Finite Group Databasebeta

← C2×#26042021 #37547841

Subgroups — C2×#26042021 #37547841

Order 23168.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#26042021C2116yesmaximal, direct product factor
2C2×#19153805C212yesmaximal, complemented
3C2×#19153804C212yesmaximal, complemented
4C2×#19153802C214yesmaximal, complemented
5C2×#19153786C214yesmaximal, complemented
6C2×#19153784C212yesmaximal, complemented
7C2×#124×C181C211yesmaximal, characteristic, complemented
8C24×D181C2212yescomplemented
9C22×Dic362C2212yescomplemented
10C22×D724C2212yes—
11C22×D724C2212yes—
12C22×C22⋊D181C2214yes—
13#19153805C22116yescomplemented
14#19153804C22116yescomplemented
15#19153802C22132yescomplemented
16#19153786C22132yescomplemented
17#19153784C22116yescomplemented
18C2×C22⋊Dic181C2212yescomplemented
19C2×#15668426C2214yescomplemented
20C2×#15668426C2214yescomplemented
21C2×#15668426C2214yescomplemented
22C2×C4×Dic181C2212yescomplemented
23C22×Q8×C181C2211yescharacteristic, complemented
24C22×D4×C181C2211yescharacteristic
25#124×C181C2218yescomplemented
26C2×C22⋊C4×C181C2214yescomplemented
27C2×C42×C181C2211yescharacteristic, complemented
28C23×D181—44no—
29C23×D181—48no—
30C23×D181C2314yes—
31C23×D181C2318yescomplemented
32C2×Dic362—44no—
33C2×Dic362C2318yescomplemented
34C2×D724—44no—
35C2×D724—44no—
36C2×D724C2318yes—
37C2×D724C2318yes—
38C2×C22⋊D181—48no—
39C2×C22⋊D181C23116yes—
40C22×Dic181C2314yes—
41C22⋊Dic181C2318yescomplemented
42#15668426C23116yescomplemented
43#15668426C23116yescomplemented
44#15668426C23116yescomplemented
45C4×Dic181C2318yescomplemented
46C24×C181C2312yes—
47C2×Q8×C181—42no—
48C2×Q8×C181C2314yescomplemented
49C2×D4×C181—42no—
50C2×D4×C181C2314yes—
51C22×C4×C181C2312yes—
52C22×C4×C181C2311yescharacteristic
53C22×C4×C181C2312yes—
54C22⋊C4×C181C23116yescomplemented
55C42×C181C2314yescomplemented
56C22×D181—28no—
57C22×D181—28no—
58C22×D181—28no—
59C22×D181—416no—
60C22×D181—48no—
61C22×D181—432no—
62C22×D181C2418yes—
63Dic362—48no—
64D724—48no—
65D724—48no—
66C22⋊D181—416no—
67C2×Dic181—28no—
68C2×Dic181C2418yes—
69C23×C181—24no—
70C23×C181—28no—
71C23×C181C2414yes—
72C23×C181C2411yescharacteristic
73Q8×C181—44no—
74D4×C181—44no—
75C2×C4×C181—22no—
76C2×C4×C181—24no—
77C2×C4×C181—24no—
78C2×C4×C181C2414yes—
79C2×C4×C181C2412yes—
80C2×C4×C181C2414yes—
81C22×C181—24no—
82C22×C181—28no—
83C22×C181—48no—
84C22×C181H(5,𝔽2)12yes—
85C22×C181H(5,𝔽2)14yes—
86C22×C181C2511yescharacteristic
87C2×D181—28no—
88C2×D181—28no—
89C2×D181—28no—
90C2×D181—432no—
91C2×D181—416no—
92C4×C181—22no—
93C4×C181—24no—
94C4×C181—24no—
95Dic181—28no—
96C2×C181—44no—
97C2×C181H(3,𝔽4)14yescomplemented
98C2×C181C2×H(5,𝔽2)11yescharacteristic
99C2×C181C2×H(5,𝔽2)12yes—
100D181—48no—
101C181C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
102C2×H(3,𝔽4)—1811nomaximal, Sylow
103H(3,𝔽4)—18116no—
104C2×#124—1812no—
105C2×#124—1814no—
106C2×#124—1812no—
107C2×#124D18111yescharacteristic, complemented
108C2×#120—1812no—
109C2×#120—1814no—
110C25—1812no—
111C22×Q8—1812no—
112C22×Q8C2×D18111yescharacteristic, complemented
113C22×D4—1814no—
114C22×D4—1812no—
115C22×D4—1812no—
116C22×D4C2×D18111yescharacteristic
117#124—18116no—
118#124—18132no—
119#124—18116no—
120#124C2×D18118yescomplemented
