Finite Group Databasebeta

← #37548806

Subgroups — #37548806

Order 23168.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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×#19153876C211yesmaximal, characteristic
2C2×#19153874C211yesmaximal, characteristic
3#26041156C211yesmaximal, characteristic
4C2×#15668458—21no—
5C2×C22⋊#10624382—21no—
6#19153876—21no—
7#19153876C412yescomplemented
8#19153874—21no—
9#19153874C412yescomplemented
10C22×C22⋊D181C2211yescharacteristic
11C23×Dic181C411yescharacteristic, complemented
12C2×#15668426—21no—
13#19153723—21no—
14C2×C22⋊C4×C181C411yescharacteristic, complemented
15C22×#10624382—41no—
16#15668458—22no—
17#15668458—41no—
18C2×#13089685—41no—
19C22⋊#10624382—22no—
20C22⋊#10624382—41no—
21C23×D181D411yescharacteristic
22C2×C22⋊D181—41no—
23C2×C22⋊D181—41no—
24C2×C22⋊D181C2×C411yescharacteristic
25C2×C22⋊D181C2×C411yescharacteristic
26C2×C22⋊D181D411yescharacteristic
27C2×C22⋊D181Q811yescharacteristic
28C22×Dic181—21no—
29C22×Dic181—41no—
30C22×Dic181—42no—
31C22×Dic181D411yescharacteristic
32#15668426—42no—
33C24×C181C2×C411yescharacteristic
34C22×C4×C181—21no—
35C22⋊C4×C181—41no—
36C2×#10624382—42no—
37C2×#10624382—81no—
38C2×#10624382—81no—
39#13089685—42no—
40C22×D181—21no—
41C22×D181—21no—
42C22×D181—41no—
43C22×D181—41no—
44C22×D181C22⋊C411yescharacteristic
45C22×D181C4⋊C411yescharacteristic
46C22⋊D181—41no—
47C22⋊D181—41no—
48C22⋊D181—41no—
49C22⋊D181—41no—
50C2×Dic181—21no—
51C2×Dic181—21no—
52C2×Dic181—22no—
53C2×Dic181—41no—
54C2×Dic181—42no—
55C2×Dic181—41no—
56C2×Dic181—42no—
57C2×Dic181C22⋊C411yescharacteristic
58C2×Dic181C4⋊C411yescharacteristic
59C23×C181—21no—
60C23×C181—21no—
61C23×C181—41no—
62C23×C181—41no—
63C23×C181C4211yescharacteristic
64C23×C181C22⋊C411yescharacteristic
65C23×C181C4⋊C411yescharacteristic
66C2×C4×C181—41no—
67C2×C4×C181—41no—
68C2×C4×C181—41no—
69C22×C181—21no—
70C22×C181—21no—
71C22×C181—21no—
72C22×C181—21no—
73C22×C181—41no—
74C22×C181—41no—
75C22×C181—42no—
76C22×C181—41no—
77C22×C181—41no—
78C22×C181#9511yescharacteristic
79C22×C181#9911yescharacteristic
80C22×C181#9911yescharacteristic
81C2×D181—21no—
82C2×D181—21no—
83C2×D181—41no—
84C2×D181—41no—
85C2×D181—41no—
86C2×D181—41no—
87C2×D181—81no—
88#10624382—82no—
89C4×C181—81no—
90Dic181—21no—
91Dic181—21no—
92Dic181—42no—
93C2×C181—21no—
94C2×C181—21no—
95C2×C181—41no—
96C2×C181—41no—
97C2×C181#34211yescharacteristic
98C2×C181AΣL(1,𝔽16)11yescharacteristic
99C2×C181#35211yescharacteristic
100D181—41no—
101D181—41no—
102C181#139011yescharacteristic, complemented, Sylow
103#1390—1811nomaximal, Sylow
104C2×#100—1811no—
105C2×#99—1811no—
106#386—1811no—
107C22×D4—1811no—
108C23×C4—1811no—
109C2×M16—3621no—
110C2×C22⋊C4—3621no—
111C2×C22⋊C4—3621no—
112C2×C22⋊C4#1062438211yescharacteristic, complemented
113#100—3621no—
114#100—1812no—
115#99—3621no—
116#99—1812no—
117#95—3621no—
118C24—1811no—
119C24C2×#1062438211yescharacteristic
120C2×D4—1811no—
121C2×D4—1811no—
122C2×D4—1811no—
123C2×D4—1811no—
124C2×D4—7241no—
125C2×D4—7241no—
126C22×C4—21no—
127C22×C4—3621no—
128C22×C4—1811no—
129C22×C4—7242no—
130C22×C4—7241no—
131C22×C4—7241no—
132M16—3622no—
133M16—7241no—
134C2×C8—7241no—
135C22⋊C4—41no—
136C22⋊C4—3622no—
137C22⋊C4—7242no—
138C22⋊C4—7241no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—3621no—
144C23—3621no—
145C23—1811no—
146C23—1811no—
147C23—7241no—
148C23—7241no—
149C23C4×#1062438211yescharacteristic
150C23C22⋊#1062438211yescharacteristic
151C23C4⋊#1062438211yescharacteristic
152D4—7241no—
153D4—7241no—
154D4—7241no—
155D4—7241no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3621no—
160C2×C4—3621no—
161C2×C4—3622no—
162C2×C4—14481no—
163C2×C4—14481no—
164C2×C4—1811no—
165C2×C4—1811no—
166C2×C4—7242no—
167C2×C4—7241no—
168C2×C4—7242no—
169C2×C4—7241no—
170C2×C4—7242no—
171C8—7242no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—3621no—
182C22—3621no—
183C22—14481no—
184C22—7241no—
185C22—7241no—
186C22—7241no—
187C22—7241no—
188C22(C2×#10624382)⋊C411yescharacteristic
189C22#1915387411yescharacteristic
190C22#1915387511yescharacteristic
191C4—81no—
192C4—3621no—
193C4—3621no—
194C4—14482no—
195C4—7242no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—7241no—
201C2—7241no—
202C2#2604215711yescharacteristic
203C2#2604218011yescharacteristic
204C2#2604218211yescharacteristic
subgroupG/H
centerC22#19153875
commutatorC23×C181C42
FrattiniC24C2×#10624382
FittingC2×C22⋊C4×C181C4
socleC22×C181#99
prime radical, cosocleC22×C22⋊D181C22
hypercenterC2×C22⋊C4#10624382
hypocenterC181#1390
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #19153876C4semidirect
9. #19153874C4semidirect
11. C23×Dic181C4semidirect
14. C2×C22⋊C4×C181C4semidirect
102. C181#1390semidirect
112. C2×C22⋊C4#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538762
C2×#191538742
#260411562
#1390181