Finite Group Databasebeta

← C2×#26012233 #36367683

Subgroups — C2×#26012233 #36367683

Order 22144.

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#26012233C2116yesmaximal, direct product factor
2C2×#19149668C212yesmaximal, complemented
3C2×#19149667C212yesmaximal, complemented
4C2×#19149665C214yesmaximal, complemented
5C2×#19149649C214yesmaximal, complemented
6C2×#19149647C212yesmaximal, complemented
7C2×#124×C173C211yesmaximal, characteristic, complemented
8C24×D173C2212yescomplemented
9C22×Dic346C2212yescomplemented
10C22×D692C2212yes—
11C22×D692C2212yes—
12C22×C22⋊D173C2214yes—
13#19149668C22116yescomplemented
14#19149667C22116yescomplemented
15#19149665C22132yescomplemented
16#19149649C22132yescomplemented
17#19149647C22116yescomplemented
18C2×C22⋊Dic173C2212yescomplemented
19C2×#14583803C2214yescomplemented
20C2×#14583803C2214yescomplemented
21C2×#14583803C2214yescomplemented
22C2×C4×Dic173C2212yescomplemented
23C22×Q8×C173C2211yescharacteristic, complemented
24C22×D4×C173C2211yescharacteristic
25#124×C173C2218yescomplemented
26C2×C22⋊C4×C173C2214yescomplemented
27C2×C42×C173C2211yescharacteristic, complemented
28C23×D173—48no—
29C23×D173—44no—
30C23×D173C2314yes—
31C23×D173C2318yescomplemented
32C2×Dic346—44no—
33C2×Dic346C2318yescomplemented
34C2×D692—44no—
35C2×D692—44no—
36C2×D692C2318yes—
37C2×D692C2318yes—
38C2×C22⋊D173—48no—
39C2×C22⋊D173C23116yes—
40C22×Dic173C2314yes—
41C22⋊Dic173C2318yescomplemented
42#14583803C23116yescomplemented
43#14583803C23116yescomplemented
44#14583803C23116yescomplemented
45C4×Dic173C2318yescomplemented
46C24×C173C2312yes—
47C2×Q8×C173—42no—
48C2×Q8×C173C2314yescomplemented
49C2×D4×C173—42no—
50C2×D4×C173C2314yes—
51C22×C4×C173C2311yescharacteristic
52C22×C4×C173C2312yes—
53C22×C4×C173C2312yes—
54C22⋊C4×C173C23116yescomplemented
55C42×C173C2314yescomplemented
56C22×D173—28no—
57C22×D173—28no—
58C22×D173—28no—
59C22×D173—416no—
60C22×D173—48no—
61C22×D173—432no—
62C22×D173C2418yes—
63Dic346—48no—
64D692—48no—
65D692—48no—
66C22⋊D173—416no—
67C2×Dic173—28no—
68C2×Dic173C2418yes—
69C23×C173—28no—
70C23×C173—24no—
71C23×C173C2414yes—
72C23×C173C2411yescharacteristic
73Q8×C173—44no—
74D4×C173—44no—
75C2×C4×C173—24no—
76C2×C4×C173—24no—
77C2×C4×C173—22no—
78C2×C4×C173C2414yes—
79C2×C4×C173C2414yes—
80C2×C4×C173C2412yes—
81C22×C173—24no—
82C22×C173—28no—
83C22×C173—48no—
84C22×C173H(5,𝔽2)12yes—
85C22×C173H(5,𝔽2)14yes—
86C22×C173C2511yescharacteristic
87C2×D173—28no—
88C2×D173—28no—
89C2×D173—28no—
90C2×D173—432no—
91C2×D173—416no—
92C4×C173—22no—
93C4×C173—24no—
94C4×C173—24no—
95Dic173—28no—
96C2×C173—44no—
97C2×C173H(3,𝔽4)14yescomplemented
98C2×C173C2×H(5,𝔽2)11yescharacteristic
99C2×C173C2×H(5,𝔽2)12yes—
100D173—48no—
101C173C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
102C2×H(3,𝔽4)—1731nomaximal, Sylow
103H(3,𝔽4)—17316no—
104C2×#124—1732no—
105C2×#124—1734no—
106C2×#124—1732no—
107C2×#124D17311yescharacteristic, complemented
108C2×#120—1732no—
109C2×#120—1734no—
110C25—1732no—
111C22×Q8—1732no—
112C22×Q8C2×D17311yescharacteristic, complemented
113C22×D4—1734no—
114C22×D4—1732no—
115C22×D4—1732no—
116C22×D4C2×D17311yescharacteristic
117#124—17316no—
118#124—17332no—
119#124—17316no—
120#124C2×D17318yescomplemented