121#120—18116no—
122#120—18132no—
123C2×C22⋊C4—1812no—
124C2×C22⋊C4—1814no—
125C2×C22⋊C4—1814no—
126C2×C22⋊C4—1814no—
127C2×C22⋊C4C2×D18114yescomplemented
128C2×C42—1812no—
129C2×C42C2×D18111yescharacteristic, complemented
130C24—1814no—
131C24—1818no—
132C24—7244no—
133C24—7248no—
134C24C22×D18112yes—
135C2×Q8—42no—
136C2×Q8—1818no—
137C2×Q8—7244no—
138C2×Q8C22×D18114yescomplemented
139C2×D4—42no—
140C2×D4—18116no—
141C2×D4—1818no—
142C2×D4—1818no—
143C2×D4—7244no—
144C2×D4—7244no—
145C2×D4—7248no—
146C2×D4C22×D18114yes—
147C22×C4—1814no—
148C22×C4C22×D18112yes—
149C22×C4C22×D18111yescharacteristic
150C22×C4C22×D18112yes—
151C22⋊C4—1818no—
152C22⋊C4—18116no—
153C22⋊C4—18116no—
154C22⋊C4—18116no—
155C22⋊C4C22×D181116yescomplemented
156C42—1818no—
157C42C22×D18114yescomplemented
158C23—24no—
159C23—28no—
160C23—3628no—
161C23—3628no—
162C23—3628no—
163C23—1818no—
164C23—7248no—
165C23—72416no—
166C23—72432no—
167C23C23×D18114yes—
168C23C23×D18111yescharacteristic
169Q8—44no—
170Q8—7248no—
171D4—44no—
172D4—7248no—
173D4—7248no—
174D4—72416no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—3628no—
179C2×C4—1818no—
180C2×C4C23×D18114yes—
181C2×C4C23×D18112yes—
182C2×C4C23×D18114yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—3628no—
187C22—3628no—
188C22—3628no—
189C22—72432no—
190C22—72416no—
191C22#1915386112yes—
192C22#1915386214yes—
193C22C24×D18111yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—3628no—
198C2—44no—
199C2—7248no—
200C2#2604202114yesdirect product factor
201C2C2×#1915386111yescharacteristic
202C2C2×#1915386212yes—
subgroupG/H
centerC23C23×D181
commutator, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC2×#124×C181C2
socleC23×C181C24
hypercenterC2×#124D181
hypocenterC181C2×H(3,𝔽4)
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #26042021C2direct (N × H)
2. C2×#19153805C2semidirect
3. C2×#19153804C2semidirect
4. C2×#19153802C2semidirect
5. C2×#19153786C2semidirect
6. C2×#19153784C2semidirect
7. C2×#124×C181C2semidirect
8. C24×D181C22semidirect
9. C22×Dic362C22semidirect
13. #19153805C22semidirect
14. #19153804C22semidirect
15. #19153802C22semidirect
16. #19153786C22semidirect
17. #19153784C22semidirect
18. C2×C22⋊Dic181C22semidirect
19. C2×#15668426C22semidirect
20. C2×#15668426C22semidirect
21. C2×#15668426C22semidirect
22. C2×C4×Dic181C22semidirect
23. C22×Q8×C181C22semidirect
25. #124×C181C22semidirect
26. C2×C22⋊C4×C181C22semidirect
27. C2×C42×C181C22semidirect
31. C23×D181C23semidirect
33. C2×Dic362C23semidirect
41. C22⋊Dic181C23semidirect
42. #15668426C23semidirect
43. #15668426C23semidirect
44. #15668426C23semidirect
45. C4×Dic181C23semidirect
48. C2×Q8×C181C23semidirect
54. C22⋊C4×C181C23semidirect
55. C42×C181C23semidirect
97. C2×C181H(3,𝔽4)semidirect
101. C181C2×H(3,𝔽4)semidirect
107. C2×#124D181semidirect
112. C22×Q8C2×D181semidirect
120. #124C2×D181semidirect
127. C2×C22⋊C4C2×D181semidirect
129. C2×C42C2×D181semidirect
138. C2×Q8C22×D181semidirect
155. C22⋊C4C22×D181semidirect
157. C42C22×D181semidirect
200. C2#26042021direct (N × H)

Maximal subgroups

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

subgroupindex
#260420212
C2×#191538052
C2×#191538042
C2×#191538022
C2×#191537862
C2×#191537842
C2×#124×C1812
C2×H(3,𝔽4)181