C22×C173C2×C22⋊C414yes—
81C22×C173C23×C411yescharacteristic
82C22×C173C22×D412yes—
83C22×C173H(5,𝔽2)14yes—
84C2×D173—28no—
85C2×D173—28no—
86C2×D173—28no—
87C2×D173—416no—
88C2×D173C2×C22⋊C418yes—
89C4×C173—44no—
90Dic173—24no—
91Dic173—44no—
92C2×C173—24no—
93C2×C173—44no—
94C2×C173C22×C22⋊C411yescharacteristic
95C2×C173#51812yes—
96C2×C173#53414yes—
97D173—28no—
98C173#230811yescharacteristic, complemented, Sylow
99#2308—1731nomaximal, Sylow
100C23×D4—1731no—
101C22×C22⋊C4—1732no—
102#386—1738no—
103#381—1732no—
104#379—1731no—
105#379D17311yescharacteristic, complemented
106C25—1731no—
107C25C2×D17311yescharacteristic
108C22×D4—3468no—
109C22×D4—1738no—
110C22×D4—1734no—
111C23×C4—1731no—
112C23×C4—1734no—
113C2×C22⋊C4—3468no—
114C2×C22⋊C4—1732no—
115C2×C22⋊C4—1734no—
116C2×C22⋊C4—1734no—
117C2×C22⋊C4C2×D17312yes—
118C2×C22⋊C4C2×D17314yescomplemented
119#95—1738no—
120C24—24no—
121C24—44no—
122C24—1732no—
123C24—1734no—
124C24—1738no—
125C24—6924no—
126C24C4×D17314yescomplemented
127C24C22×D17312yes—
128C24C22×D17311yescharacteristic
129C2×D4—34632no—
130C2×D4—3468no—
131C2×D4—3468no—
132C2×D4—17316no—
133C22×C4—3468no—
134C22×C4—3464no—
135C22×C4—1734no—
136C22×C4—1734no—
137C22×C4—1732no—
138C22×C4—6928no—
139C22×C4C22×D17314yes—
140C22⋊C4—24no—
141C22⋊C4—44no—
142C22⋊C4—3468no—
143C22⋊C4—3464no—
144C22⋊C4—6924no—
145C23—24no—
146C23—28no—
147C23—24no—
148C23—24no—
149C23—24no—
150C23—48no—
151C23—416no—
152C23—3468no—
153C23—1734no—
154C23—1734no—
155C23—1734no—
156C23—17316no—
157C23—6928no—
158C23—69216no—
159C23C2×C4×D17314yes—
160C23C2×C4×D17312yes—
161C23D4×D17314yes—
162C23C23×D17311yescharacteristic
163D4—34632no—
164C2×C4—24no—
165C2×C4—28no—
166C2×C4—3468no—
167C2×C4—3464no—
168C2×C4—3464no—
169C2×C4—3468no—
170C2×C4—1734no—
171C2×C4—69216no—
172C22—24no—
173C22—28no—
174C22—24no—
175C22—24no—
176C22—24no—
177C22—416no—
178C22—48no—
179C22—3468no—
180C22—3468no—
181C22—3468no—
182C22—1738no—
183C22—69216no—
184C22C22⋊C4×D17314yes—
185C22C22×C4×D17311yescharacteristic
186C22C2×D4×D17312yes—
187C22#1914972414yes—
188C4—44no—
189C4—3464no—
190C4—6924no—
191C2—24no—
192C2—44no—
193C2—3468no—
194C2C2×C22⋊C4×D17311yescharacteristic
195C2#2601196412yes—
196C2#2601205114yes—
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#379×C173C2
socle, cosocleC23×C173C24
prime radicalC23×C22⋊D173C2
hypercenter#379D173
hypocenterC173#2308
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D173C2semidirect
3. #26011367C2semidirect
4. #26011352C2semidirect
5. #26011347C2semidirect
6. #379×C173C2semidirect
9. C22×C22⋊D173C4semidirect
12. C22×C4×D173C22semidirect
14. C2×C22⋊Dic173C22semidirect
15. C2×#14583803C22semidirect
17. #19149586C22semidirect
20. C2×C22⋊C4×C173C22semidirect
28. C2×C22⋊D173C2×C4semidirect
30. C22×Dic173C2×C4semidirect
40. C24×C173C2×C4semidirect
49. C22×D173C22⋊C4semidirect
98. C173#2308semidirect
105. #379D173semidirect
118. C2×C22⋊C4C2×D173semidirect
126. C24C4×D173semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1732
C2×C22⋊C4×D1732
#260113672
#260113522
#260113472
#379×C1732
#2308173