Finite Group Databasebeta

← C22×#14581837 #24922264

Subgroups — C22×#14581837 #24922264

Order 10624.

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×#14581837C2148yesmaximal, direct product factor
2C22×#13050590C212yesmaximal, complemented
3C22×#13050589C214yesmaximal, complemented
4C23×C22⋊D83C214yesmaximal, complemented
5C22×D4×D83C214yesmaximal, complemented
6C23×D4×C83C211yesmaximal, characteristic, complemented
7C24×D83C2214yescomplemented
8#14581837C221256yesdirect product factor
9C2×#13050590C22148yescomplemented
10C2×#13050589C22196yescomplemented
11C22×Dic166C2212yescomplemented
12C22×D332C2212yescomplemented
13C22×C22⋊D83C2218yescomplemented
14C22×C22⋊D83C2214yescomplemented
15C22×C22⋊D83C22196yescomplemented
16C2×D4×D83C22196yescomplemented
17C23×Dic83C2214yescomplemented
18C22×C4×D83C2214yescomplemented
19C25×C83C2212yescomplemented
20C22×D4×C83C2214yescomplemented
21C22×D4×C83C22124yescomplemented
22C23×C4×C83C2211yescharacteristic, complemented
23C23×D83—416no—
24C23×D83C2314yes—
25C23×D83C23148yescomplemented
26#13050590C231128yescomplemented
27#13050589C231256yescomplemented
28C2×Dic166C23124yescomplemented
29C2×D332C23124yescomplemented
30C2×C22⋊D83C23196yescomplemented
31C2×C22⋊D83C23148yescomplemented
32C2×C22⋊D83C231256yescomplemented
33D4×D83C231256yescomplemented
34C22×Dic83C2314yes—
35C22×Dic83C23148yescomplemented
36C2×C4×D83C23148yescomplemented
37C24×C83—48no—
38C24×C83C2314yes—
39C24×C83C2311yescharacteristic
40C24×C83C23124yescomplemented
41C2×D4×C83C23148yescomplemented
42C2×D4×C83C23164yescomplemented
43C22×C4×C83C2312yes—
44C22×C4×C83C23112yescomplemented
45C23×C83—24no—
46C23×C83—216no—
47C23×C83—448no—
48C23×C83C2416yes—
49C23×C83C24124yes—
50C23×C83C2411yescharacteristic
51C23×C83C24132yescomplemented
52C22×D83—216no—
53C22×D83—496no—
54C22×D83C24124yes—
55C22×D83C24164yescomplemented
56D4×C83C24164yescomplemented
57C2×C4×C83C24112yes—
58C2×C4×C83C24116yescomplemented
59C22⋊D83C241128yescomplemented
60C22⋊D83C24164yescomplemented
61C2×Dic83C24124yes—
62C2×Dic83C24164yescomplemented
63D332C24132yescomplemented
64C4×D83C24164yescomplemented
65Dic166C24132yescomplemented
66C22×C83—248no—
67C22×C83—212no—
68C22×C83—432no—
69C22×C83H(5,𝔽2)14yescomplemented
70C22×C83C25116yes—
71C22×C83C2514yes—
72C22×C83C2513yes—
73C2×D83—248no—
74C2×D83—464no—
75C2×D83C25116yes—
76C4×C83C2518yes—
77Dic83C25116yes—
78C2×C83—216no—
79C2×C83—24no—
80C2×C83C2×H(5,𝔽2)16yescomplemented
81C2×C83C2611yescharacteristic
82D83—216no—
83C22×H(5,𝔽2)—831nomaximal, Sylow
84C83C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
85C2×H(5,𝔽2)—8348no—
86C22×#41—834no—
87C22×#41—832no—
88C23×D4—834no—
89C23×D4—834no—
90C23×D4D8311yescharacteristic, complemented
91C25—834no—
92C25C2×D8312yescomplemented
93H(5,𝔽2)—83256no—
94C2×#41—8396no—
95C2×#41—8348no—
96C22×Q8—832no—
97C22×D4—838no—
98C22×D4—834no—
99C22×D4—8396no—
100C22×D4—8396no—
101C22×D4—832no—
102C22×D4C2×D8314yescomplemented
103C22×D4C2×D83124yescomplemented
104C23×C4—834no—
105C23×C4—834no—
106C23×C4C2×D8311yescharacteristic, complemented
107C24—48no—
108C24—33216no—
109C24—8348no—
110C24—834no—
111C24C22×D8314yes—
112C24C22×D8311yescharacteristic
