Finite Group Databasebeta

← #37542043

Subgroups — #37542043

Order 23168.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19153799C211yesmaximal, characteristic
2#26041170C214yesmaximal, complemented
3#26041163C214yesmaximal, complemented
4#26041158C212yesmaximal
5#26041152C212yesmaximal, complemented
6#26041142C211yesmaximal, characteristic, complemented
7C22×C42×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2211yescharacteristic
9#19153799—24no—
10C2×#15668428C2214yes—
11C2×#15668428C2212yes—
12C2×C22⋊Dic181C2212yes—
13C2×C22⋊Dic181C2211yescharacteristic
14C2×#15668426C2214yes—
15C2×#15668426C2212yes—
16#19153723C2214yes—
17#19153723C2214yes—
18#19153723C2214yes—
19C23×C4×C181C2212yes—
20C23×C4×C181C2211yescharacteristic
21C2×C42×C181—24no—
22C2×C42×C181C2214yes—
23C23×D181C2311yescharacteristic
24C2×C22⋊D181—22no—
25C2×C22⋊D181—24no—
26C22×Dic181C2314yes—
27C22×Dic181C2311yescharacteristic
28C22×Dic181C2312yes—
29#15668428—24no—
30#15668428—28no—
31C22⋊Dic181—22no—
32C22⋊Dic181—24no—
33#15668426—24no—
34#15668426—28no—
35C24×C181C2311yescharacteristic
36C22×C4×C181—24no—
37C22×C4×C181—24no—
38C22×C4×C181—28no—
39C22×C4×C181D414yes—
40C22×C4×C181C2312yes—
41C22×C4×C181C2314yes—
42C42×C181—28no—
43C22×D181—21no—
44C22×D181—24no—
45C22×D181—22no—
46C22⋊D181—44no—
47C2×Dic181—22no—
48C2×Dic181—21no—
49C2×Dic181—24no—
50C2×Dic181—24no—
51C2×Dic181—24no—
52C2×Dic181—24no—
53C2×Dic181—22no—
54C23×C181—24no—
55C23×C181C2×D412yes—
56C23×C181#4114yes—
57C23×C181C2411yescharacteristic
58C2×C4×C181—216no—
59C2×C4×C181—28no—
60C2×C4×C181—24no—
61C2×C4×C181—24no—
62C2×C4×C181—24no—
63C2×C4×C181C2×D414yes—
64C2×C4×C181#4118yes—
65C22×C181—24no—
66C22×C181—28no—
67C22×C181#12314yes—
68C22×C181C22×D411yescharacteristic
69C22×C181C2×#4112yes—
70C22×C181C2×#4114yes—
71C2×D181—41no—
72C2×D181—42no—
73C2×D181—44no—
74C4×C181—24no—
75C4×C181—28no—
76Dic181—41no—
77Dic181—42no—
78Dic181—44no—
79C2×C181—24no—
80C2×C181C2×#12311yescharacteristic
81C2×C181#52512yes—
82C2×C181#52914yes—
83D181—81no—
84C181#285311yescharacteristic, complemented, Sylow
85#2853—1811nomaximal, Sylow
86C2×#123—1811no—
87C22×C42D18111yescharacteristic, complemented
88#388—1814no—
89#387—1814no—
90#386—1812no—
91#385—1812no—
92#381—1811no—
93C22×D4—1811no—
94C23×C4C2×D18112yes—
95C23×C4C2×D18111yescharacteristic
96#123—3624no—
97C2×C4⋊C4—1812no—
98C2×C4⋊C4—1814no—
99C2×C22⋊C4—1811no—
100C2×C22⋊C4—1812no—
101C2×C22⋊C4—1812no—
102C2×C22⋊C4—1814no—
103C2×C42—24no—
104C2×C42C2×D18114yes—
105#95—1814no—
106#95—1814no—
107#95—1814no—
108C24—1811no—
109C24C22×D18111yescharacteristic
110C2×D4—3622no—
111C2×D4—3624no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—1814no—
116C22×C4—1811no—
117C22×C4—1812no—
118C22×C4C22⋊D18114yes—
119C22×C4C22×D18112yes—
120C22×C4C22×D18114yes—
121C4⋊C4—3624no—
122C4⋊C4—3628no—
123C22⋊C4—3622no—
124C22⋊C4—3624no—
125C22⋊C4—3624no—
126C22⋊C4—3628no—
127C42—28no—
128C23—24no—
129C23—3621no—
130C23—3624no—
131C23—3622no—
132C23C2×C22⋊D18112yes—
133C23#1566844914yes—
134C23C23×D18111yescharacteristic
135D4—7244no—
136C2×C4—216no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—3622no—
142C2×C4—3621no—
143C2×C4—3624no—
144C2×C4—3624no—
145C2×C4—3624no—
146C2×C4—3624no—
147C2×C4—3622no—
148C2×C4C2×C22⋊D18114yes—
149C2×C4#1566844918yes—
150C22—24no—
151C22—28no—
152C22—7241no—
153C22—7242no—
154C22—7244no—
155C22#1915379914yes—
156C22C22×C22⋊D18111yescharacteristic
157C22C2×#1566844912yes—
158C22C2×#1566844914yes—
159C4—24no—
160C4—28no—
161C4—7241no—
162C4—7242no—
163C4—7244no—
164C2—24no—
165C2—14481no—
166C2C2×#1915379911yescharacteristic
167C2#2604179012yes—
168C2#2604181314yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
FittingC22×C42×C181C2
prime radicalC22×C22⋊D181C22
hypercenterC22×C42D181
hypocenterC181#2853
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #26041170C2semidirect
3. #26041163C2semidirect
5. #26041152C2semidirect
6. #26041142C2semidirect
7. C22×C42×C181C2semidirect
84. C181#2853semidirect
87. C22×C42D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537992
#260411702
#260411632
#260411582
#260411522
#260411422
C22×C42×C1812
#2853181