Finite Group Databasebeta

← C22×D3×C7×Q8 #13062355

Subgroups — C22×D3×C7×Q8 #13062355

Order 1344.

1700 conjugacy classes of subgroups; only the 1006 normal subgroups are shown.

normal subgroups only

This group has 1700 conjugacy classes of subgroups — too many to draw. Only its 1006 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×D3×C7×Q8C2124yesmaximal, direct product factor
3C22×C4×D3×C7C213yesmaximal
4C22×C7×Dic6C213yesmaximal, complemented
5C23×C7×Q8—31nomaximal, Hall
6C23×D3×C7C2211yescharacteristic
7C2×C3×C7×Q8C22112yescomplemented
8C22×C3×C4×C7C2213yes—
9C22×C7×Dic3C2213yes—
10D3×C7×Q8C22164yesdirect product factor
11C2×C4×D3×C7C22136yes—
12C2×C7×Dic6C22136yescomplemented
13C22×C7×Q8—324no—
14C22×C7×Q8—33no—
15C22×C7×Q8D311yescharacteristic, direct product factor
16C23×C4×C7—33no—
17C22×D3×Q8C711yesmaximal, characteristic, direct product factor, Hall
18C23×C3×C7C2311yescharacteristic
19C22×D3×C7Q818yesdirect product factor
20C22×D3×C7C2316yes—
21C3×C7×Q8C23116yescomplemented
22C2×C3×C4×C7C23118yes—
23C2×C7×Dic3C23118yes—
24C4×D3×C7C23148yes—
25C7×Dic6C23148yescomplemented
26C24×C7—31no—
27C2×C7×Q8—364no—
28C2×C7×Q8—336no—
29C2×C7×Q8C2×D3112yesdirect product factor
30C22×C4×C7—33no—
31C22×C4×C7—336no—
32C22×C4×C7C2×D313yes—
33C22×C3×Q8C2×C711yescharacteristic, complemented
34C2×D3×Q8C2×C7124yesdirect product factor
35C22×C4×D3C2×C713yes—
36C22×Dic6C2×C713yescomplemented
37C22×C3×C7C2×Q814yescomplemented
38C22×C3×C7C2413yes—
39C2×D3×C7C2×Q8124yesdirect product factor
40C2×D3×C7C2414yes—
41C3×C4×C7C24112yes—
42C7×Dic3C24112yes—
43C23×Q8—31noSylow
44C23×C7—38no—
45C23×C7—36no—
46C23×C7C22×D311yescharacteristic
47C7×Q8—348no—
48C7×Q8C22×D3116yesdirect product factor
49C2×C4×C7—318no—
50C2×C4×C7—348no—
51C2×C4×C7C22×D3118yes—
52C23×D3C22×C711yescharacteristic
53C2×C3×Q8C22×C7112yescomplemented
54C22×C3×C4C22×C713yes—
55C22×Dic3C22×C713yes—
56D3×Q8C22×C7164yesdirect product factor
57C2×C4×D3C22×C7136yes—
58C2×Dic6C22×C7136yescomplemented
59C2×C3×C7C22×Q816yescomplemented
60C2×C3×C7C2511yescharacteristic
61D3×C7C22×Q818yesdirect product factor
62C22×Q8—324no—
63C22×Q8—33no—
64C22×Q8D3×C711yescharacteristic, direct product factor
65C23×C4—33no—
66C22×C7—324no—
67C22×C7—34no—
68C22×C7D3×Q814yesdirect product factor
69C22×C7C23×D313yes—
70C4×C7—312no—
71C4×C7C23×D3112yes—
72C23×C3C23×C711yescharacteristic
73C22×D3C7×Q818yesdirect product factor
74C22×D3C23×C716yes—
75C3×Q8C23×C7116yescomplemented
76C2×C3×C4C23×C7118yes—
77C2×Dic3C23×C7118yes—
78C4×D3C23×C7148yes—
79Dic6C23×C7148yescomplemented
