Finite Group Databasebeta

← C22×C7×#239 #13062356

Subgroups — C22×C7×#239 #13062356

Order 1344.

1780 conjugacy classes of subgroups; only the 926 normal subgroups are shown.

normal subgroups only

This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 926 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
1C22×C3×C7×Q8C211yesmaximal, characteristic, complemented
2C2×C7×#239C2124yesmaximal, direct product factor
3C22×C7×D12C213yesmaximal, complemented
4C22×C4×D3×C7C213yesmaximal, complemented
5C22×C7×#41—31nomaximal, Hall
6C23×D3×C7C2213yes—
7C2×C3×C7×Q8C22112yescomplemented
8C22×C3×C4×C7C2213yes—
9C22×C7×Dic3C2211yescharacteristic
10C7×#239C22164yesdirect product factor
11C2×C7×D12C22136yescomplemented
12C2×C4×D3×C7C22136yescomplemented
13C2×C7×#41—324no—
14C22×C7×Q8D311yescharacteristic, complemented
15C22×C7×D4—33no—
16C23×C4×C7—33no—
17C22×#239C711yesmaximal, characteristic, direct product factor, Hall
18C23×C3×C7C2311yescharacteristic
19C22×D3×C7—212no—
20C22×D3×C7C23118yes—
21C3×C7×Q8C23116yescomplemented
22C2×C3×C4×C7C23118yes—
23C2×C7×Dic3C2316yes—
24C7×D12C23148yescomplemented
25C4×D3×C7C23148yescomplemented
26C24×C7—33no—
27C7×#41—364no—
28C2×C7×Q8C2×D3112yescomplemented
29C2×C7×D4—336no—
30C22×C4×C7—31no—
31C22×C4×C7—336no—
32C22×C4×C7C2×D313yes—
33C22×C3×Q8C2×C711yescharacteristic, complemented
34C2×#239C2×C7124yesdirect product factor
35C22×D12C2×C713yescomplemented
36C22×C4×D3C2×C713yescomplemented
37C22×C3×C7#4114yescomplemented
38C22×C3×C7C2413yes—
39C2×D3×C7—236no—
40C2×D3×C7C24112yes—
41C3×C4×C7C24112yes—
42C7×Dic3C2414yes—
43C22×#41—31noSylow
44C23×C7—318no—
45C23×C7—612no—
46C23×C7C22×D311yescharacteristic
47C7×Q8C22×D3116yescomplemented
48C7×D4—348no—
49C2×C4×C7—36no—
50C2×C4×C7—348no—
51C2×C4×C7C22×D3118yes—
52C23×D3C22×C713yes—
53C2×C3×Q8C22×C7112yescomplemented
54C22×C3×C4C22×C713yes—
55C22×Dic3C22×C711yescharacteristic
56#239C22×C7164yesdirect product factor
57C2×D12C22×C7136yescomplemented
58C2×C4×D3C22×C7136yescomplemented
59C2×C3×C7C2×#4116yescomplemented
60C2×C3×C7C2511yescharacteristic
61D3×C7—212no—
62C2×#41—324no—
63C22×Q8D3×C711yescharacteristic, complemented
64C22×D4—33no—
65C23×C4—33no—
66C22×C7—312no—
67C22×C7—636no—
68C22×C7#23914yesdirect product factor
69C22×C7C23×D313yes—
70C4×C7—34no—
71C4×C7C23×D3112yes—
72C23×C3C23×C711yescharacteristic
73C22×D3—212no—
74C22×D3C23×C7118yes—
75C3×Q8C23×C7116yescomplemented
76C2×C3×C4C23×C7118yes—
77C2×Dic3C23×C716yes—
78D12C23×C7148yescomplemented
79C4×D3C23×C7148yescomplemented
80C3×C7C22×#4111yescharacteristic, complemented, Hall
81C24—33no—
82#41—364no—
83C2×Q8C2×D3×C7112yescomplemented
84C2×D4—336no—
85C22×C4—31no—
86C22×C4—336no—
87C22×C4C2×D3×C713yes—
88C2×C7—612no—
89C2×C7C2×#23916yesdirect product factor
90C2×C7C24×D311yescharacteristic
91C22×C3C7×#4114yescomplemented
92C22×C3C24×C713yes—
93C2×D3—236no—
94C2×D3C24×C7112yes—
95C3×C4C24×C7112yes—
96Dic3C24×C714yes—
97C23—318no—
98C23—612no—
99C23C22×D3×C711yescharacteristic
100Q8C22×D3×C7116yescomplemented
101D4—348no—
102C2×C4—36no—
103C2×C4—348no—
104C2×C4C22×D3×C7118yes—
105C7C22×#23911yescharacteristic, direct product factor, Sylow
106C2×C3C2×C7×#4116yescomplemented
107C2×C3C25×C711yescharacteristic
108D3—212no—
109C22—312no—
110C22—636no—
111C22C7×#23914yesdirect product factor
112C22C23×D3×C713yes—
113C4—34no—
114C4C23×D3×C7112yes—
115C3C22×C7×#4111yescharacteristic, complemented, Sylow
116C2—612no—
117C2C2×C7×#23916yesdirect product factor
118C2C24×D3×C711yescharacteristic
subgroupG/H
centerC23×C7C22×D3
commutator, cosocleC2×C3C25×C7
FrattiniC2C24×D3×C7
FittingC22×C3×C7×Q8C2
socleC23×C3×C7C23
hypercenterC22×C7×Q8D3
hypocenterC3C22×C7×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C7 ◁ C22×C7×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×Q8C2semidirect
2. C2×C7×#239C2direct (N × H)
3. C22×C7×D12C2semidirect
4. C22×C4×D3×C7C2semidirect
7. C2×C3×C7×Q8C22semidirect
10. C7×#239C22direct (N × H)
11. C2×C7×D12C22semidirect
12. C2×C4×D3×C7C22semidirect
14. C22×C7×Q8D3semidirect
17. C22×#239C7direct (N × H)
21. C3×C7×Q8C23semidirect
24. C7×D12C23semidirect
25. C4×D3×C7C23semidirect
28. C2×C7×Q8C2×D3semidirect
33. C22×C3×Q8C2×C7semidirect
34. C2×#239C2×C7direct (N × H)
35. C22×D12C2×C7semidirect
36. C22×C4×D3C2×C7semidirect
37. C22×C3×C7#41semidirect
47. C7×Q8C22×D3semidirect
53. C2×C3×Q8C22×C7semidirect
56. #239C22×C7direct (N × H)
57. C2×D12C22×C7semidirect
58. C2×C4×D3C22×C7semidirect
59. C2×C3×C7C2×#41semidirect
63. C22×Q8D3×C7semidirect
68. C22×C7#239direct (N × H)
75. C3×Q8C23×C7semidirect
78. D12C23×C7semidirect
79. C4×D3C23×C7semidirect
80. C3×C7C22×#41semidirect
83. C2×Q8C2×D3×C7semidirect
89. C2×C7C2×#239direct (N × H)
91. C22×C3C7×#41semidirect
100. Q8C22×D3×C7semidirect
105. C7C22×#239direct (N × H)
106. C2×C3C2×C7×#41semidirect
111. C22C7×#239direct (N × H)
115. C3C22×C7×#41semidirect
117. C2C2×C7×#239direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×Q82
C2×C7×#2392
C22×C7×D122
C22×C4×D3×C72
C22×C7×#413
C22×#2397