Finite Group Databasebeta

← #37534164

Subgroups — #37534164

Order 23168.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#26041790C211yesmaximal, characteristic
2C2×#19153744C212yesmaximal, complemented
3#26041243C211yesmaximal, characteristic, complemented
4#26041039C212yesmaximal, complemented
5C22×C42×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C2211yescharacteristic
7C2×#15668433C2212yes—
8#19153799—21no—
9#19153795C412yescomplemented
10#19153790C412yescomplemented
11#19153785—21no—
12C4×C22⋊D181—22no—
13#19153775C2211yescharacteristic
14#19153744—24no—
15C22⋊#13089678—22no—
16C23×C4×C181—21no—
17C23×C4×C181C2211yescharacteristic
18C2×C42×C181—21no—
19C2×C42×C181—24no—
20C2×C42×C181C2212yes—
21#15668449—22no—
22#15668449—22no—
23C2×Dic362C2×C411yescharacteristic, complemented
24C2×D724C2×C411yescharacteristic, complemented
25C2×C22⋊D181—22no—
26C2×C22⋊D181—21no—
27C2×C4×D181—21no—
28#15668433—22no—
29#15668433—22no—
30C2×#13089678—22no—
31C4⋊Dic181C2×C412yescomplemented
32#15668428—22no—
33C22⋊Dic181—22no—
34C22⋊Dic181—21no—
35#15668426—22no—
36C4×Dic181—21no—
37C24×C181D411yescharacteristic
38C22×C4×C181—24no—
39C22×C4×C181—28no—
40C22×C4×C181—22no—
41C22×C4×C181—22no—
42C22×C4×C181—22no—
43C22×C4×C181C2×C412yes—
44C22×C4×C181D411yescharacteristic
45C22×C4×C181D412yes—
46C22×C4×C181C2311yescharacteristic
47C42×C181—24no—
48C42×C181—22no—
49C42×C181D414yescomplemented
50C22×D181—21no—
51Dic362—22no—
52D724—22no—
53C22⋊D181—44no—
54C22⋊D181—41no—
55C22⋊D181—41no—
56C2×Dic181—22no—
57C2×Dic181—21no—
58C4×D181—42no—
59#13089678—42no—
60C23×C181—24no—
61C23×C181—21no—
62C23×C181—21no—
63C23×C181C22⋊C412yes—
64C23×C181C2×D411yescharacteristic
65C2×C4×C181—216no—
66C2×C4×C181—28no—
67C2×C4×C181—21no—
68C2×C4×C181—24no—
69C2×C4×C181—21no—
70C2×C4×C181—24no—
71C2×C4×C181—24no—
72C2×C4×C181C22⋊C412yes—
73C2×C4×C181C22×C411yescharacteristic
74C2×C4×C181C2×D412yes—
75C2×C4×C181C2×D411yescharacteristic
76C2×C4×C181#4112yes—
77C22×C181—24no—
78C22×C181—24no—
79C22×C181—22no—
80C22×C181—22no—
81C22×C181—22no—
82C22×C181#10414yescomplemented
83C22×C181C2×C22⋊C411yescharacteristic
84C22×C181#12011yescharacteristic
85C22×C181#12311yescharacteristic
86C2×D181—41no—
87C2×D181—42no—
88C4×C181—28no—
89C4×C181—22no—
90C4×C181C4×D412yes—
91C4×C181#12112yes—
92Dic181—41no—
93Dic181—42no—
94C2×C181—21no—
95C2×C181—24no—
96C2×C181—21no—
97C2×C181#38611yescharacteristic
98C2×C181C2×#10412yes—
99D181—81no—
100C181#189611yescharacteristic, complemented, Sylow
101#1896—1811nomaximal, Sylow
102#525—1811no—
103C22×C42D18111yescharacteristic, complemented
104C2×#104—1812no—
105#407—1811no—
106#339—1812no—
107C2×#41—1811no—
108C23×C4—21no—
109C23×C4C2×D18111yescharacteristic
110C2×M16—1812no—
111#123—3621no—
112C22⋊Q8—1812no—
113#121—1812no—
114#120—3621no—
115C4×D4—3622no—
116#117—1811no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D18112yes—
120#104—3624no—
121C22⋊C8—3622no—
122C24C22⋊D18111yescharacteristic
123#41—3622no—
124#41—3622no—
125C2×Q8—1811no—
126C2×D4—3622no—
127C2×D4—3621no—
128C2×D4—1811no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—3621no—
135C22×C4C4×D18112yes—
136C22×C4C22⋊D18111yescharacteristic
137C22×C4D72412yes—
138C22×C4C22×D18111yescharacteristic
139M16—3622no—
140M16—3622no—
141C2×C8—3622no—
142C4⋊C4—3622no—
143C4⋊C4—1812no—
144C22⋊C4—3622no—
145C22⋊C4—3621no—
146C22⋊C4—3622no—
147C42—24no—
148C42—22no—
149C42—3621no—
150C42C22⋊D18114yescomplemented
151C23—24no—
152C23—21no—
153C23—21no—
154C23—3621no—
155C23#1566842612yes—
156C23C2×C22⋊D18111yescharacteristic
157Q8—3622no—
158D4—3622no—
159D4—7244no—
160D4—7241no—
161D4—7241no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—3622no—
170C2×C4—3621no—
171C2×C4—7242no—
172C2×C4#1566842612yes—
173C2×C4C2×C4×D18111yescharacteristic
174C2×C4C2×C22⋊D18112yes—
175C2×C4C2×D72411yescharacteristic
176C2×C4#1566844912yes—
177C8—7242no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—7241no—
184C22—7242no—
185C22#1915374414yescomplemented
186C22C2×#1566842611yescharacteristic
187C22#1915378511yescharacteristic
188C22#1915379911yescharacteristic
189C4—28no—
190C4—22no—
191C4—7241no—
192C4—7242no—
193C4C4×C22⋊D18112yes—
194C4#1915379012yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—14481no—
199C2#2604115811yescharacteristic
200C2C2×#1915374412yes—
subgroupG/H
centerC2×C4C2×D724
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
FittingC22×C42×C181C2
socleC22×C181C2×C22⋊C4
prime radical#26041790C2
cosocleC22×C4×C181C23
hypercenterC22×C42D181
hypocenterC181#1896
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#19153744C2semidirect
3. #26041243C2semidirect
4. #26041039C2semidirect
5. C22×C42×C181C2semidirect
9. #19153795C4semidirect
10. #19153790C4semidirect
23. C2×Dic362C2×C4semidirect
24. C2×D724C2×C4semidirect
31. C4⋊Dic181C2×C4semidirect
49. C42×C181D4semidirect
82. C22×C181#104semidirect
100. C181#1896semidirect
103. C22×C42D181semidirect
150. C42C22⋊D181semidirect
185. C22#19153744semidirect

Maximal subgroups

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

subgroupindex
#260417902
C2×#191537442
#260412432
#260410392
C22×C42×C1812
#1896181