Finite Group Databasebeta

← #37542181

Subgroups — #37542181

Order 23168.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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×#19153861C211yesmaximal, characteristic
2#26041754C211yesmaximal, characteristic, complemented
3#26041252C214yesmaximal, complemented
4C2×#19153730C218yesmaximal, complemented
5#518×C181C211yesmaximal, characteristic, complemented
6#19153861—24no—
7#19153861—24no—
8C2×#15668449C2212yes—
9C2×#15668448C414yescomplemented
10C22×C22⋊D181C2214yes—
11C2×D4×D181C414yescomplemented
12D4×Dic181—24no—
13#19153775C2212yescomplemented
14C2×C22⋊Dic181C2214yescomplemented
15#19153730C22116yescomplemented
16C22×D4×C181C2211yescharacteristic
17C4×D4×C181—24no—
18#117×C181C2212yescomplemented
19C2×C22⋊C4×C181C2214yescomplemented
20C23×D181C2×C414yescomplemented
21#15668449—24no—
22#15668449—24no—
23#15668448—216no—
24C2×Dic362C2×C412yescomplemented
25C2×D724C2×C412yescomplemented
26C2×C22⋊D181—28no—
27C2×C22⋊D181—28no—
28C2×C22⋊D181C2×C418yescomplemented
29C2×C22⋊D181C2314yes—
30D4×D181—216no—
31C22×Dic181—24no—
32C22×Dic181C2×C414yescomplemented
33C2×C4×D181C2×C414yescomplemented
34C4⋊Dic181—22no—
35C22⋊Dic181—24no—
36C22⋊Dic181C2314yes—
37C4×Dic181—22no—
38C24×C181C2312yes—
39C2×D4×C181C2×C414yes—
40C2×D4×C181D414yes—
41C2×D4×C181D414yes—
42C22×C4×C181—24no—
43C22×C4×C181C2311yescharacteristic
44C4⋊C4×C181—22no—
45C22⋊C4×C181—24no—
46C22⋊C4×C181C2314yes—
47C42×C181—22no—
48C22×D181—28no—
49C22×D181—84no—
50C22×D181C22×C414yes—
51Dic362—24no—
52D724—24no—
53C22⋊D181—216no—
54C22⋊D181—24no—
55C22⋊D181—24no—
56C2×Dic181—28no—
57C2×Dic181—24no—
58C2×Dic181—84no—
59C2×Dic181C22×C414yes—
60C4×D181—28no—
61C23×C181—82no—
62C23×C181C22×C414yes—
63C23×C181C2×D414yes—
64C23×C181C2×D414yes—
65C23×C181C2411yescharacteristic
66D4×C181C22⋊C4116yes—
67C2×C4×C181—24no—
68C2×C4×C181—84no—
69C2×C4×C181C22×C412yes—
70C2×C4×C181C2×D412yes—
71C2×C4×C181C2×D412yes—
72C22×C181—44no—
73C22×C181—41no—
74C22×C181—44no—
75C22×C181—44no—
76C22×C181C2×C22⋊C418yes—
77C22×C181C23×C411yescharacteristic
78C22×C181C22×D411yescharacteristic
79C22×C181C22×D411yescharacteristic
80C2×D181—24no—
81C2×D181—44no—
82C2×D181—88no—
83C4×C181—44no—
84C4×C181C2×C22⋊C414yes—
85Dic181—24no—
86Dic181—44no—
87C2×C181—21no—
88C2×C181—21no—
89C2×C181—21no—
90C2×C181—28no—
91C2×C181C22×C22⋊C411yescharacteristic
92D181—44no—
93C181#288311yescharacteristic, complemented, Sylow
94#2883—1811nomaximal, Sylow
95C2×H(5,𝔽2)—1811no—
96#518—1811no—
97#518D18111yescharacteristic, complemented
98#410—1814no—
99C2×#99—1818no—
100H(5,𝔽2)—3624no—
101H(5,𝔽2)—3624no—
102C2×#41—1814no—
103C2×#41—1812no—
104C22×D4—1814no—
105C22×D4—1814no—
106C22×D4C2×D18111yescharacteristic
107C4×D4—24no—
108C4×D4—3624no—
109#117—1812no—
110#117C2×D18112yescomplemented
111C2×C22⋊C4—1814no—
112C2×C22⋊C4C2×D18114yescomplemented
113#99—18116no—
114C24—1814no—
115C24C22×D18112yes—
116#41—36216no—
117#41—3624no—
118#41—3624no—
119C2×Q8—1812no—
120C2×D4—3628no—
121C2×D4—3628no—
122C2×D4—36216no—
123C2×D4—1818no—
124C2×D4—1814no—
125C2×D4—1812no—
126C2×D4C4×D18114yes—
127C2×D4C22⋊D18114yes—
128C2×D4D72414yes—
129C22×C4—24no—
130C22×C4—3624no—
131C22×C4—1814no—
132C22×C4—1814no—
133C22×C4C22×D18111yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—3622no—
136C22⋊C4—24no—
137C22⋊C4—3624no—
138C22⋊C4—1814no—
139C22⋊C4C22×D18114yes—
140C42—22no—
141C42—3622no—
142C23—82no—
143C23—3628no—
144C23—14484no—
145C23—1814no—
146C23C2×C4×D18114yes—
147C23C2×C22⋊D18114yes—
148C23C2×D72414yes—
149C23C23×D18111yescharacteristic
150Q8—3624no—
151D4—36216no—
152D4—3624no—
153D4—3624no—
154D4—3624no—
155D4#15668426116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—3628no—
159C2×C4—3628no—
160C2×C4—3624no—
161C2×C4—14484no—
162C2×C4—1814no—
163C2×C4C2×C4×D18112yes—
164C2×C4C2×C22⋊D18112yes—
165C2×C4C2×D72412yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—3624no—
171C22—14488no—
172C22—7244no—
173C22C2×#1566842618yes—
174C22C22×C4×D18111yescharacteristic
175C22C22×C22⋊D18111yescharacteristic
176C22C22×D72411yescharacteristic
177C4—44no—
178C4—3624no—
179C4—7244no—
180C4C2×#1566842614yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—7244no—
186C2C22×#1566842611yescharacteristic
subgroupG/H
centerC2C22×#15668426
commutatorC22×C181C23×C4
FrattiniC23C23×D181
Fitting#518×C181C2
socleC2×C181C22×C22⋊C4
prime radicalC2×#19153861C2
cosocleC23×C181C24
hypercenter#518D181
hypocenterC181#2883
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #26041754C2semidirect
3. #26041252C2semidirect
4. C2×#19153730C2semidirect
5. #518×C181C2semidirect
9. C2×#15668448C4semidirect
11. C2×D4×D181C4semidirect
13. #19153775C22semidirect
14. C2×C22⋊Dic181C22semidirect
15. #19153730C22semidirect
18. #117×C181C22semidirect
19. C2×C22⋊C4×C181C22semidirect
20. C23×D181C2×C4semidirect
24. C2×Dic362C2×C4semidirect
25. C2×D724C2×C4semidirect
28. C2×C22⋊D181C2×C4semidirect
32. C22×Dic181C2×C4semidirect
33. C2×C4×D181C2×C4semidirect
93. C181#2883semidirect
97. #518D181semidirect
110. #117C2×D181semidirect
112. C2×C22⋊C4C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538612
#260417542
#260412522
C2×#191537302
#518×C1812
#2883181