80C3×C7C23×Q811yescharacteristic, complemented, Hall
81C24—31no—
82C2×Q8—364no—
83C2×Q8—336no—
84C2×Q8C2×D3×C7112yesdirect product factor
85C22×C4—33no—
86C22×C4—336no—
87C22×C4C2×D3×C713yes—
88C2×C7—38no—
89C2×C7C2×D3×Q816yesdirect product factor
90C2×C7C24×D311yescharacteristic
91C22×C3C2×C7×Q814yescomplemented
92C22×C3C24×C713yes—
93C2×D3C2×C7×Q8124yesdirect product factor
94C2×D3C24×C714yes—
95C3×C4C24×C7112yes—
96Dic3C24×C7112yes—
97C23—38no—
98C23—36no—
99C23C22×D3×C711yescharacteristic
100Q8—348no—
101Q8C22×D3×C7116yesdirect product factor
102C2×C4—318no—
103C2×C4—348no—
104C2×C4C22×D3×C7118yes—
105C7C22×D3×Q811yescharacteristic, direct product factor, Sylow
106C2×C3C22×C7×Q816yescomplemented
107C2×C3C25×C711yescharacteristic
108D3C22×C7×Q818yesdirect product factor
109C22—34no—
110C22—324no—
111C22D3×C7×Q814yesdirect product factor
112C22C23×D3×C713yes—
113C4—312no—
114C4C23×D3×C7112yes—
115C3C23×C7×Q811yescharacteristic, complemented, Sylow
116C2—38no—
117C2C2×D3×C7×Q816yesdirect product factor
118C2C24×D3×C711yescharacteristic
subgroupG/H
centerC23×C7C22×D3
commutator, cosocleC2×C3C25×C7
FrattiniC2C24×D3×C7
FittingC22×C3×C7×Q8C2
socleC23×C3×C7C23
prime radicalC23×D3×C7C22
hypercenterC22×C7×Q8D3
hypocenterC3C23×C7×Q8
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×D3×C7×Q8C2direct (N × H)
4. C22×C7×Dic6C2semidirect
7. C2×C3×C7×Q8C22semidirect
10. D3×C7×Q8C22direct (N × H)
12. C2×C7×Dic6C22semidirect
15. C22×C7×Q8D3direct (N × H)
17. C22×D3×Q8C7direct (N × H)
19. C22×D3×C7Q8direct (N × H)
21. C3×C7×Q8C23semidirect
25. C7×Dic6C23semidirect
29. C2×C7×Q8C2×D3direct (N × H)
33. C22×C3×Q8C2×C7semidirect
34. C2×D3×Q8C2×C7direct (N × H)
36. C22×Dic6C2×C7semidirect
37. C22×C3×C7C2×Q8semidirect
39. C2×D3×C7C2×Q8direct (N × H)
48. C7×Q8C22×D3direct (N × H)
53. C2×C3×Q8C22×C7semidirect
56. D3×Q8C22×C7direct (N × H)
58. C2×Dic6C22×C7semidirect
59. C2×C3×C7C22×Q8semidirect
61. D3×C7C22×Q8direct (N × H)
64. C22×Q8D3×C7direct (N × H)
68. C22×C7D3×Q8direct (N × H)
73. C22×D3C7×Q8direct (N × H)
75. C3×Q8C23×C7semidirect
79. Dic6C23×C7semidirect
80. C3×C7C23×Q8semidirect
84. C2×Q8C2×D3×C7direct (N × H)
89. C2×C7C2×D3×Q8direct (N × H)
91. C22×C3C2×C7×Q8semidirect
93. C2×D3C2×C7×Q8direct (N × H)
101. Q8C22×D3×C7direct (N × H)
105. C7C22×D3×Q8direct (N × H)
106. C2×C3C22×C7×Q8semidirect
108. D3C22×C7×Q8direct (N × H)
111. C22D3×C7×Q8direct (N × H)
115. C3C23×C7×Q8semidirect
117. C2C2×D3×C7×Q8direct (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×D3×C7×Q82
C22×C4×D3×C72
C22×C7×Dic62
C23×C7×Q83
C22×D3×Q87