Finite Group Databasebeta

← #37542001

Subgroups — #37542001

Order 23168.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic181C211yesmaximal, characteristic, complemented
2C2×#19153788C213yesmaximal, complemented
3C2×#19153785C211yesmaximal, characteristic, complemented
4#26041159C213yesmaximal, complemented
5#26041156C213yesmaximal, complemented
6#26041137C213yesmaximal, complemented
7C2×#120×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2213yes—
9C22×C22⋊D181C2211yescharacteristic, complemented
10C23×Dic181—212no—
11C23×Dic181C2213yes—
12C23×Dic181C2213yes—
13#19153788—212no—
14#19153785—24no—
15C2×#15668428C2213yescomplemented
16C2×C22⋊Dic181C2213yes—
17C2×C22⋊Dic181C2213yescomplemented
18C2×#15668426C2213yescomplemented
19#19153723C2216yescomplemented
20C25×C181C2211yescharacteristic
21C22×D4×C181C2213yes—
22#120×C181—24no—
23C2×C22⋊C4×C181C2213yes—
24C23×D181C2311yescharacteristic
25C2×C22⋊D181—26no—
26C2×C22⋊D181—26no—
27C2×C22⋊D181—26no—
28C2×C22⋊D181—26no—
29C22×Dic181—232no—
30C22×Dic181—212no—
31C22×Dic181—212no—
32C22×Dic181—26no—
33C22×Dic181D4112yescomplemented
34C22×Dic181C2313yes—
35C22×Dic181C2313yes—
36C22×Dic181C2311yescharacteristic
37#15668428—26no—
38C22⋊Dic181—26no—
39C22⋊Dic181—26no—
40#15668426—26no—
41C24×C181—26no—
42C24×C181—26no—
43C24×C181D414yescomplemented
44C24×C181C2311yescharacteristic
45C24×C181C2313yes—
46C2×D4×C181—26no—
47C2×D4×C181—26no—
48C2×D4×C181—26no—
49C22×C4×C181C2313yes—
50C22⋊C4×C181—26no—
51C22×D181—23no—
52C22×D181—21no—
53C22×D181—23no—
54C22⋊D181—412no—
55C22⋊D181—44no—
56C2×Dic181—224no—
57C2×Dic181—26no—
58C2×Dic181—224no—
59C2×Dic181—23no—
60C2×Dic181—23no—
61C2×Dic181—23no—
62C2×Dic181C2×D416yes—
63C2×Dic181C2×D416yes—
64C23×C181—224no—
65C23×C181—28no—
66C23×C181—224no—
67C23×C181—26no—
68C23×C181—26no—
69C23×C181—21no—
70C23×C181—23no—
71C23×C181—23no—
72C23×C181C2×D416yes—
73C23×C181C2×D416yes—
74C23×C181#4116yes—
75C23×C181C2411yescharacteristic
76D4×C181—412no—
77C2×C4×C181—23no—
78C2×C4×C181—23no—
79C2×C4×C181—23no—
80C22×C181—232no—
81C22×C181—26no—
82C22×C181—212no—
83C22×C181—26no—
84C22×C181—212no—
85C22×C181—41no—
86C22×C181—43no—
87C22×C181—43no—
88C22×C181#121112yescomplemented
89C22×C181C22×D413yes—
90C22×C181C22×D411yescharacteristic
91C22×C181C2×#4113yes—
92C2×D181—41no—
93C2×D181—43no—
94C2×D181—43no—
95C4×C181—43no—
96Dic181—26no—
97Dic181—43no—
98Dic181#12014yescomplemented
99C2×C181—212no—
100C2×C181—81no—
101C2×C181C2×#12011yescharacteristic
102C2×C181C2×#12113yes—
103C2×C181#52513yes—
104D181—81no—
105C181#284711yescharacteristic, complemented, Sylow
106#2847—1811nomaximal, Sylow
107C24×C4—1811no—
108C2×#121—1813no—
109C2×#120—1811no—
110C2×#120D18111yescharacteristic, complemented
111#386—1813no—
112#386—1813no—
113#380—1813no—
114C25C2×D18111yescharacteristic
115C22×D4—1813no—
116C22×D4—1811no—
117C22×D4C2×D18113yes—
118C23×C4—36212no—
119C23×C4—1813no—
120C23×C4—1813no—
121#121—36212no—
122#120—24no—
123#120—3624no—
124C2×C4⋊C4—1813no—
125C2×C22⋊C4—1813no—
126C2×C22⋊C4—1813no—
127C2×C22⋊C4—1813no—
128C2×C22⋊C4C2×D18113yes—
129#95—1816no—
130C24—26no—
131C24—26no—
132C24—1811no—
133C24C22⋊D18114yescomplemented
134C24C22×D18111yescharacteristic
135C24C22×D18113yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—3626no—
140C2×D4—3626no—
141C2×D4—3626no—
142C2×D4—3626no—
143C22×C4—36232no—
144C22×C4—3626no—
145C22×C4—36212no—
146C22×C4—36212no—
147C22×C4—1813no—
148C22×C4—18112no—
149C22×C4—1813no—
150C22×C4—1811no—
151C22×C4C22×D18113yes—
152C4⋊C4—3626no—
153C22⋊C4—26no—
154C22⋊C4—3626no—
155C22⋊C4—3626no—
156C22⋊C4—3626no—
157C23—28no—
158C23—224no—
159C23—224no—
160C23—26no—
161C23—26no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—3623no—
166C23—3621no—
167C23—3623no—
168C23D4×D18116yes—
169C23C2×C22⋊D18116yes—
170C23#1566844816yes—
171C23C23×D18111yescharacteristic
172D4—412no—
173D4—7244no—
174D4—72412no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—36224no—
179C2×C4—3626no—
180C2×C4—36224no—
181C2×C4—3623no—
182C2×C4—3623no—
183C2×C4—3623no—
184C2×C4—1816no—
185C2×C4—1816no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—41no—
192C22—43no—
193C22—43no—
194C22—7241no—
195C22—7243no—
196C22—7243no—
197C22#19153788112yescomplemented
198C22C2×D4×D18113yes—
199C22C22×C22⋊D18111yescharacteristic
200C22C2×#1566844813yes—
201C4—43no—
202C4—3626no—
203C4—1814no—
204C4—7243no—
205C2—212no—
206C2—81no—
207C2—14481no—
208C2#120×D18111yescharacteristic
209C2C2×#1915378813yes—
210C2#2604178513yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
FittingC2×#120×C181C2
hypercenterC2×#120D181
hypocenterC181#2847
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic181C2semidirect
2. C2×#19153788C2semidirect
3. C2×#19153785C2semidirect
4. #26041159C2semidirect
5. #26041156C2semidirect
6. #26041137C2semidirect
7. C2×#120×C181C2semidirect
9. C22×C22⋊D181C22semidirect
15. C2×#15668428C22semidirect
17. C2×C22⋊Dic181C22semidirect
18. C2×#15668426C22semidirect
19. #19153723C22semidirect
33. C22×Dic181D4semidirect
43. C24×C181D4semidirect
88. C22×C181#121semidirect
98. Dic181#120semidirect
105. C181#2847semidirect
110. C2×#120D181semidirect
133. C24C22⋊D181semidirect
197. C22#19153788semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1812
C2×#191537882
C2×#191537852
#260411592
#260411562
#260411372
C2×#120×C1812
#2847181