Finite Group Databasebeta

← C2×C7×D5×A4 #13132957

Subgroups — C2×C7×D5×A4 #13132957

Order 1680.

156 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 42 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×C5×C7×A4C211yesmaximal, characteristic, complemented
2C7×D5×A4C212yesmaximal, direct product factor
3C23×C7×D5C311yesmaximal, characteristic, complemented, Hall
4C2×C3×C7×D5—41nomaximal
5C5×C7×A4C2211yescharacteristic, complemented
6C22×C7×A4—51nomaximal, Hall
7C23×C5×C7C2×C311yescharacteristic, complemented
8C22×C7×D5—31no—
9C22×C7×D5—31no—
10C22×C7×D5—32no—
11C22×C7×D5C2×C312yescomplemented
12C2×D5×A4C711yesmaximal, characteristic, direct product factor, Hall
13C2×C3×C5×C7—41no—
14C3×C7×D5—42no—
15C2×C7×A4—52no—
16C2×C7×A4D511yescharacteristic, direct product factor
17C22×C5×C7—31no—
18C22×C5×C7—31no—
19C22×C5×C7C22×C311yescharacteristic, complemented
20C2×C7×D5—32no—
21C2×C7×D5—31no—
22C2×C7×D5—32no—
23C2×C7×D5—32no—
24C2×C7×D5—32no—
25C2×C7×D5A411yescharacteristic, direct product factor
26C2×C5×A4C2×C711yescharacteristic, complemented
27D5×A4C2×C712yesdirect product factor
28C24×C7—51noHall
29C3×C5×C7—41noHall
30C22×C3×C7—201no—
31C7×A4C2×D511yescharacteristic, direct product factor
32C23×D5C3×C711yescharacteristic, complemented, Hall
33C2×C5×C7—31no—
34C2×C5×C7—31no—
35C2×C5×C7C2×A411yescharacteristic, complemented
36C7×D5—32no—
37C7×D5C2×A412yesdirect product factor
38C2×C3×D5—41no—
39C5×A4C22×C711yescharacteristic, complemented
40C23×C7—52no—
41C23×C7—151no—
42C23×C7—152no—
43C23×C7—151no—
44C23×C7C3×D511yescharacteristic, complemented
45C22×A4—51noHall
46C2×C3×C7—41no—
47C2×C3×C7—202no—
48C23×C5C2×C3×C711yescharacteristic, complemented
49C22×D5—31no—
50C22×D5—31no—
51C22×D5—32no—
52C22×D5C2×C3×C712yescomplemented
53C5×C7C22×A411yescharacteristic, complemented, Hall
54C2×C3×C5—41no—
55C3×D5—42no—
56C22×C7—31no—
57C22×C7—31no—
58C22×C7—51no—
59C22×C7—152no—
60C22×C7—152no—
61C22×C7—152no—
62C22×C7—152no—
63C22×C7—151no—
64C22×C7C2×C3×D511yescharacteristic, complemented
65C2×A4—52no—
66C2×A4C7×D511yescharacteristic, direct product factor
67C3×C7—41noHall
68C22×C5—31no—
69C22×C5—31no—
70C22×C5C22×C3×C711yescharacteristic, complemented
71C2×D5—32no—
72C2×D5—31no—
73C2×D5—32no—
74C2×D5—32no—
75C2×D5—32no—
76C2×D5C7×A411yescharacteristic, direct product factor
77C24—51noSylow
78C3×C5—41noHall
79C2×C7—31no—
80C2×C7—31no—
81C2×C7—52no—
82C2×C7—152no—
83C2×C7D5×A411yescharacteristic, direct product factor
84C22×C3—201no—
85A4C2×C7×D511yescharacteristic, direct product factor
86C2×C5—31no—
87C2×C5—31no—
88C2×C5C2×C7×A411yescharacteristic, complemented
89D5—32no—
90D5C2×C7×A412yesdirect product factor
91C23—52no—
92C23—151no—
93C23—152no—
94C23—151no—
95C23C3×C7×D511yescharacteristic, complemented
96C7C2×D5×A411yescharacteristic, direct product factor, Sylow
97C2×C3—41no—
98C2×C3—202no—
99C5C22×C7×A411yescharacteristic, complemented, Sylow
100C22—31no—
101C22—31no—
102C22—51no—
103C22—152no—
104C22—152no—
105C22—152no—
106C22—152no—
107C22—151no—
108C22C2×C3×C7×D511yescharacteristic, complemented
109C3—41noSylow
110C2—31no—
111C2—31no—
112C2—52no—
113C2—152no—
114C2C7×D5×A411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C7D5×A4
commutator, cosocle, hypocenterC22×C5C22×C3×C7
FrattiniC1G
Fitting, socleC23×C5×C7C2×C3
LCSG ▷ C22×C5
UCSC1 ◁ C2×C7

Complements

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

NH ≅ G/Ntype
1. C2×C5×C7×A4C2semidirect
2. C7×D5×A4C2direct (N × H)
3. C23×C7×D5C3semidirect
5. C5×C7×A4C22semidirect
7. C23×C5×C7C2×C3semidirect
11. C22×C7×D5C2×C3semidirect
12. C2×D5×A4C7direct (N × H)
16. C2×C7×A4D5direct (N × H)
19. C22×C5×C7C22×C3semidirect
25. C2×C7×D5A4direct (N × H)
26. C2×C5×A4C2×C7semidirect
27. D5×A4C2×C7direct (N × H)
31. C7×A4C2×D5direct (N × H)
32. C23×D5C3×C7semidirect
35. C2×C5×C7C2×A4semidirect
37. C7×D5C2×A4direct (N × H)
39. C5×A4C22×C7semidirect
44. C23×C7C3×D5semidirect
48. C23×C5C2×C3×C7semidirect
52. C22×D5C2×C3×C7semidirect
53. C5×C7C22×A4semidirect
64. C22×C7C2×C3×D5semidirect
66. C2×A4C7×D5direct (N × H)
70. C22×C5C22×C3×C7semidirect
76. C2×D5C7×A4direct (N × H)
83. C2×C7D5×A4direct (N × H)
85. A4C2×C7×D5direct (N × H)
88. C2×C5C2×C7×A4semidirect
90. D5C2×C7×A4direct (N × H)
95. C23C3×C7×D5semidirect
96. C7C2×D5×A4direct (N × H)
99. C5C22×C7×A4semidirect
108. C22C2×C3×C7×D5semidirect
114. C2C7×D5×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C7×A42
C7×D5×A42
C23×C7×D53
C2×C3×C7×D54
C22×C7×A45
C2×D5×A47