Finite Group Databasebeta

← #37542013

Subgroups — #37542013

Order 23168.

1102 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C2×#19153799C212yesmaximal, complemented
2C2×#19153785C211yesmaximal, characteristic
3#26041158C214yesmaximal, complemented
4#26041152C214yesmaximal, complemented
5#26041142C212yesmaximal, complemented
6#26041135C211yesmaximal, characteristic, complemented
7C24×C4×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2212yes—
9C22×C22⋊D181C2211yescharacteristic
10#19153799—28no—
11#19153785—24no—
12C2×#15668428C2214yescomplemented
13C2×C22⋊Dic181C2212yes—
14C2×C22⋊Dic181C2214yes—
15C2×C22⋊Dic181C2211yescharacteristic
16C2×C22⋊Dic181C2212yes—
17C2×#15668426C2214yescomplemented
18#19153723C2218yescomplemented
19C25×C181C2211yescharacteristic
20C23×C4×C181—24no—
21C23×C4×C181—28no—
22C23×C4×C181C2214yes—
23C23×C4×C181C2212yes—
24C23×D181C2311yescharacteristic
25C2×C22⋊D181—22no—
26C2×C22⋊D181—24no—
27C2×C22⋊D181—24no—
28C2×C22⋊D181—28no—
29C22×Dic181C2312yes—
30C22×Dic181C2314yes—
31C22×Dic181C2311yescharacteristic
32#15668428—28no—
33C22⋊Dic181—22no—
34C22⋊Dic181—24no—
35C22⋊Dic181—24no—
36C22⋊Dic181—28no—
37#15668426—28no—
38C24×C181—24no—
39C24×C181—28no—
40C24×C181D414yes—
41C24×C181C2311yescharacteristic
42C24×C181C2312yes—
43C22×C4×C181—232no—
44C22×C4×C181—28no—
45C22×C4×C181—24no—
46C22×C4×C181—24no—
47C22×C4×C181—216no—
48C22×C4×C181D418yescomplemented
49C22×C4×C181C2314yes—
50C22×D181—22no—
51C22×D181—21no—
52C22×D181—24no—
53C22⋊D181—44no—
54C22⋊D181—48no—
55C2×Dic181—24no—
56C2×Dic181—21no—
57C2×Dic181—22no—
58C2×Dic181—22no—
59C2×Dic181—28no—
60C2×Dic181—24no—
61C23×C181—232no—
62C23×C181—28no—
63C23×C181—216no—
64C23×C181—24no—
65C23×C181—24no—
66C23×C181—24no—
67C23×C181C2×D412yes—
68C23×C181C2×D414yes—
69C23×C181C2×D414yes—
70C23×C181#4118yes—
71C23×C181C2411yescharacteristic
72C2×C4×C181—232no—
73C2×C4×C181—216no—
74C2×C4×C181—24no—
75C2×C4×C181—24no—
76C2×C4×C181C2×D418yes—
77C22×C181—232no—
78C22×C181—24no—
79C22×C181—24no—
80C22×C181—24no—
81C22×C181—28no—
82C22×C181—216no—
83C22×C181#12014yes—
84C22×C181#12318yescomplemented
85C22×C181C22×D411yescharacteristic
86C22×C181C22×D412yes—
87C22×C181C2×#4114yes—
88C2×D181—41no—
89C2×D181—42no—
90C2×D181—44no—
91C4×C181—28no—
92Dic181—41no—
93Dic181—42no—
94Dic181—44no—
95C2×C181—24no—
96C2×C181—28no—
97C2×C181C2×#12011yescharacteristic
98C2×C181C2×#12312yes—
99C2×C181#52514yes—
100D181—81no—
101C181#284911yescharacteristic, complemented, Sylow
102#2849—1811nomaximal, Sylow
103C24×C4D18111yescharacteristic, complemented
104C2×#123—1812no—
105C2×#120—1811no—
106#386—1814no—
107#385—1814no—
108#381—1812no—
109#379—1811no—
110C25C2×D18111yescharacteristic
111C22×D4—1811no—
112C22×D4—1812no—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4C2×D18114yes—
116C23×C4C2×D18112yes—
117#123—3628no—
118#120—3624no—
119C2×C4⋊C4—1814no—
120C2×C22⋊C4—1811no—
121C2×C22⋊C4—1812no—
122C2×C22⋊C4—1812no—
123C2×C22⋊C4—1814no—
124C2×C22⋊C4—1814no—
125#95—1818no—
126C24—24no—
127C24—28no—
128C24—1811no—
129C24C22⋊D18114yes—
130C24C22×D18111yescharacteristic
131C24C22×D18112yes—
132C2×D4—3622no—
133C2×D4—3624no—
134C2×D4—3624no—
135C2×D4—3628no—
136C22×C4—232no—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—216no—
141C22×C4—1812no—
142C22×C4—1814no—
143C22×C4—1811no—
144C22×C4C22⋊D18118yescomplemented
145C22×C4C22×D18114yes—
146C4⋊C4—3628no—
147C22⋊C4—3622no—
148C22⋊C4—3624no—
149C22⋊C4—3624no—
150C22⋊C4—3628no—
151C22⋊C4—3628no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—3622no—
159C23—3621no—
160C23—3624no—
161C23C2×C22⋊D18112yes—
162C23C2×C22⋊D18114yes—
163C23C2×C22⋊D18114yes—
164C23#1566844918yes—
165C23C23×D18111yescharacteristic
166D4—7244no—
167D4—7248no—
168C2×C4—232no—
169C2×C4—216no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—3624no—
173C2×C4—3621no—
174C2×C4—3622no—
175C2×C4—3622no—
176C2×C4—3628no—
177C2×C4—3624no—
178C2×C4C2×C22⋊D18118yes—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—7241no—
186C22—7242no—
187C22—7244no—
188C22#1915378514yes—
189C22#1915379918yescomplemented
190C22C22×C22⋊D18111yescharacteristic
191C22C22×C22⋊D18112yes—
192C22C2×#1566844914yes—
193C4—28no—
194C4—7241no—
195C4—7242no—
196C4—7244no—
197C2—24no—
198C2—28no—
199C2—14481no—
200C2C2×#1915378511yescharacteristic
201C2C2×#1915379912yes—
202C2#2604179014yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
FittingC24×C4×C181C2
prime radicalC2×#19153785C2
hypercenterC24×C4D181
hypocenterC181#2849
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C2×#19153799C2semidirect
3. #26041158C2semidirect
4. #26041152C2semidirect
5. #26041142C2semidirect
6. #26041135C2semidirect
7. C24×C4×C181C2semidirect
12. C2×#15668428C22semidirect
17. C2×#15668426C22semidirect
18. #19153723C22semidirect
48. C22×C4×C181D4semidirect
84. C22×C181#123semidirect
101. C181#2849semidirect
103. C24×C4D181semidirect
144. C22×C4C22⋊D181semidirect
189. C22#19153799semidirect

Maximal subgroups

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

subgroupindex
C2×#191537992
C2×#191537852
#260411582
#260411522
#260411422
#260411352
C24×C4×C1812
#2849181