Finite Group Databasebeta

← #37549117

Subgroups — #37549117

Order 23168.

346 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 64 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#26042250C211yesmaximal, characteristic, complemented
2C4×C22⋊#10624382C211yesmaximal, characteristic, complemented
3#26042130C214yesmaximal, complemented
4C2×#19153787C211yesmaximal, characteristic
5C2×#15668459—21no—
6C2×C22⋊#10624382—21no—
7C2×C4×#10624382—21no—
8#19153873C2212yescomplemented
9C22⋊#13089685—21no—
10(C2×#10624382)⋊C4C2212yescomplemented
11C22×C22⋊D181—21no—
12C22×C4×D181C2211yescharacteristic
13#19153787—21no—
14#19153787—21no—
15#19153787C412yescomplemented
16#19153787C2212yes—
17C2×C4⋊Dic181C411yescharacteristic, complemented
18C2×C22⋊Dic181—21no—
19C22×D4×C181C411yescharacteristic, complemented
20C22×#10624382—22no—
21#15668459—42no—
22C2×#13089685—22no—
23C22⋊#10624382—24no—
24C4×#10624382—42no—
25C23×D181D411yescharacteristic
26C2×C22⋊D181—22no—
27C2×C22⋊D181—22no—
28C2×C22⋊D181—44no—
29C22×Dic181—21no—
30C22×Dic181D411yescharacteristic
31C2×C4×D181—42no—
32C2×C4×D181C2×C411yescharacteristic
33C2×C4×D181D411yescharacteristic
34C2×C4×D181D411yescharacteristic
35C2×C4×D181C2311yescharacteristic
36C4⋊Dic181—21no—
37C4⋊Dic181C2×C412yes—
38C22⋊Dic181—22no—
39C22⋊Dic181—22no—
40C24×C181—21no—
41C2×D4×C181—21no—
42C2×D4×C181—42no—
43C2×D4×C181C2×C412yes—
44C22×C4×C181C2×C411yescharacteristic
45C2×#10624382—24no—
46C2×#10624382—42no—
47C2×#10624382—42no—
48#13089685—42no—
49C22×D181—22no—
50C22×D181—22no—
51C22×D181—41no—
52C22×D181C22⋊C411yescharacteristic
53C22×D181C2×D411yescharacteristic
54C22⋊D181—44no—
55C22⋊D181—44no—
56C2×Dic181—22no—
57C2×Dic181—41no—
58C2×Dic181—42no—
59C2×Dic181C22⋊C411yescharacteristic
60C2×Dic181C2×D411yescharacteristic
61C4×D181—22no—
62C4×D181—22no—
63C4×D181—41no—
64C4×D181—41no—
65C23×C181—22no—
66C23×C181—42no—
67C23×C181—42no—
68C23×C181—42no—
69C23×C181C22⋊C411yescharacteristic
70D4×C181—42no—
71D4×C181—42no—
72C2×C4×C181—22no—
73C2×C4×C181C22⋊C411yescharacteristic
74C2×C4×C181C22×C411yescharacteristic
75C22×C181—21no—
76C22×C181—21no—
77C22×C181—42no—
78C22×C181—42no—
79C22×C181—42no—
80C22×C181—84no—
81C22×C181#10412yes—
82C22×C181C2×C22⋊C411yescharacteristic
83C2×D181—21no—
84C2×D181—21no—
85C2×D181—21no—
86C2×D181—21no—
87C2×D181—21no—
88C2×D181—44no—
89C2×D181AGL(1,ℤ/8)12yes—
90#10624382—44no—
91C4×C181—21no—
92C4×C181#9912yes—
93Dic181—21no—
94Dic181—21no—
95Dic181—42no—
96C2×C181—22no—
97C2×C181—82no—
98C2×C181C2×#9911yescharacteristic
99C2×C181#41811yescharacteristic
100C2×C181C2×#10411yescharacteristic
101D181—22no—
102D181—41no—
103C181#150711yescharacteristic, complemented, Sylow
104#1507—1811nomaximal, Sylow
105C2×#121—1811no—
106#407—1811no—
107C4×C22⋊C4—1811no—
108#327—1814no—
109C22×D4—3621no—
110C22×D4#1062438211yescharacteristic, complemented
111C23×C4—1811no—
112C2×M16—3621no—
113#121—3621no—
114#121—3621no—
115#121—1812no—
116#121—1812no—
117C2×C4⋊C4—1811no—
118C2×C22⋊C4—3621no—
119C2×C22⋊C4—3621no—
120C2×C42—3621no—
121C22⋊C8—3621no—
122C22⋊C8—1812no—
123#95—1812no—
124C24—21no—
125C24—1811no—
126C2×D4—21no—
127C2×D4—42no—
128C2×D4—3622no—
129C2×D4—3622no—
130C2×D4—7244no—
131C2×D4C2×#1062438212yes—
132C22×C4—3622no—
133C22×C4—3621no—
134C22×C4—1811no—
135C22×C4—1811no—
136C22×C4—1811no—
137C22×C4—1811no—
138C22×C4—1811no—
139C22×C4—7242no—
140C22×C4C2×#1062438211yescharacteristic
141M16—7242no—
142C2×C8—3622no—
143C4⋊C4—3621no—
144C4⋊C4—1812no—
145C22⋊C4—3624no—
146C22⋊C4—3622no—
147C22⋊C4—3622no—
148C42—7242no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—3622no—
154C23—3622no—
155C23—1811no—
156C23—1811no—
157C23—7241no—
158C23C22⋊#1062438211yescharacteristic
159D4—42no—
160D4—42no—
161D4—7244no—
162D4—7244no—
163C2×C4—22no—
164C2×C4—3622no—
165C2×C4—3622no—
166C2×C4—3624no—
167C2×C4—3622no—
168C2×C4—1811no—
169C2×C4—1811no—
170C2×C4—7241no—
171C2×C4—7241no—
172C2×C4—7241no—
173C2×C4—7242no—
174C2×C4—7242no—
175C2×C4—7242no—
176C2×C4C22⋊#1062438211yescharacteristic
177C2×C4C22×#1062438211yescharacteristic
178C8—7242no—
179C22—21no—
180C22—21no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22—3621no—
186C22—3621no—
187C22—3621no—
188C22—3621no—
189C22—3621no—
190C22—1812no—
191C22—7244no—
192C22#1915388012yes—
193C22C2×C22⋊#1062438211yescharacteristic
194C4—21no—
195C4—3621no—
196C4—3621no—
197C4—7244no—
198C4—7242no—
199C4#1915387412yes—
200C2—22no—
201C2—82no—
202C2—3622no—
203C2—7241no—
204C2C2×#1915387411yescharacteristic
205C2#2604227111yescharacteristic
206C2C2×#1915388011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC22×D4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC2×#19153787C2
cosocleC2×C4×D181C23
hypercenterC22×D4#10624382
hypocenterC181#1507
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26042250C2semidirect
2. C4×C22⋊#10624382C2semidirect
3. #26042130C2semidirect
8. #19153873C22semidirect
10. (C2×#10624382)⋊C4C22semidirect
15. #19153787C4semidirect
17. C2×C4⋊Dic181C4semidirect
19. C22×D4×C181C4semidirect
103. C181#1507semidirect
110. C22×D4#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422502
C4×C22⋊#106243822
#260421302
C2×#191537872
#1507181