Finite Group Databasebeta

← #37542040

Subgroups — #37542040

Order 23168.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C4×Dic181C211yesmaximal, characteristic, complemented
2#26041167C214yesmaximal, complemented
3#26041160C214yesmaximal, complemented
4#26041159C212yesmaximal
5#26041151C212yesmaximal, complemented
6#26041142C211yesmaximal, characteristic, complemented
7C2×#123×C181C211yesmaximal, characteristic
8C23×Dic181C2212yes—
9C2×#15668428C2214yes—
10C2×C22⋊Dic181C2212yes—
11C2×C22⋊Dic181C2214yes—
12C2×C4×Dic181—24no—
13C2×C4×Dic181C2214yes—
14#19153723C2214yes—
15#19153723C2214yes—
16#19153723C2214yes—
17C22×D4×C181C2211yescharacteristic
18C23×C4×C181C2211yescharacteristic
19#123×C181—24no—
20C2×C4⋊C4×C181C2212yes—
21C2×C22⋊C4×C181C2211yescharacteristic
22C2×C22⋊C4×C181C2212yes—
23C22×Dic181—28no—
24C22×Dic181—24no—
25C22×Dic181C2314yes—
26C22×Dic181C2314yes—
27#15668428—28no—
28C22⋊Dic181—28no—
29C22⋊Dic181—24no—
30C4×Dic181—28no—
31C24×C181C2311yescharacteristic
32C24×C181C2311yescharacteristic
33C2×D4×C181—22no—
34C2×D4×C181—24no—
35C22×C4×C181—24no—
36C22×C4×C181D414yes—
37C22×C4×C181C2312yes—
38C22×C4×C181C2311yescharacteristic
39C22×C4×C181C2312yes—
40C4⋊C4×C181—24no—
41C22⋊C4×C181—22no—
42C22⋊C4×C181—24no—
43C2×Dic181—216no—
44C2×Dic181—24no—
45C2×Dic181—24no—
46C2×Dic181—24no—
47C2×Dic181—24no—
48C2×Dic181—24no—
49C2×Dic181#4118yes—
50C23×C181—24no—
51C23×C181—21no—
52C23×C181—22no—
53C23×C181—24no—
54C23×C181C2×D412yes—
55C23×C181#4114yes—
56C23×C181C2411yescharacteristic
57D4×C181—44no—
58C2×C4×C181—28no—
59C2×C4×C181—24no—
60C2×C4×C181—22no—
61C2×C4×C181—21no—
62C2×C4×C181—22no—
63C2×C4×C181—24no—
64C2×C4×C181C2×D414yes—
65C22×C181—24no—
66C22×C181—28no—
67C22×C181—41no—
68C22×C181—42no—
69C22×C181—44no—
70C22×C181#12314yes—
71C22×C181C22×D411yescharacteristic
72C22×C181C2×#4114yes—
73C22×C181C2×#4112yes—
74C4×C181—24no—
75C4×C181—41no—
76C4×C181—42no—
77Dic181—28no—
78Dic181—44no—
79C2×C181—24no—
80C2×C181—81no—
81C2×C181C2×#12311yescharacteristic
82C2×C181#52512yes—
83C2×C181#52914yes—
84C181#285311yescharacteristic, complemented, Sylow
85#2853—1811nomaximal, Sylow
86C2×#123D18111yescharacteristic
87C22×C42—1811no—
88#388—1814no—
89#387—1814no—
90#386—1812no—
91#385—1812no—
92#381—1811no—
93C22×D4C2×D18111yescharacteristic
94C23×C4—1812no—
95C23×C4C2×D18111yescharacteristic
96#123—24no—
97C2×C4⋊C4—1814no—
98C2×C4⋊C4C2×D18112yes—
99C2×C22⋊C4—1812no—
100C2×C22⋊C4—1814no—
101C2×C22⋊C4C2×D18111yescharacteristic
102C2×C22⋊C4C2×D18112yes—
103C2×C42—3624no—
104C2×C42—1814no—
105#95—1814no—
106#95—1814no—
107#95—1814no—
108C24C22×D18111yescharacteristic
109C24C22×D18111yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—3624no—
114C22×C4—3628no—
115C22×C4—1814no—
116C22×C4—1814no—
117C22×C4C22⋊D18114yes—
118C22×C4C22×D18111yescharacteristic
119C22×C4C22×D18112yes—
120C22×C4C22×D18112yes—
121C4⋊C4—24no—
122C4⋊C4—3628no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—3624no—
126C22⋊C4—3628no—
127C42—3628no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×C22⋊D18112yes—
133C23#1566844814yes—
134C23C23×D18111yescharacteristic
135D4—44no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—36216no—
143C2×C4—3624no—
144C2×C4—3624no—
145C2×C4—3624no—
146C2×C4—3624no—
147C2×C4—3624no—
148C2×C4—1818no—
149C2×C4C2×C22⋊D18114yes—
150C22—24no—
151C22—28no—
152C22—41no—
153C22—42no—
154C22—44no—
155C22#1915379714yes—
156C22C22×C22⋊D18111yescharacteristic
157C22#41×D18114yes—
158C22C2×#1566844812yes—
159C4—24no—
160C4—41no—
161C4—42no—
162C4—3628no—
163C4—7244no—
164C2—24no—
165C2—81no—
166C2C2×#1915379711yescharacteristic
167C2#2604178612yes—
168C2#2604180614yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
FittingC2×#123×C181C2
prime radicalC22×D4×C181C22
hypercenterC2×#123D181
hypocenterC181#2853
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic181C2semidirect
2. #26041167C2semidirect
3. #26041160C2semidirect
5. #26041151C2semidirect
6. #26041142C2semidirect
84. C181#2853semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1812
#260411672
#260411602
#260411592
#260411512
#260411422
C2×#123×C1812
#2853181