Finite Group Databasebeta

← C22×#19149724 #36368364

Subgroups — C22×#19149724 #36368364

Order 22144.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#19149724C2148yesmaximal, direct product factor
2C22×#14583826C212yesmaximal, complemented
3C22×#14583825C214yesmaximal, complemented
4C23×C22⋊D173C214yesmaximal, complemented
5C22×D4×D173C214yesmaximal, complemented
6C23×D4×C173C211yesmaximal, characteristic, complemented
7C24×D173C2214yescomplemented
8#19149724C221256yesdirect product factor
9C2×#14583826C22148yescomplemented
10C2×#14583825C22196yescomplemented
11C22×Dic346C2212yescomplemented
12C22×D692C2212yescomplemented
13C22×C22⋊D173C2218yescomplemented
14C22×C22⋊D173C2214yescomplemented
15C22×C22⋊D173C22196yescomplemented
16C2×D4×D173C22196yescomplemented
17C23×Dic173C2214yescomplemented
18C22×C4×D173C2214yescomplemented
19C25×C173C2212yescomplemented
20C22×D4×C173C2214yescomplemented
21C22×D4×C173C22124yescomplemented
22C23×C4×C173C2211yescharacteristic, complemented
23C23×D173—416no—
24C23×D173C2314yes—
25C23×D173C23148yescomplemented
26#14583826C231128yescomplemented
27#14583825C231256yescomplemented
28C2×Dic346C23124yescomplemented
29C2×D692C23124yescomplemented
30C2×C22⋊D173C23196yescomplemented
31C2×C22⋊D173C23148yescomplemented
32C2×C22⋊D173C231256yescomplemented
33D4×D173C231256yescomplemented
34C22×Dic173C2314yes—
35C22×Dic173C23148yescomplemented
36C2×C4×D173C23148yescomplemented
37C24×C173—48no—
38C24×C173C2311yescharacteristic
39C24×C173C2314yes—
40C24×C173C23124yescomplemented
41C2×D4×C173C23148yescomplemented
42C2×D4×C173C23164yescomplemented
43C22×C4×C173C2312yes—
44C22×C4×C173C23112yescomplemented
45C22×D173—216no—
46C22×D173—496no—
47C22×D173C24124yes—
48C22×D173C24164yescomplemented
49Dic346C24132yescomplemented
50D692C24132yescomplemented
51C22⋊D173C241128yescomplemented
52C22⋊D173C24164yescomplemented
53C2×Dic173C24124yes—
54C2×Dic173C24164yescomplemented
55C4×D173C24164yescomplemented
56C23×C173—216no—
57C23×C173—24no—
58C23×C173—448no—
59C23×C173C24124yes—
60C23×C173C2416yes—
61C23×C173C2411yescharacteristic
62C23×C173C24132yescomplemented
63D4×C173C24164yescomplemented
64C2×C4×C173C24112yes—
65C2×C4×C173C24116yescomplemented
66C22×C173—212no—
67C22×C173—248no—
68C22×C173—432no—
69C22×C173H(5,𝔽2)14yescomplemented
70C22×C173C2514yes—
71C22×C173C25116yes—
72C22×C173C2513yes—
73C2×D173—248no—
74C2×D173—464no—
75C2×D173C25116yes—
76C4×C173C2518yes—
77Dic173C25116yes—
78C2×C173—216no—
79C2×C173—24no—
80C2×C173C2×H(5,𝔽2)16yescomplemented
81C2×C173C2611yescharacteristic
82D173—216no—
83C173C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—1731nomaximal, Sylow
85C2×H(5,𝔽2)—17348no—
86C22×#41—1734no—
87C22×#41—1732no—
88C23×D4—1734no—
89C23×D4—1734no—
90C23×D4D17311yescharacteristic, complemented
91C25—1734no—
92C25C2×D17312yescomplemented
93H(5,𝔽2)—173256no—
94C2×#41—17396no—
95C2×#41—17348no—
96C22×Q8—1732no—
97C22×D4—1738no—
98C22×D4—1734no—
99C22×D4—17396no—
100C22×D4—17396no—
101C22×D4—1732no—
102C22×D4C2×D17314yescomplemented
103C22×D4C2×D173124yescomplemented
104C23×C4—1734no—
105C23×C4—1734no—
106C23×C4C2×D17311yescharacteristic, complemented
107C24—48no—
108C24—1734no—
109C24—17348no—
110C24—69216no—
111C24C22×D17314yes—
112C24C22×D17311yescharacteristic
