Finite Group Databasebeta

← C2×#26042020 #37547839

Subgroups — C2×#26042020 #37547839

Order 23168.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 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#26042020C2116yesmaximal, direct product factor
2C2×#19153813C211yesmaximal, characteristic, complemented
3C2×#19153810C211yesmaximal, characteristic, complemented
4C2×#19153801C213yesmaximal, complemented
5C2×#19153791C213yesmaximal, complemented
6C2×#19153789C213yesmaximal, complemented
7C2×#19153786C213yesmaximal, complemented
8C2×#126×C181C211yesmaximal, characteristic, complemented
9C24×D181C2211yescharacteristic
10C22×D724C2213yes—
11C22×D724C2213yes—
12C22×D724C2213yescomplemented
13C22×C22⋊D181C2213yes—
14C22×C4×D181C2213yescomplemented
15#19153813C2218yescomplemented
16#19153810C2218yescomplemented
17#19153801C22124yescomplemented
18#19153791C22124yescomplemented
19#19153789C22124yescomplemented
20#19153786C22124yescomplemented
21C2×#15668428C2213yescomplemented
22C2×#15668426C2213yes—
23C2×#15668426C2213yes—
24C2×#15668426C2213yescomplemented
25#126×C181C2218yescomplemented
26C2×C4⋊C4×C181C2213yescomplemented
27C2×C22⋊C4×C181C2213yes—
28C2×C42×C181C2211yescharacteristic, complemented
29C23×D181—46no—
30C23×D181C2311yescharacteristic
31C23×D181C2311yescharacteristic
32C23×D181C2313yes—
33C23×D181C2314yes—
34C2×D724—46no—
35C2×D724—46no—
36C2×D724—46no—
37C2×D724C23112yes—
38C2×D724C23112yes—
39C2×D724C23112yescomplemented
40C2×C22⋊D181—46no—
41C2×C22⋊D181C23112yes—
42C22×Dic181C2313yes—
43C2×C4×D181—46no—
44C2×C4×D181C23112yescomplemented
45#15668428C23112yescomplemented
46#15668426C23112yes—
47#15668426C23112yes—
48#15668426C23112yescomplemented
49C24×C181C2311yescharacteristic
50C22×C4×C181C2313yes—
51C22×C4×C181C2313yes—
52C4⋊C4×C181C23112yescomplemented
53C22⋊C4×C181C23112yes—
54C42×C181C2314yescomplemented
55C22×D181—26no—
56C22×D181—26no—
57C22×D181—26no—
58C22×D181—26no—
59C22×D181—26no—
60C22×D181—412no—
61C22×D181—416no—
62C22×D181C2412yes—
63C22×D181C2412yes—
64C22×D181C2416yes—
65D724—412no—
66D724—412no—
67D724—412no—
68C22⋊D181—412no—
69C2×Dic181—26no—
70C2×Dic181C2416yes—
71C4×D181—412no—
72C23×C181—26no—
73C23×C181C2412yes—
74C23×C181C2411yescharacteristic
75C2×C4×C181—26no—
76C2×C4×C181—26no—
77C2×C4×C181C2416yes—
78C2×C4×C181C2416yes—
79C22×C181—26no—
80C22×C181—44no—
81C22×C181H(5,𝔽2)16yes—
82C22×C181C2511yescharacteristic
83C2×D181—26no—
84C2×D181—26no—
85C2×D181—26no—
86C2×D181—26no—
87C2×D181—26no—
88C2×D181—416no—
89C2×D181—412no—
90C2×D181—44no—
91C2×D181—44no—
92C4×C181—26no—
93C4×C181—26no—
94Dic181—26no—
95C2×C181—42no—
96C2×C181#56014yescomplemented
97C2×C181C2×H(5,𝔽2)13yes—
98D181—46no—
99D181—42no—
100D181—42no—
101C181C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—1811nomaximal, Sylow
103#560—18116no—
104C2×#127—1811no—
105C2×#126—1811no—
106C2×#126D18111yescharacteristic, complemented
107C2×#123—1813no—
108C2×#121—1813no—
109C2×#121—1813no—
110C2×#120—1813no—
111C25—1811no—
112C22×D4—1813no—
113C22×D4—1813no—
114C22×D4—1813no—
115C22×D4—1813no—
116C23×C4—1813no—
