Finite Group Databasebeta

← C22×#15668058 #26035054

Subgroups — C22×#15668058 #26035054

Order 11392.

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×#15668058C2148yesmaximal, direct product factor
2C22×#13083575C212yesmaximal, complemented
3C22×#13083574C213yesmaximal, complemented
4C23×D356C213yesmaximal, complemented
5C22×D4×D89C216yesmaximal, complemented
6C22×#41×C89C211yesmaximal, characteristic, complemented
7C24×D89C2216yescomplemented
8#15668058C221256yesdirect product factor
9C2×#13083575C22148yescomplemented
10C2×#13083574C22172yescomplemented
11C22×Dic178C2211yescharacteristic, complemented
12C22×D356C2216yescomplemented
13C22×D356C2213yescomplemented
14C22×D356C22172yescomplemented
15C22×C22⋊D89C2216yescomplemented
16C2×D4×D89C221144yescomplemented
17C22×C4×D89C2216yescomplemented
18C2×#41×C89C22124yescomplemented
19C22×Q8×C89C2211yescharacteristic, complemented
20C22×D4×C89C2213yescomplemented
21C23×C4×C89C2213yescomplemented
22C23×D89—424no—
23C23×D89C2316yes—
24C23×D89C23172yescomplemented
25#13083575C231128yescomplemented
26#13083574C231192yescomplemented
27C2×Dic178C23112yescomplemented
28C2×D356C23172yescomplemented
29C2×D356C23136yescomplemented
30C2×D356C231192yescomplemented
31C2×C22⋊D89C23172yescomplemented
32D4×D89C231384yescomplemented
33C22×Dic89C2312yes—
34C2×C4×D89C23172yescomplemented
35C24×C89C2313yes—
36#41×C89C23164yescomplemented
37C2×Q8×C89C23112yescomplemented
38C2×D4×C89C23136yescomplemented
39C22×C4×C89C2313yes—
40C22×C4×C89C2311yescharacteristic
41C22×C4×C89C23136yescomplemented
42C23×C89—212no—
43C23×C89C24118yes—
44C23×C89C2411yescharacteristic
45C22×D89—224no—
46C22×D89—4144no—
47C22×D89C24136yes—
48C22×D89C24196yescomplemented
49Q8×C89C24116yescomplemented
50D4×C89C24148yescomplemented
51C2×C4×C89C2416yes—
52C2×C4×C89C24118yes—
53C2×C4×C89C24148yescomplemented
54C22⋊D89C24196yescomplemented
55C2×Dic89C24112yes—
56D356C24196yescomplemented
57D356C24148yescomplemented
58C4×D89C24196yescomplemented
59Dic178C24116yescomplemented
60C22×C89—236no—
61C22×C89H(5,𝔽2)14yescomplemented
62C22×C89C25112yes—
63C22×C89C2513yes—
64C2×D89—272no—
65C2×D89—496no—
66C2×D89C25124yes—
67C4×C89C2514yes—
68C4×C89C25112yes—
69Dic89C2518yes—
70C2×C89—212no—
71C2×C89C2×H(5,𝔽2)16yescomplemented
72C2×C89C2611yescharacteristic
73D89—224no—
74C22×H(5,𝔽2)—891nomaximal, Sylow
75C89C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
76C2×H(5,𝔽2)—8948no—
77C22×#41—892no—
78C22×#41—893no—
79C22×#41D8911yescharacteristic, complemented
80C23×D4—896no—
81C23×D4—893no—
82C25—896no—
83H(5,𝔽2)—89256no—
84C2×#41—8948no—
85C2×#41—8972no—
86C2×#41C2×D89124yescomplemented
87C22×Q8—891no—
88C22×Q8C2×D8911yescharacteristic, complemented
89C22×D4—896no—
90C22×D4—89144no—
91C22×D4—896no—
92C22×D4—893no—
93C22×D4—8972no—
94C22×D4C2×D8913yescomplemented
95C23×C4—896no—
96C23×C4C2×D8913yescomplemented
97C24—8972no—
98C24—896no—
99C24—35624no—
100C24C22×D8913yes—
101#41—89128no—
102#41—89192no—
