Finite Group Databasebeta

← #2196×C181 #37530139

Subgroups — #2196×C181 #37530139

Order 23168.

354 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 56 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
1D42×C181C211yesmaximal, characteristic, complemented
2B(4,𝔽2)×C181C212yesmaximal, complemented
3#453×C181C211yesmaximal, characteristic, complemented
4#353×C181C211yesmaximal, characteristic, complemented
5AΣL(1,𝔽16)×C181C212yesmaximal, complemented
6H(5,𝔽2)×C181C2211yescharacteristic, complemented
7C22×D4×C181—22no—
8AGL(1,ℤ/8)×C181—21no—
9#127×C181C2211yescharacteristic
10#121×C181—22no—
11#120×C181—21no—
12#120×C181—22no—
13#120×C181C2212yes—
14C4×D4×C181—21no—
15#104×C181—21no—
16#100×C181C2211yescharacteristic, complemented
17#99×C181—22no—
18#99×C181C2212yescomplemented
19C24×C181—21no—
20C24×C181D412yescomplemented
21#41×C181—21no—
22#41×C181—41no—
23C2×D4×C181—21no—
24C2×D4×C181—22no—
25C2×D4×C181—22no—
26C2×D4×C181—22no—
27C2×D4×C181—22no—
28C2×D4×C181—41no—
29C2×D4×C181—48no—
30C2×D4×C181—42no—
31C2×D4×C181D411yescharacteristic, complemented
32C2×D4×C181C2311yescharacteristic
33C22×C4×C181—22no—
34SD16×C181—41no—
35D8×C181—41no—
36M16×C181—21no—
37C4⋊C4×C181—21no—
38C22⋊C4×C181—21no—
39C22⋊C4×C181—22no—
40C22⋊C4×C181—22no—
41C22⋊C4×C181—42no—
42C22⋊C4×C181D412yescomplemented
43C42×C181D411yescharacteristic, complemented
44C23×C181—22no—
45C23×C181—22no—
46C23×C181—21no—
47C23×C181—42no—
48C23×C181—44no—
49C23×C181—44no—
50C23×C181—81no—
51C23×C181C2×D412yes—
52Q8×C181—21no—
53D4×C181—24no—
54D4×C181—21no—
55D4×C181—41no—
56D4×C181—41no—
57D4×C181—44no—
58D4×C181—44no—
59D4×C181—44no—
60D4×C181—42no—
61D4×C181—41no—
62D4×C181—84no—
63C2×C4×C181—21no—
64C2×C4×C181—22no—
65C2×C4×C181—22no—
66C2×C4×C181—44no—
67C2×C4×C181—41no—
68C2×C4×C181—42no—
69C2×C4×C181C2×D411yescharacteristic
70C8×C181—41no—
71C22×C181—24no—
72C22×C181—22no—
73C22×C181—42no—
74C22×C181—42no—
75C22×C181—42no—
76C22×C181—44no—
77C22×C181—42no—
78C22×C181—41no—
79C22×C181—41no—
80C22×C181—82no—
81C22×C181—84no—
82C22×C181—82no—
83C22×C181#12011yescharacteristic
84C4×C181—22no—
85C4×C181—21no—
86C4×C181—42no—
87C4×C181—41no—
88C4×C181—82no—
89C2×C181—21no—
90C2×C181—42no—
91C2×C181—44no—
92C2×C181—81no—
93C2×C181—81no—
94C2×C181B(4,𝔽2)11yescharacteristic
95C181#219611yescharacteristic, direct product factor, Sylow
96#2196C18111yesmaximal, characteristic, direct product factor, Sylow
97D42C2×C18111yescharacteristic, complemented
98B(4,𝔽2)C2×C18112yescomplemented
99#453C2×C18111yescharacteristic, complemented
100#353C2×C18111yescharacteristic, complemented
101AΣL(1,𝔽16)C2×C18112yescomplemented
102H(5,𝔽2)C22×C18111yescharacteristic, complemented
103C22×D4—22no—
104AGL(1,ℤ/8)—21no—
105#127C22×C18111yescharacteristic
106#121—22no—
107#120—21no—
108#120—22no—
109#120C22×C18112yes—
110C4×D4—21no—
111#104—21no—
112#100C22×C18111yescharacteristic, complemented
113#99—22no—
114#99C22×C18112yescomplemented
115C24—21no—
116C24D4×C18112yescomplemented
117#41—21no—
118#41—41no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—41no—
125C2×D4—48no—
126C2×D4—42no—
127C2×D4D4×C18111yescharacteristic, complemented
128C2×D4C23×C18111yescharacteristic
129C22×C4—22no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—42no—
138C22⋊C4D4×C18112yescomplemented
139C42D4×C18111yescharacteristic, complemented
140C23—22no—
141C23—22no—
142C23—21no—
143C23—42no—
144C23—44no—
145C23—44no—
146C23—81no—
147C23C2×D4×C18112yes—
148Q8—21no—
149D4—24no—
150D4—21no—
151D4—41no—
152D4—41no—
153D4—44no—
154D4—44no—
155D4—44no—
156D4—42no—
157D4—41no—
158D4—84no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4C2×D4×C18111yescharacteristic
166C8—41no—
167C22—24no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—44no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22—82no—
177C22—84no—
178C22—82no—
179C22#120×C18111yescharacteristic
180C4—22no—
181C4—21no—
182C4—42no—
183C4—41no—
184C4—82no—
185C2—21no—
186C2—42no—
187C2—44no—
188C2—81no—
189C2—81no—
190C2B(4,𝔽2)×C18111yescharacteristic
subgroupG/H
center, socleC2×C181B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C181
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C181 ◁ C22×C181 ◁ C2×D4×C181 ◁ G

Complements

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

NH ≅ G/Ntype
1. D42×C181C2semidirect
2. B(4,𝔽2)×C181C2semidirect
3. #453×C181C2semidirect
4. #353×C181C2semidirect
5. AΣL(1,𝔽16)×C181C2semidirect
6. H(5,𝔽2)×C181C22semidirect
16. #100×C181C22semidirect
18. #99×C181C22semidirect
20. C24×C181D4semidirect
31. C2×D4×C181D4semidirect
42. C22⋊C4×C181D4semidirect
43. C42×C181D4semidirect
95. C181#2196direct (N × H)
96. #2196C181direct (N × H)
97. D42C2×C181semidirect
98. B(4,𝔽2)C2×C181semidirect
99. #453C2×C181semidirect
100. #353C2×C181semidirect
101. AΣL(1,𝔽16)C2×C181semidirect
102. H(5,𝔽2)C22×C181semidirect
112. #100C22×C181semidirect
114. #99C22×C181semidirect
116. C24D4×C181semidirect
127. C2×D4D4×C181semidirect
138. C22⋊C4D4×C181semidirect
139. C42D4×C181semidirect

Maximal subgroups

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

subgroupindex
D42×C1812
B(4,𝔽2)×C1812
#453×C1812
#353×C1812
AΣL(1,𝔽16)×C1812
#2196181