Finite Group Databasebeta

← #37534249

Subgroups — #37534249

Order 23168.

344 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 93 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#26042046C211yesmaximal, characteristic, complemented
2#26041284C212yesmaximal, complemented
3#26041253C211yesmaximal, characteristic, complemented
4#26041035C212yesmaximal, complemented
5#535×C181C211yesmaximal, characteristic
6#19153823—24no—
7C2×#15668433—22no—
8C2×#15668433C2212yes—
9#19153775C2212yes—
10#19153737—24no—
11#19153731—22no—
12C2×#41×C181C2211yescharacteristic
13C22×D4×C181C2211yescharacteristic
14#127×C181C412yescomplemented
15#124×C181C412yescomplemented
16#121×C181—22no—
17#120×C181—22no—
18#117×C181C2211yescharacteristic
19#15668433—24no—
20#15668433—22no—
21#15668433—22no—
22C2×#13089678—24no—
23C2×#13089678D414yescomplemented
24C4⋊Dic181—22no—
25C22⋊Dic181—22no—
26C4×Dic181—22no—
27C24×C181—21no—
28#41×C181—22no—
29#41×C181—22no—
30C2×Q8×C181C2×C411yescharacteristic
31C2×D4×C181—24no—
32C2×D4×C181—22no—
33C2×D4×C181—21no—
34C2×D4×C181—22no—
35C2×D4×C181C2×C412yes—
36C2×D4×C181C2×C411yescharacteristic
37C2×D4×C181D412yes—
38C22×C4×C181D412yes—
39C22×C4×C181C2311yescharacteristic
40C4⋊C4×C181—21no—
41C22⋊C4×C181—21no—
42C22⋊C4×C181—24no—
43C42×C181C2×C412yes—
44C2×Dic181—42no—
45#13089678—24no—
46C23×C181—22no—
47C23×C181—42no—
48C23×C181—82no—
49C23×C181C22⋊C412yes—
50C23×C181C2×D411yescharacteristic
51Q8×C181—22no—
52D4×C181—24no—
53D4×C181—22no—
54D4×C181—24no—
55D4×C181—41no—
56D4×C181—41no—
57D4×C181—84no—
58C2×C4×C181—22no—
59C2×C4×C181—22no—
60C2×C4×C181—22no—
61C2×C4×C181C22⋊C412yes—
62C2×C4×C181C22×C411yescharacteristic
63C2×C4×C181C2×D412yes—
64C2×C4×C181C2×D411yescharacteristic
65C2×C4×C181#4112yes—
66C22×C181—22no—
67C22×C181—41no—
68C22×C181—42no—
69C22×C181—44no—
70C22×C181—42no—
71C22×C181—84no—
72C22×C181C2×C22⋊C411yescharacteristic
73C22×C181#12112yes—
74C4×C181—22no—
75C4×C181—42no—
76C4×C181C4×D412yes—
77C4×C181#12011yescharacteristic
78C4×C181#12311yescharacteristic
79Dic181—82no—
80C2×C181—22no—
81C2×C181—21no—
82C2×C181—42no—
83C2×C181—82no—
84C2×C181#38611yescharacteristic
85C181#191011yescharacteristic, complemented, Sylow
86#1910—1811nomaximal, Sylow
87#568—1811no—
88#535D18111yescharacteristic
89#418—1812no—
90#410—1811no—
91H(3,ℤ/4)—1812no—
92C2×#41C2×D18111yescharacteristic
93C22×D4C2×D18111yescharacteristic
94#131—3624no—
95C2×M16—3622no—
96C2×M16—1812no—
97#127Dic18112yescomplemented
98#124Dic18112yescomplemented
99#121—22no—
100#120—22no—
101#117—1812no—
102#117C2×D18111yescharacteristic
103#102—3624no—
104#99—3622no—
105C24—21no—
106#41—22no—
107#41—22no—
108C2×Q8C2×Dic18111yescharacteristic
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4C2×Dic18112yes—
114C2×D4C2×Dic18111yescharacteristic
115C2×D4C22⋊D18112yes—
116C22×C4C22⋊D18112yes—
117C22×C4C22×D18111yescharacteristic
118M16—3624no—
119M16—3622no—
120M16—3622no—
121C2×C8—3624no—
122C2×C8—1814no—
123C4⋊C4—21no—
124C4⋊C4—3622no—
125C22⋊C4—21no—
126C22⋊C4—24no—
127C22⋊C4—3622no—
128C42—3622no—
129C42C2×Dic18112yes—
130C23—22no—
131C23—42no—
132C23—82no—
133C23C22⋊Dic18112yes—
134C23C2×C22⋊D18111yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—24no—
139D4—41no—
140D4—41no—
141D4—84no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—7242no—
146C2×C4C22⋊Dic18112yes—
147C2×C4C22×Dic18111yescharacteristic
148C2×C4D4×D18112yes—
149C2×C4C2×C22⋊D18111yescharacteristic
150C2×C4#1566844812yes—
151C8—3624no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—84no—
158C22C2×C22⋊Dic18111yescharacteristic
159C22#1915378812yes—
160C4—22no—
161C4—42no—
162C4—14482no—
163C4D4×Dic18112yes—
164C4#1915378411yescharacteristic
165C4#1915379711yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2604115911yescharacteristic
subgroupG/H
centerC2#26041159
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting, prime radical#535×C181C2
socleC2×C181#386
cosocleC22×C4×C181C23
hypercenter#535D181
hypocenterC181#1910
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26042046C2semidirect
2. #26041284C2semidirect
3. #26041253C2semidirect
4. #26041035C2semidirect
14. #127×C181C4semidirect
15. #124×C181C4semidirect
23. C2×#13089678D4semidirect
85. C181#1910semidirect
97. #127Dic181semidirect
98. #124Dic181semidirect

Maximal subgroups

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

subgroupindex
#260420462
#260412842
#260412532
#260410352
#535×C1812
#1910181