113C24C22×D83124yescomplemented
114#41—83256no—
115#41—83128no—
116C2×Q8—8324no—
117C2×D4—8396no—
118C2×D4—8348no—
119C2×D4—83256no—
120C2×D4—83256no—
121C2×D4—8324no—
122C2×D4C22×D83148yescomplemented
123C2×D4C22×D83164yescomplemented
124C22×C4—8348no—
125C22×C4—834no—
126C22×C4—8348no—
127C22×C4C22×D8312yes—
128C22×C4C22×D83112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—33296no—
133C23—8364no—
134C23—8324no—
135C23—16616no—
136C23C23×D83124yes—
137C23C23×D8316yes—
138C23C23×D8311yescharacteristic
139C23C23×D83132yescomplemented
140Q8—8332no—
141D4—83128no—
142D4—8364no—
143D4—8332no—
144D4C23×D83164yescomplemented
145C2×C4—8324no—
146C2×C4—8364no—
147C2×C4—8364no—
148C2×C4C23×D83112yes—
149C2×C4C23×D83116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—33264no—
154C22—8316no—
155C22—16648no—
156C22#1458183714yesdirect product factor
157C22C24×D83116yes—
158C22C24×D8314yes—
159C22C24×D8313yes—
160C4—8316no—
161C4C24×D8318yes—
162C2—24no—
163C2—216no—
164C2—16616no—
165C2C2×#1458183716yesdirect product factor
166C2C25×D8311yescharacteristic
subgroupG/H
centerC23C23×D83
commutator, cosocleC2×C83C26
FrattiniC2C25×D83
FittingC23×D4×C83C2
socleC23×C83C24
hypercenterC23×D4D83
hypocenterC83C22×H(5,𝔽2)
LCSG ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581837C2direct (N × H)
2. C22×#13050590C2semidirect
3. C22×#13050589C2semidirect
4. C23×C22⋊D83C2semidirect
5. C22×D4×D83C2semidirect
6. C23×D4×C83C2semidirect
7. C24×D83C22semidirect
8. #14581837C22direct (N × H)
9. C2×#13050590C22semidirect
10. C2×#13050589C22semidirect
11. C22×Dic166C22semidirect
12. C22×D332C22semidirect
13. C22×C22⋊D83C22semidirect
14. C22×C22⋊D83C22semidirect
15. C22×C22⋊D83C22semidirect
16. C2×D4×D83C22semidirect
17. C23×Dic83C22semidirect
18. C22×C4×D83C22semidirect
19. C25×C83C22semidirect
20. C22×D4×C83C22semidirect
21. C22×D4×C83C22semidirect
22. C23×C4×C83C22semidirect
25. C23×D83C23semidirect
26. #13050590C23semidirect
27. #13050589C23semidirect
28. C2×Dic166C23semidirect
29. C2×D332C23semidirect
30. C2×C22⋊D83C23semidirect
31. C2×C22⋊D83C23semidirect
32. C2×C22⋊D83C23semidirect
33. D4×D83C23semidirect
35. C22×Dic83C23semidirect
36. C2×C4×D83C23semidirect
40. C24×C83C23semidirect
41. C2×D4×C83C23semidirect
42. C2×D4×C83C23semidirect
44. C22×C4×C83C23semidirect
51. C23×C83C24semidirect
55. C22×D83C24semidirect
56. D4×C83C24semidirect
58. C2×C4×C83C24semidirect
59. C22⋊D83C24semidirect
60. C22⋊D83C24semidirect
62. C2×Dic83C24semidirect
63. D332C24semidirect
64. C4×D83C24semidirect
65. Dic166C24semidirect
69. C22×C83H(5,𝔽2)semidirect
80. C2×C83C2×H(5,𝔽2)semidirect
84. C83C22×H(5,𝔽2)semidirect
90. C23×D4D83semidirect
92. C25C2×D83semidirect
102. C22×D4C2×D83semidirect
103. C22×D4C2×D83semidirect
106. C23×C4C2×D83semidirect
113. C24C22×D83semidirect
122. C2×D4C22×D83semidirect
123. C2×D4C22×D83semidirect
128. C22×C4C22×D83semidirect
139. C23C23×D83semidirect
144. D4C23×D83semidirect
149. C2×C4C23×D83semidirect
156. C22#14581837direct (N × H)
165. C2C2×#14581837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145818372
C22×#130505902
C22×#130505892
C23×C22⋊D832
C22×D4×D832
C23×D4×C832
C22×H(5,𝔽2)83