Finite Group Databasebeta

← C22×C31×#543 #21519429

Subgroups — C22×C31×#543 #21519429

Order 7936.

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×C31×#543C2148yesmaximal, direct product factor
2C22×C31×C22⋊Q8C2112yesmaximal, complemented
3C23×C22⋊C4×C31C213yesmaximal
4C26×C31C2211yescharacteristic
5C23×Q8×C31C2214yescomplemented
6C24×C4×C31C2216yes—
7C31×#543C221256yesdirect product factor
8C2×C31×C22⋊Q8C221288yescomplemented
9C22×C4⋊C4×C31C22112yescomplemented
10C22×C22⋊C4×C31C22112yes—
11C22×C22⋊C4×C31C22172yes—
12C25×C31—48no—
13C25×C31Q8116yescomplemented
14C25×C31C2313yes—
15C25×C31C23112yes—
16C22×Q8×C31—416no—
17C22×Q8×C31C23148yescomplemented
18C23×C4×C31—424no—
19C23×C4×C31C23112yes—
20C23×C4×C31C23172yes—
21C31×C22⋊Q8C231768yescomplemented
22C2×C4⋊C4×C31C231144yescomplemented
23C2×C22⋊C4×C31C231144yes—
24C2×C22⋊C4×C31C231192yes—
25C24×C31—224no—
26C24×C31—464no—
27C24×C31—448no—
28C24×C31C2×Q8124yes—
29C24×C31C2×Q8196yescomplemented
30C24×C31C24118yes—
31C24×C31C24116yes—
32C24×C31C2411yescharacteristic
33C2×Q8×C31—496no—
34C2×Q8×C31C24164yescomplemented
35C22×C4×C31—248no—
36C22×C4×C31—4144no—
37C22×C4×C31C24172yes—
38C22×C4×C31C24196yes—
39C4⋊C4×C31C241192yescomplemented
40C22⋊C4×C31C241192yes—
41C22×#543C3111yesmaximal, characteristic, direct product factor, Sylow
42C23×C31—296no—
43C23×C31—272no—
44C23×C31—4192no—
45C23×C31—432no—
46C23×C31C22×Q8172yes—
47C23×C31C22×Q814yes—
48C23×C31C22×Q8164yescomplemented
49C23×C31H(5,𝔽2)18yes—
50C23×C31C25112yes—
51C23×C31C2513yes—
52Q8×C31—464no—
53C2×C4×C31—2144no—
54C2×C4×C31—496no—
55C2×C4×C31C25148yes—
56C2×#543C2×C31148yesdirect product factor
57C22×C22⋊Q8C2×C31112yescomplemented
58C23×C22⋊C4C2×C3113yes—
59C22×C31—224no—
60C22×C31—2144no—
61C22×C31—464no—
62C22×C31#543116yesdirect product factor
63C22×C31C23×Q8124yes—
64C22×C31C23×Q816yes—
65C22×C31C2×H(5,𝔽2)112yes—
66C22×C31C2611yescharacteristic
67C4×C31—248no—
68C26C22×C3111yescharacteristic
69C23×Q8C22×C3114yescomplemented
70C24×C4C22×C3116yes—
71#543C22×C311256yesdirect product factor
72C2×C22⋊Q8C22×C311288yescomplemented
73C22×C4⋊C4C22×C31112yescomplemented
74C22×C22⋊C4C22×C31112yes—
75C22×C22⋊C4C22×C31172yes—
76C2×C31—224no—
77C2×C31C2×#543112yesdirect product factor
78C2×C31C24×Q811yescharacteristic
79C2×C31C22×H(5,𝔽2)12yes—
80C25—48no—
81C25Q8×C31116yescomplemented
82C25C23×C3113yes—
83C25C23×C31112yes—
84C22×Q8—416no—
85C22×Q8C23×C31148yescomplemented
86C23×C4—424no—
87C23×C4C23×C31112yes—
88C23×C4C23×C31172yes—
89C22⋊Q8C23×C311768yescomplemented
90C2×C4⋊C4C23×C311144yescomplemented
91C2×C22⋊C4C23×C311144yes—
92C2×C22⋊C4C23×C311192yes—
93C31C22×#54311yescharacteristic, direct product factor, Sylow
94C24—224no—
95C24—464no—
96C24—448no—
97C24C2×Q8×C31124yes—
98C24C2×Q8×C31196yescomplemented
99C24C24×C31118yes—
100C24C24×C31116yes—
101C24C24×C3111yescharacteristic
102C2×Q8—496no—
103C2×Q8C24×C31164yescomplemented
104C22×C4—248no—
105C22×C4—4144no—
106C22×C4C24×C31172yes—
107C22×C4C24×C31196yes—
108C4⋊C4C24×C311192yescomplemented
109C22⋊C4C24×C311192yes—
110C23—296no—
111C23—272no—
112C23—4192no—
113C23—432no—
114C23C22×Q8×C31172yes—
115C23C22×Q8×C3114yes—
116C23C22×Q8×C31164yescomplemented
117C23C31×H(5,𝔽2)18yes—
118C23C25×C31112yes—
119C23C25×C3113yes—
120Q8—464no—
121C2×C4—2144no—
122C2×C4—496no—
123C2×C4C25×C31148yes—
124C22—224no—
125C22—2144no—
126C22—464no—
127C22C31×#543116yesdirect product factor
128C22C23×Q8×C31124yes—
129C22C23×Q8×C3116yes—
130C22C2×C31×H(5,𝔽2)112yes—
131C22C26×C3111yescharacteristic
132C4—248no—
133C2—224no—
134C2C2×C31×#543112yesdirect product factor
135C2C24×Q8×C3111yescharacteristic
136C2C22×C31×H(5,𝔽2)12yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC22C26×C31
prime radicalC26×C31C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#543C2direct (N × H)
2. C22×C31×C22⋊Q8C2semidirect
5. C23×Q8×C31C22semidirect
7. C31×#543C22direct (N × H)
8. C2×C31×C22⋊Q8C22semidirect
9. C22×C4⋊C4×C31C22semidirect
13. C25×C31Q8semidirect
17. C22×Q8×C31C23semidirect
21. C31×C22⋊Q8C23semidirect
22. C2×C4⋊C4×C31C23semidirect
29. C24×C31C2×Q8semidirect
34. C2×Q8×C31C24semidirect
39. C4⋊C4×C31C24semidirect
41. C22×#543C31direct (N × H)
48. C23×C31C22×Q8semidirect
56. C2×#543C2×C31direct (N × H)
57. C22×C22⋊Q8C2×C31semidirect
62. C22×C31#543direct (N × H)
69. C23×Q8C22×C31semidirect
71. #543C22×C31direct (N × H)
72. C2×C22⋊Q8C22×C31semidirect
73. C22×C4⋊C4C22×C31semidirect
77. C2×C31C2×#543direct (N × H)
81. C25Q8×C31semidirect
85. C22×Q8C23×C31semidirect
89. C22⋊Q8C23×C31semidirect
90. C2×C4⋊C4C23×C31semidirect
93. C31C22×#543direct (N × H)
98. C24C2×Q8×C31semidirect
103. C2×Q8C24×C31semidirect
108. C4⋊C4C24×C31semidirect
116. C23C22×Q8×C31semidirect
127. C22C31×#543direct (N × H)
134. C2C2×C31×#543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5432
C22×C31×C22⋊Q82
C23×C22⋊C4×C312
C22×#54331