103#41C22×D89164yescomplemented
104C2×Q8—8912no—
105C2×Q8C22×D89112yescomplemented
106C2×D4—8972no—
107C2×D4—89384no—
108C2×D4—8972no—
109C2×D4—8936no—
110C2×D4—89192no—
111C2×D4C22×D89136yescomplemented
112C22×C4—892no—
113C22×C4—8972no—
114C22×C4C22×D8913yes—
115C22×C4C22×D8911yescharacteristic
116C22×C4C22×D89136yescomplemented
117C23—212no—
118C23—8996no—
119C23—8936no—
120C23—356144no—
121C23—17824no—
122C23C23×D89118yes—
123C23C23×D8911yescharacteristic
124Q8—8916no—
125Q8C23×D89116yescomplemented
126D4—8996no—
127D4—8996no—
128D4—8948no—
129D4C23×D89148yescomplemented
130C2×C4—8912no—
131C2×C4—8996no—
132C2×C4C23×D89118yes—
133C2×C4C23×D8916yes—
134C2×C4C23×D89148yescomplemented
135C22—236no—
136C22—8924no—
137C22—35696no—
138C22—17872no—
139C22#1566805814yesdirect product factor
140C22C24×D89112yes—
141C22C24×D8913yes—
142C4—898no—
143C4C24×D89112yes—
144C4C24×D8914yes—
145C2—212no—
146C2—17824no—
147C2C2×#1566805816yesdirect product factor
148C2C25×D8911yescharacteristic
subgroupG/H
centerC23C23×D89
commutator, cosocleC2×C89C26
FrattiniC2C25×D89
FittingC22×#41×C89C2
socleC23×C89C24
hypercenterC22×#41D89
hypocenterC89C22×H(5,𝔽2)
LCSG ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15668058C2direct (N × H)
2. C22×#13083575C2semidirect
3. C22×#13083574C2semidirect
4. C23×D356C2semidirect
5. C22×D4×D89C2semidirect
6. C22×#41×C89C2semidirect
7. C24×D89C22semidirect
8. #15668058C22direct (N × H)
9. C2×#13083575C22semidirect
10. C2×#13083574C22semidirect
11. C22×Dic178C22semidirect
12. C22×D356C22semidirect
13. C22×D356C22semidirect
14. C22×D356C22semidirect
15. C22×C22⋊D89C22semidirect
16. C2×D4×D89C22semidirect
17. C22×C4×D89C22semidirect
18. C2×#41×C89C22semidirect
19. C22×Q8×C89C22semidirect
20. C22×D4×C89C22semidirect
21. C23×C4×C89C22semidirect
24. C23×D89C23semidirect
25. #13083575C23semidirect
26. #13083574C23semidirect
27. C2×Dic178C23semidirect
28. C2×D356C23semidirect
29. C2×D356C23semidirect
30. C2×D356C23semidirect
31. C2×C22⋊D89C23semidirect
32. D4×D89C23semidirect
34. C2×C4×D89C23semidirect
36. #41×C89C23semidirect
37. C2×Q8×C89C23semidirect
38. C2×D4×C89C23semidirect
41. C22×C4×C89C23semidirect
48. C22×D89C24semidirect
49. Q8×C89C24semidirect
50. D4×C89C24semidirect
53. C2×C4×C89C24semidirect
54. C22⋊D89C24semidirect
56. D356C24semidirect
57. D356C24semidirect
58. C4×D89C24semidirect
59. Dic178C24semidirect
61. C22×C89H(5,𝔽2)semidirect
71. C2×C89C2×H(5,𝔽2)semidirect
75. C89C22×H(5,𝔽2)semidirect
79. C22×#41D89semidirect
86. C2×#41C2×D89semidirect
88. C22×Q8C2×D89semidirect
94. C22×D4C2×D89semidirect
96. C23×C4C2×D89semidirect
103. #41C22×D89semidirect
105. C2×Q8C22×D89semidirect
111. C2×D4C22×D89semidirect
116. C22×C4C22×D89semidirect
125. Q8C23×D89semidirect
129. D4C23×D89semidirect
134. C2×C4C23×D89semidirect
139. C22#15668058direct (N × H)
147. C2C2×#15668058direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156680582
C22×#130835752
C22×#130835742
C23×D3562
C22×D4×D892
C22×#41×C892
C22×H(5,𝔽2)89