113C24C22×D173124yescomplemented
114#41—173256no—
115#41—173128no—
116C2×Q8—17324no—
117C2×D4—17396no—
118C2×D4—17348no—
119C2×D4—173256no—
120C2×D4—173256no—
121C2×D4—17324no—
122C2×D4C22×D173148yescomplemented
123C2×D4C22×D173164yescomplemented
124C22×C4—17348no—
125C22×C4—1734no—
126C22×C4—17348no—
127C22×C4C22×D17312yes—
128C22×C4C22×D173112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—34616no—
133C23—17324no—
134C23—17364no—
135C23—69296no—
136C23C23×D173124yes—
137C23C23×D17316yes—
138C23C23×D17311yescharacteristic
139C23C23×D173132yescomplemented
140Q8—17332no—
141D4—173128no—
142D4—17364no—
143D4—17332no—
144D4C23×D173164yescomplemented
145C2×C4—17324no—
146C2×C4—17364no—
147C2×C4—17364no—
148C2×C4C23×D173112yes—
149C2×C4C23×D173116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—34648no—
154C22—17316no—
155C22—69264no—
156C22#1914972414yesdirect product factor
157C22C24×D173116yes—
158C22C24×D17314yes—
159C22C24×D17313yes—
160C4—17316no—
161C4C24×D17318yes—
162C2—24no—
163C2—216no—
164C2—34616no—
165C2C2×#1914972416yesdirect product factor
166C2C25×D17311yescharacteristic
subgroupG/H
centerC23C23×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC23×D4×C173C2
socleC23×C173C24
hypercenterC23×D4D173
hypocenterC173C22×H(5,𝔽2)
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19149724C2direct (N × H)
2. C22×#14583826C2semidirect
3. C22×#14583825C2semidirect
4. C23×C22⋊D173C2semidirect
5. C22×D4×D173C2semidirect
6. C23×D4×C173C2semidirect
7. C24×D173C22semidirect
8. #19149724C22direct (N × H)
9. C2×#14583826C22semidirect
10. C2×#14583825C22semidirect
11. C22×Dic346C22semidirect
12. C22×D692C22semidirect
13. C22×C22⋊D173C22semidirect
14. C22×C22⋊D173C22semidirect
15. C22×C22⋊D173C22semidirect
16. C2×D4×D173C22semidirect
17. C23×Dic173C22semidirect
18. C22×C4×D173C22semidirect
19. C25×C173C22semidirect
20. C22×D4×C173C22semidirect
21. C22×D4×C173C22semidirect
22. C23×C4×C173C22semidirect
25. C23×D173C23semidirect
26. #14583826C23semidirect
27. #14583825C23semidirect
28. C2×Dic346C23semidirect
29. C2×D692C23semidirect
30. C2×C22⋊D173C23semidirect
31. C2×C22⋊D173C23semidirect
32. C2×C22⋊D173C23semidirect
33. D4×D173C23semidirect
35. C22×Dic173C23semidirect
36. C2×C4×D173C23semidirect
40. C24×C173C23semidirect
41. C2×D4×C173C23semidirect
42. C2×D4×C173C23semidirect
44. C22×C4×C173C23semidirect
48. C22×D173C24semidirect
49. Dic346C24semidirect
50. D692C24semidirect
51. C22⋊D173C24semidirect
52. C22⋊D173C24semidirect
54. C2×Dic173C24semidirect
55. C4×D173C24semidirect
62. C23×C173C24semidirect
63. D4×C173C24semidirect
65. C2×C4×C173C24semidirect
69. C22×C173H(5,𝔽2)semidirect
80. C2×C173C2×H(5,𝔽2)semidirect
83. C173C22×H(5,𝔽2)semidirect
90. C23×D4D173semidirect
92. C25C2×D173semidirect
102. C22×D4C2×D173semidirect
103. C22×D4C2×D173semidirect
106. C23×C4C2×D173semidirect
113. C24C22×D173semidirect
122. C2×D4C22×D173semidirect
123. C2×D4C22×D173semidirect
128. C22×C4C22×D173semidirect
139. C23C23×D173semidirect
144. D4C23×D173semidirect
149. C2×C4C23×D173semidirect
156. C22#19149724direct (N × H)
165. C2C2×#19149724direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497242
C22×#145838262
C22×#145838252
C23×C22⋊D1732
C22×D4×D1732
C23×D4×C1732
C22×H(5,𝔽2)173