Finite Group Databasebeta

← C2×C3×C42×D7 #13060813

Subgroups — C2×C3×C42×D7 #13060813

Order 1344.

996 conjugacy classes of subgroups; only the 606 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 606 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×C4×D7C213yesmaximal
2C2×C3×C42×C7C211yesmaximal, characteristic, complemented
3C2×C3×C4×Dic7C213yesmaximal, complemented
4C3×C42×D7C218yesmaximal, direct product factor
5C2×C42×D7C311yesmaximal, characteristic, direct product factor, Hall
6C23×C3×D7C2211yescharacteristic
7C22×C3×C4×C7C2213yes—
8C22×C3×Dic7C2213yes—
9C2×C3×C4×D7C4124yesdirect product factor
10C2×C3×C4×D7C22112yes—
11C3×C42×C7C2214yescomplemented
12C3×C4×Dic7C22112yescomplemented
13C22×C4×D7C2×C313yes—
14C2×C42×C7C2×C311yescharacteristic, complemented
15C2×C4×Dic7C2×C313yescomplemented
16C42×D7C2×C318yesdirect product factor
17C22×C3×C42—71nomaximal, Hall
18C23×C3×C7C2311yescharacteristic
19C22×C3×D7C2×C4112yes—
20C22×C3×D7C2312yes—
21C2×C3×C4×C7C2×C4112yescomplemented
22C2×C3×C4×C7C2316yes—
23C2×C3×Dic7C2×C4112yescomplemented
24C2×C3×Dic7C2316yes—
25C3×C4×D7C2×C4148yesdirect product factor
26C23×D7C22×C311yescharacteristic
27C22×C4×C7C22×C313yes—
28C22×Dic7C22×C313yes—
29C2×C4×D7C3×C4124yesdirect product factor
30C2×C4×D7C22×C3112yes—
31C42×C7C22×C314yescomplemented
32C4×Dic7C22×C3112yescomplemented
33C23×C3×C4—73no—
34C2×C3×C42—73no—
35C2×C3×C42—78no—
36C2×C3×C42D711yescharacteristic, direct product factor
37C22×C3×C7C22×C416yes—
38C22×C3×C7C2411yescharacteristic
39C2×C3×D7C42116yesdirect product factor
40C2×C3×D7C22×C4112yes—
41C3×C4×C7C22×C4112yescomplemented
42C3×Dic7C22×C4112yescomplemented
43C22×C42—71noSylow
44C23×C7C23×C311yescharacteristic
45C22×D7C2×C3×C4112yes—
46C22×D7C23×C312yes—
47C2×C4×C7C2×C3×C4112yescomplemented
48C2×C4×C7C23×C316yes—
49C2×Dic7C2×C3×C4112yescomplemented
50C2×Dic7C23×C316yes—
51C4×D7C2×C3×C4148yesdirect product factor
52C24×C3—71no—
53C22×C3×C4—73no—
54C22×C3×C4—724no—
55C22×C3×C4—712no—
56C22×C3×C4C2×D713yes—
57C3×C42—712no—
58C3×C42C2×D714yesdirect product factor
59C2×C3×C7C2×C4214yescomplemented
60C2×C3×C7C23×C413yes—
61C3×D7C2×C4218yesdirect product factor
62C23×C4—73no—
63C2×C42—73no—
64C2×C42—78no—
65C2×C42C3×D711yescharacteristic, direct product factor
66C22×C7C22×C3×C416yes—
67C22×C7C24×C311yescharacteristic
68C2×D7C3×C42116yesdirect product factor
69C2×D7C22×C3×C4112yes—
70C4×C7C22×C3×C4112yescomplemented
71Dic7C22×C3×C4112yescomplemented
72C23×C3—712no—
73C23×C3—72no—
74C23×C3C22×D711yescharacteristic
75C2×C3×C4—712no—
76C2×C3×C4—76no—
77C2×C3×C4—748no—
78C2×C3×C4C4×D7112yesdirect product factor
79C2×C3×C4C22×D716yes—
