Finite Group Databasebeta

← C22×C37×#543 #22683923

Subgroups — C22×C37×#543 #22683923

Order 9472.

9310 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.

normal subgroups only

This group has 9310 conjugacy classes of subgroups — too many to draw. Only its 6366 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×C37×#543C2148yesmaximal, direct product factor
2C22×C22⋊Q8×C37C2112yesmaximal, complemented
3C23×C22⋊C4×C37C213yesmaximal
4C26×C37C2211yescharacteristic
5C23×Q8×C37C2214yescomplemented
6C24×C4×C37C2216yes—
7C37×#543C221256yesdirect product factor
8C2×C22⋊Q8×C37C221288yescomplemented
9C22×C4⋊C4×C37C22112yescomplemented
10C22×C22⋊C4×C37C22112yes—
11C22×C22⋊C4×C37C22172yes—
12C25×C37—48no—
13C25×C37Q8116yescomplemented
14C25×C37C2313yes—
15C25×C37C23112yes—
16C22×Q8×C37—416no—
17C22×Q8×C37C23148yescomplemented
18C23×C4×C37—424no—
19C23×C4×C37C23112yes—
20C23×C4×C37C23172yes—
21C22⋊Q8×C37C231768yescomplemented
22C2×C4⋊C4×C37C231144yescomplemented
23C2×C22⋊C4×C37C231144yes—
24C2×C22⋊C4×C37C231192yes—
25C24×C37—224no—
26C24×C37—464no—
27C24×C37—448no—
28C24×C37C2×Q8124yes—
29C24×C37C2×Q8196yescomplemented
30C24×C37C24118yes—
31C24×C37C24116yes—
32C24×C37C2411yescharacteristic
33C2×Q8×C37—496no—
34C2×Q8×C37C24164yescomplemented
35C22×C4×C37—248no—
36C22×C4×C37—4144no—
37C22×C4×C37C24172yes—
38C22×C4×C37C24196yes—
39C4⋊C4×C37C241192yescomplemented
40C22⋊C4×C37C241192yes—
41C23×C37—296no—
42C23×C37—272no—
43C23×C37—4192no—
44C23×C37—432no—
45C23×C37C22×Q8172yes—
46C23×C37C22×Q814yes—
47C23×C37C22×Q8164yescomplemented
48C23×C37H(5,𝔽2)18yes—
49C23×C37C25112yes—
50C23×C37C2513yes—
51Q8×C37—464no—
52C2×C4×C37—2144no—
53C2×C4×C37—496no—
54C2×C4×C37C25148yes—
55C22×#543C3711yesmaximal, characteristic, direct product factor, Sylow
56C22×C37—224no—
57C22×C37—2144no—
58C22×C37—464no—
59C22×C37#543116yesdirect product factor
60C22×C37C23×Q8124yes—
61C22×C37C23×Q816yes—
62C22×C37C2×H(5,𝔽2)112yes—
63C22×C37C2611yescharacteristic
64C4×C37—248no—
65C2×#543C2×C37148yesdirect product factor
66C22×C22⋊Q8C2×C37112yescomplemented
67C23×C22⋊C4C2×C3713yes—
68C2×C37—224no—
69C2×C37C2×#543112yesdirect product factor
70C2×C37C24×Q811yescharacteristic
71C2×C37C22×H(5,𝔽2)12yes—
72C26C22×C3711yescharacteristic
73C23×Q8C22×C3714yescomplemented
74C24×C4C22×C3716yes—
75#543C22×C371256yesdirect product factor
76C2×C22⋊Q8C22×C371288yescomplemented
77C22×C4⋊C4C22×C37112yescomplemented
78C22×C22⋊C4C22×C37112yes—
79C22×C22⋊C4C22×C37172yes—
80C37C22×#54311yescharacteristic, direct product factor, Sylow
81C25—48no—
82C25Q8×C37116yescomplemented
83C25C23×C3713yes—
84C25C23×C37112yes—
85C22×Q8—416no—
86C22×Q8C23×C37148yescomplemented
87C23×C4—424no—
88C23×C4C23×C37112yes—
89C23×C4C23×C37172yes—
90C22⋊Q8C23×C371768yescomplemented
91C2×C4⋊C4C23×C371144yescomplemented
92C2×C22⋊C4C23×C371144yes—
93C2×C22⋊C4C23×C371192yes—
94C24—224no—
95C24—464no—
96C24—448no—
97C24C2×Q8×C37124yes—
98C24C2×Q8×C37196yescomplemented
99C24C24×C37118yes—
100C24C24×C37116yes—
101C24C24×C3711yescharacteristic
102C2×Q8—496no—
103C2×Q8C24×C37164yescomplemented
104C22×C4—248no—
105C22×C4—4144no—
106C22×C4C24×C37172yes—
107C22×C4C24×C37196yes—
108C4⋊C4C24×C371192yescomplemented
109C22⋊C4C24×C371192yes—
110C23—296no—
111C23—272no—
112C23—4192no—
113C23—432no—
114C23C22×Q8×C37172yes—
115C23C22×Q8×C3714yes—
116C23C22×Q8×C37164yescomplemented
117C23H(5,𝔽2)×C3718yes—
118C23C25×C37112yes—
119C23C25×C3713yes—
120Q8—464no—
121C2×C4—2144no—
122C2×C4—496no—
123C2×C4C25×C37148yes—
124C22—2144no—
125C22—224no—
126C22—464no—
127C22C37×#543116yesdirect product factor
128C22C23×Q8×C37124yes—
129C22C23×Q8×C3716yes—
130C22C2×H(5,𝔽2)×C37112yes—
131C22C26×C3711yescharacteristic
132C4—248no—
133C2—224no—
134C2C2×C37×#543112yesdirect product factor
135C2C24×Q8×C3711yescharacteristic
136C2C22×H(5,𝔽2)×C3712yes—
subgroupG/H
center, socleC24×C37C24
commutator, Frattini, cosocleC22C26×C37
prime radicalC26×C37C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C37×#543C2direct (N × H)
2. C22×C22⋊Q8×C37C2semidirect
5. C23×Q8×C37C22semidirect
7. C37×#543C22direct (N × H)
8. C2×C22⋊Q8×C37C22semidirect
9. C22×C4⋊C4×C37C22semidirect
13. C25×C37Q8semidirect
17. C22×Q8×C37C23semidirect
21. C22⋊Q8×C37C23semidirect
22. C2×C4⋊C4×C37C23semidirect
29. C24×C37C2×Q8semidirect
34. C2×Q8×C37C24semidirect
39. C4⋊C4×C37C24semidirect
47. C23×C37C22×Q8semidirect
55. C22×#543C37direct (N × H)
59. C22×C37#543direct (N × H)
65. C2×#543C2×C37direct (N × H)
66. C22×C22⋊Q8C2×C37semidirect
69. C2×C37C2×#543direct (N × H)
73. C23×Q8C22×C37semidirect
75. #543C22×C37direct (N × H)
76. C2×C22⋊Q8C22×C37semidirect
77. C22×C4⋊C4C22×C37semidirect
80. C37C22×#543direct (N × H)
82. C25Q8×C37semidirect
86. C22×Q8C23×C37semidirect
90. C22⋊Q8C23×C37semidirect
91. C2×C4⋊C4C23×C37semidirect
98. C24C2×Q8×C37semidirect
103. C2×Q8C24×C37semidirect
108. C4⋊C4C24×C37semidirect
116. C23C22×Q8×C37semidirect
127. C22C37×#543direct (N × H)
134. C2C2×C37×#543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#5432
C22×C22⋊Q8×C372
C23×C22⋊C4×C372
C22×#54337