121#120—17316no—
122#120—17332no—
123C2×C22⋊C4—1732no—
124C2×C22⋊C4—1734no—
125C2×C22⋊C4—1734no—
126C2×C22⋊C4—1734no—
127C2×C22⋊C4C2×D17314yescomplemented
128C2×C42—1732no—
129C2×C42C2×D17311yescharacteristic, complemented
130C24—1734no—
131C24—1738no—
132C24—6924no—
133C24—6928no—
134C24C22×D17312yes—
135C2×Q8—42no—
136C2×Q8—1738no—
137C2×Q8—6924no—
138C2×Q8C22×D17314yescomplemented
139C2×D4—42no—
140C2×D4—17316no—
141C2×D4—1738no—
142C2×D4—1738no—
143C2×D4—6924no—
144C2×D4—6924no—
145C2×D4—6928no—
146C2×D4C22×D17314yes—
147C22×C4—1734no—
148C22×C4C22×D17312yes—
149C22×C4C22×D17311yescharacteristic
150C22×C4C22×D17312yes—
151C22⋊C4—1738no—
152C22⋊C4—17316no—
153C22⋊C4—17316no—
154C22⋊C4—17316no—
155C22⋊C4C22×D173116yescomplemented
156C42—1738no—
157C42C22×D17314yescomplemented
158C23—24no—
159C23—28no—
160C23—3468no—
161C23—3468no—
162C23—3468no—
163C23—1738no—
164C23—6928no—
165C23—69216no—
166C23—69232no—
167C23C23×D17314yes—
168C23C23×D17311yescharacteristic
169Q8—44no—
170Q8—6928no—
171D4—44no—
172D4—6928no—
173D4—6928no—
174D4—69216no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—3468no—
179C2×C4—1738no—
180C2×C4C23×D17314yes—
181C2×C4C23×D17312yes—
182C2×C4C23×D17314yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—3468no—
187C22—3468no—
188C22—3468no—
189C22—69232no—
190C22—69216no—
191C22#1914972412yes—
192C22#1914972514yes—
193C22C24×D17311yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—3468no—
198C2—44no—
199C2—6928no—
200C2#2601223314yesdirect product factor
201C2C2×#1914972411yescharacteristic
202C2C2×#1914972512yes—
subgroupG/H
centerC23C23×D173
commutator, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC2×#124×C173C2
socleC23×C173C24
hypercenterC2×#124D173
hypocenterC173C2×H(3,𝔽4)
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #26012233C2direct (N × H)
2. C2×#19149668C2semidirect
3. C2×#19149667C2semidirect
4. C2×#19149665C2semidirect
5. C2×#19149649C2semidirect
6. C2×#19149647C2semidirect
7. C2×#124×C173C2semidirect
8. C24×D173C22semidirect
9. C22×Dic346C22semidirect
13. #19149668C22semidirect
14. #19149667C22semidirect
15. #19149665C22semidirect
16. #19149649C22semidirect
17. #19149647C22semidirect
18. C2×C22⋊Dic173C22semidirect
19. C2×#14583803C22semidirect
20. C2×#14583803C22semidirect
21. C2×#14583803C22semidirect
22. C2×C4×Dic173C22semidirect
23. C22×Q8×C173C22semidirect
25. #124×C173C22semidirect
26. C2×C22⋊C4×C173C22semidirect
27. C2×C42×C173C22semidirect
31. C23×D173C23semidirect
33. C2×Dic346C23semidirect
41. C22⋊Dic173C23semidirect
42. #14583803C23semidirect
43. #14583803C23semidirect
44. #14583803C23semidirect
45. C4×Dic173C23semidirect
48. C2×Q8×C173C23semidirect
54. C22⋊C4×C173C23semidirect
55. C42×C173C23semidirect
97. C2×C173H(3,𝔽4)semidirect
101. C173C2×H(3,𝔽4)semidirect
107. C2×#124D173semidirect
112. C22×Q8C2×D173semidirect
120. #124C2×D173semidirect
127. C2×C22⋊C4C2×D173semidirect
129. C2×C42C2×D173semidirect
138. C2×Q8C22×D173semidirect
155. C22⋊C4C22×D173semidirect
157. C42C22×D173semidirect
200. C2#26012233direct (N × H)

Maximal subgroups

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

subgroupindex
#260122332
C2×#191496682
C2×#191496672
C2×#191496652
C2×#191496492
C2×#191496472
C2×#124×C1732
C2×H(3,𝔽4)173