Finite Group Databasebeta

← C2×C4×C5×D21 #13132871

Subgroups — C2×C4×C5×D21 #13132871

Order 1680.

216 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 102 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×C5×D21C211yesmaximal, characteristic
2C2×C3×C4×C5×C7C211yesmaximal, characteristic, complemented
3C2×C5×Dic21C211yesmaximal, characteristic, complemented
4C4×C5×D21C214yesmaximal, direct product factor
5C2×C4×C5×D7—31nomaximal, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C5×D21C414yesdirect product factor
8C2×C5×D21C2212yes—
9C3×C4×C5×C7C2212yescomplemented
10C5×Dic21C2212yescomplemented
11C2×C4×D21C511yesmaximal, characteristic, direct product factor, Hall
12C22×C5×D7—31no—
13C2×C4×C5×C7D311yescharacteristic, complemented
14C2×C5×Dic7—31no—
15C4×C5×D7—34no—
16C2×C4×C5×D3—71nomaximal, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C5×D21C2×C414yesdirect product factor
20C22×D21C2×C511yescharacteristic
21C2×C3×C4×C7C2×C511yescharacteristic, complemented
22C2×Dic21C2×C511yescharacteristic, complemented
23C4×D21C2×C514yesdirect product factor
24C22×C5×C7C2×D311yescharacteristic
25C2×C5×D7—34no—
26C2×C5×D7—32no—
27C4×C5×C7C2×D312yescomplemented
28C5×Dic7—32no—
29C22×C5×D3—71no—
30C2×C3×C4×C5D711yescharacteristic, complemented
31C2×C5×Dic3—71no—
32C4×C5×D3—74no—
33C2×C4×D7—31noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C22×C511yescharacteristic
36C2×D21C4×C514yesdirect product factor
37C2×D21C22×C512yes—
38C3×C4×C7C22×C512yescomplemented
39Dic21C22×C512yescomplemented
40C22×C4×C5—211noHall
41C2×C5×C7C4×D312yescomplemented
42C2×C5×C7C22×D311yescharacteristic
43C5×D7—34no—
44C22×C3×C5C2×D711yescharacteristic
45C2×C5×D3—74no—
46C2×C5×D3—72no—
47C3×C4×C5C2×D712yescomplemented
48C5×Dic3—72no—
49C22×D7—31no—
50C2×C4×C7C5×D311yescharacteristic, complemented
51C2×Dic7—31no—
52C4×D7—34no—
53C2×C4×D3—71noHall
54C2×C3×C7C2×C4×C512yescomplemented
55C2×C3×C7C23×C511yescharacteristic
56D21C2×C4×C514yesdirect product factor
57C23×C5—211no—
58C2×C4×C5—211no—
59C2×C4×C5—214no—
60C2×C4×C5D2111yescharacteristic, direct product factor
61C5×C7C2×C4×D311yescharacteristic, complemented, Hall
62C2×C3×C5C4×D712yescomplemented
63C2×C3×C5C22×D711yescharacteristic
64C5×D3—74no—
65C22×C7C2×C5×D311yescharacteristic
66C2×D7—34no—
67C2×D7—32no—
68C4×C7C2×C5×D312yescomplemented
69Dic7—32no—
70C22×D3—71no—
71C2×C3×C4C5×D711yescharacteristic, complemented
72C2×Dic3—71no—
73C4×D3—74no—
74C3×C7C22×C4×C511yescharacteristic, complemented, Hall
75C22×C5—214no—
76C22×C5—212no—
77C22×C5C2×D2111yescharacteristic
78C4×C5—212no—
79C4×C5C2×D2112yesdirect product factor
80C22×C4—211noSylow
81C3×C5C2×C4×D711yescharacteristic, complemented, Hall
82C2×C7C4×C5×D312yescomplemented
83C2×C7C22×C5×D311yescharacteristic
84D7—34no—
85C22×C3C2×C5×D711yescharacteristic
86C2×D3—74no—
87C2×D3—72no—
88C3×C4C2×C5×D712yescomplemented
89Dic3—72no—
90C2×C5—214no—
91C2×C5C4×D2112yesdirect product factor
92C2×C5C22×D2111yescharacteristic
93C23—211no—
94C2×C4—211no—
95C2×C4—214no—
96C2×C4C5×D2111yescharacteristic, direct product factor
97C7C2×C4×C5×D311yescharacteristic, complemented, Sylow
98C2×C3C4×C5×D712yescomplemented
99C2×C3C22×C5×D711yescharacteristic
100D3—74no—
101C5C2×C4×D2111yescharacteristic, direct product factor, Sylow
102C22—214no—
103C22—212no—
104C22C2×C5×D2111yescharacteristic
105C4—212no—
106C4C2×C5×D2112yesdirect product factor
107C3C2×C4×C5×D711yescharacteristic, complemented, Sylow
108C2—214no—
109C2C4×C5×D2112yesdirect product factor
110C2C22×C5×D2111yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C5D21
commutator, hypocenterC3×C7C22×C4×C5
FrattiniC2C22×C5×D21
FittingC2×C3×C4×C5×C7C2
socleC22×C3×C5×C7C22
prime radicalC22×C5×D21C2
cosocleC2×C3×C7C23×C5
LCSG ▷ C3×C7
UCSC1 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×C7C2semidirect
3. C2×C5×Dic21C2semidirect
4. C4×C5×D21C2direct (N × H)
7. C2×C5×D21C4direct (N × H)
9. C3×C4×C5×C7C22semidirect
10. C5×Dic21C22semidirect
11. C2×C4×D21C5direct (N × H)
13. C2×C4×C5×C7D3semidirect
17. C2×C3×C5×C7C2×C4semidirect
19. C5×D21C2×C4direct (N × H)
21. C2×C3×C4×C7C2×C5semidirect
22. C2×Dic21C2×C5semidirect
23. C4×D21C2×C5direct (N × H)
27. C4×C5×C7C2×D3semidirect
30. C2×C3×C4×C5D7semidirect
34. C3×C5×C7C22×C4semidirect
36. C2×D21C4×C5direct (N × H)
38. C3×C4×C7C22×C5semidirect
39. Dic21C22×C5semidirect
41. C2×C5×C7C4×D3semidirect
47. C3×C4×C5C2×D7semidirect
50. C2×C4×C7C5×D3semidirect
54. C2×C3×C7C2×C4×C5semidirect
56. D21C2×C4×C5direct (N × H)
60. C2×C4×C5D21direct (N × H)
61. C5×C7C2×C4×D3semidirect
62. C2×C3×C5C4×D7semidirect
68. C4×C7C2×C5×D3semidirect
71. C2×C3×C4C5×D7semidirect
74. C3×C7C22×C4×C5semidirect
79. C4×C5C2×D21direct (N × H)
81. C3×C5C2×C4×D7semidirect
82. C2×C7C4×C5×D3semidirect
88. C3×C4C2×C5×D7semidirect
91. C2×C5C4×D21direct (N × H)
96. C2×C4C5×D21direct (N × H)
97. C7C2×C4×C5×D3semidirect
98. C2×C3C4×C5×D7semidirect
101. C5C2×C4×D21direct (N × H)
106. C4C2×C5×D21direct (N × H)
107. C3C2×C4×C5×D7semidirect
109. C2C4×C5×D21direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D212
C2×C3×C4×C5×C72
C2×C5×Dic212
C4×C5×D212
C2×C4×C5×D73
C2×C4×D215
C2×C4×C5×D37