80C3×C7C22×C4211yescharacteristic, complemented, Hall
81C24—71no—
82C22×C4—73no—
83C22×C4—724no—
84C22×C4—712no—
85C22×C4C2×C3×D713yes—
86C42—712no—
87C42C2×C3×D714yesdirect product factor
88C2×C7C2×C3×C4214yescomplemented
89C2×C7C23×C3×C413yes—
90D7C2×C3×C4218yesdirect product factor
91C22×C3—716no—
92C22×C3—712no—
93C22×C3C2×C4×D716yes—
94C22×C3C23×D711yescharacteristic
95C3×C4—712no—
96C3×C4C2×C4×D7112yesdirect product factor
97C23—712no—
98C23—72no—
99C23C22×C3×D711yescharacteristic
100C2×C4—712no—
101C2×C4—76no—
102C2×C4—748no—
103C2×C4C3×C4×D7112yesdirect product factor
104C2×C4C22×C3×D716yes—
105C7C22×C3×C4211yescharacteristic, complemented, Sylow
106C2×C3—78no—
107C2×C3C42×D714yesdirect product factor
108C2×C3C22×C4×D713yes—
109C22—712no—
110C22—716no—
111C22C2×C3×C4×D716yes—
112C22C23×C3×D711yescharacteristic
113C4—712no—
114C4C2×C3×C4×D7112yesdirect product factor
115C3C2×C42×D711yescharacteristic, direct product factor, Sylow
116C2—78no—
117C2C3×C42×D714yesdirect product factor
118C2C22×C3×C4×D713yes—
subgroupG/H
center, hypercenterC2×C3×C42D7
commutator, hypocenterC7C22×C3×C42
FrattiniC22C23×C3×D7
FittingC2×C3×C42×C7C2
socleC23×C3×C7C23
prime radicalC23×C3×D7C22
cosocleC22×C7C24×C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
2. C2×C3×C42×C7C2semidirect
3. C2×C3×C4×Dic7C2semidirect
4. C3×C42×D7C2direct (N × H)
5. C2×C42×D7C3direct (N × H)
9. C2×C3×C4×D7C4direct (N × H)
11. C3×C42×C7C22semidirect
12. C3×C4×Dic7C22semidirect
14. C2×C42×C7C2×C3semidirect
15. C2×C4×Dic7C2×C3semidirect
16. C42×D7C2×C3direct (N × H)
21. C2×C3×C4×C7C2×C4semidirect
23. C2×C3×Dic7C2×C4semidirect
25. C3×C4×D7C2×C4direct (N × H)
29. C2×C4×D7C3×C4direct (N × H)
31. C42×C7C22×C3semidirect
32. C4×Dic7C22×C3semidirect
36. C2×C3×C42D7direct (N × H)
39. C2×C3×D7C42direct (N × H)
41. C3×C4×C7C22×C4semidirect
42. C3×Dic7C22×C4semidirect
47. C2×C4×C7C2×C3×C4semidirect
49. C2×Dic7C2×C3×C4semidirect
51. C4×D7C2×C3×C4direct (N × H)
58. C3×C42C2×D7direct (N × H)
59. C2×C3×C7C2×C42semidirect
61. C3×D7C2×C42direct (N × H)
65. C2×C42C3×D7direct (N × H)
68. C2×D7C3×C42direct (N × H)
70. C4×C7C22×C3×C4semidirect
71. Dic7C22×C3×C4semidirect
78. C2×C3×C4C4×D7direct (N × H)
80. C3×C7C22×C42semidirect
87. C42C2×C3×D7direct (N × H)
88. C2×C7C2×C3×C42semidirect
90. D7C2×C3×C42direct (N × H)
96. C3×C4C2×C4×D7direct (N × H)
103. C2×C4C3×C4×D7direct (N × H)
105. C7C22×C3×C42semidirect
107. C2×C3C42×D7direct (N × H)
114. C4C2×C3×C4×D7direct (N × H)
115. C3C2×C42×D7direct (N × H)
117. C2C3×C42×D7direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D72
C2×C3×C42×C72
C2×C3×C4×Dic72
C3×C42×D72
C2×C42×D73
C22×C3×C427