117#127—1818no—
118#126—1818no—
119#126C2×D18118yescomplemented
120#123—18124no—
121#121—18124no—
122#121—18124no—
123#120—18124no—
124C2×C4⋊C4—1813no—
125C2×C4⋊C4C2×D18113yescomplemented
126C2×C22⋊C4—1813no—
127C2×C22⋊C4—1813no—
128C2×C22⋊C4—1813no—
129C2×C22⋊C4C2×D18113yes—
130C2×C42C2×D18111yescharacteristic, complemented
131C24—1811no—
132C24—1811no—
133C24—1813no—
134C24—1814no—
135C24—7246no—
136C24C22×D18111yescharacteristic
137C2×D4—18112no—
138C2×D4—18112no—
139C2×D4—18112no—
140C2×D4—18112no—
141C2×D4—7246no—
142C2×D4—7246no—
143C2×D4—7246no—
144C2×D4—7246no—
145C22×C4—1813no—
146C22×C4—18112no—
147C22×C4—7246no—
148C22×C4C22×D18113yes—
149C22×C4C22×D18113yes—
150C4⋊C4—18112no—
151C4⋊C4C22×D181112yescomplemented
152C22⋊C4—18112no—
153C22⋊C4—18112no—
154C22⋊C4—18112no—
155C22⋊C4C22×D181112yes—
156C42C22×D18114yescomplemented
157C23—26no—
158C23—3626no—
159C23—3626no—
160C23—3626no—
161C23—3626no—
162C23—3626no—
163C23—1812no—
164C23—1812no—
165C23—1816no—
166C23—72412no—
167C23—72416no—
168C23C23×D18112yes—
169C23C23×D18111yescharacteristic
170D4—72412no—
171D4—72412no—
172D4—72412no—
173D4—72412no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—3626no—
177C2×C4—1816no—
178C2×C4—72412no—
179C2×C4C23×D18116yes—
180C2×C4C23×D18116yes—
181C22—26no—
182C22—44no—
183C22—3626no—
184C22—3626no—
185C22—3626no—
186C22—3626no—
187C22—3626no—
188C22—72416no—
189C22—7244no—
190C22—7244no—
191C22—72412no—
192C22#1915386216yes—
193C22C24×D18111yescharacteristic
194C4—26no—
195C4—26no—
196C4—3626no—
197C2—42no—
198C2—7242no—
199C2—7242no—
200C2—7246no—
201C2#2604202014yesdirect product factor
202C2C2×#1915386213yes—
subgroupG/H
centerC23C23×D181
commutator, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC2×#126×C181C2
socleC23×C181C24
hypercenterC2×#126D181
hypocenterC181C2×#560
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #26042020C2direct (N × H)
2. C2×#19153813C2semidirect
3. C2×#19153810C2semidirect
4. C2×#19153801C2semidirect
5. C2×#19153791C2semidirect
6. C2×#19153789C2semidirect
7. C2×#19153786C2semidirect
8. C2×#126×C181C2semidirect
12. C22×D724C22semidirect
14. C22×C4×D181C22semidirect
15. #19153813C22semidirect
16. #19153810C22semidirect
17. #19153801C22semidirect
18. #19153791C22semidirect
19. #19153789C22semidirect
20. #19153786C22semidirect
21. C2×#15668428C22semidirect
24. C2×#15668426C22semidirect
25. #126×C181C22semidirect
26. C2×C4⋊C4×C181C22semidirect
28. C2×C42×C181C22semidirect
39. C2×D724C23semidirect
44. C2×C4×D181C23semidirect
45. #15668428C23semidirect
48. #15668426C23semidirect
52. C4⋊C4×C181C23semidirect
54. C42×C181C23semidirect
96. C2×C181#560semidirect
101. C181C2×#560semidirect
106. C2×#126D181semidirect
119. #126C2×D181semidirect
125. C2×C4⋊C4C2×D181semidirect
130. C2×C42C2×D181semidirect
151. C4⋊C4C22×D181semidirect
156. C42C22×D181semidirect
201. C2#26042020direct (N × H)

Maximal subgroups

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

subgroupindex
#260420202
C2×#191538132
C2×#191538102
C2×#191538012
C2×#191537912
C2×#191537892
C2×#191537862
C2×#126×C1812
C2×#560181