Finite Group Databasebeta

← C5×#66681 #13132671

Subgroups — C5×#66681 #13132671

Order 1680.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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×D3×D7C212yesmaximal, complemented
2C4×C5×D21C211yesmaximal, characteristic, complemented
3C5×C7×D12C211yesmaximal, characteristic, complemented
4C3×C5×D28C211yesmaximal, characteristic, complemented
5C5×#4245C212yesmaximal, complemented
6C5×D4×D7—31nomaximal, Hall
7C2×C5×D21C2211yescharacteristic
8C2×C5×D3×C7C2212yescomplemented
9C2×C3×C5×D7C2212yescomplemented
10C5×D3×D7—24no—
11C3×C4×C5×C7C2211yescharacteristic, complemented
12C5×Dic21C2211yescharacteristic, complemented
13#66681C511yesmaximal, characteristic, direct product factor, Hall
14C22×C5×D7—32no—
15C5×C7×D4—31no—
16C5×C22⋊D7—32no—
17C5×D28D311yescharacteristic, complemented
18C4×C5×D7—31no—
19C5×D3×D4—71nomaximal, Hall
20C2×C3×C5×C7C2311yescharacteristic
21C5×D21D412yescomplemented
22C5×D3×C7—22no—
23C3×C5×D7—22no—
24C2×D3×D7C2×C512yescomplemented
25C4×D21C2×C511yescharacteristic, complemented
26C7×D12C2×C511yescharacteristic, complemented
27C3×D28C2×C511yescharacteristic, complemented
28#4245C2×C512yescomplemented
29C22×C5×C7—32no—
30C2×C5×D7—31no—
31C2×C5×D7—64no—
32C2×C5×D7C2×D312yescomplemented
33C4×C5×C7C2×D311yescharacteristic, complemented
34C5×Dic7—31no—
35C22×C5×D3—72no—
36C3×C5×D4—71no—
37C5×#67—72no—
38C5×D12D711yescharacteristic, complemented
39C4×C5×D3—71no—
40D4×D7—31noHall
41C3×C5×C7C2×D411yescharacteristic, complemented, Hall
42C2×D21C22×C511yescharacteristic
43C2×D3×C7C22×C512yescomplemented
44C2×C3×D7C22×C512yescomplemented
45D3×D7—24no—
46C3×C4×C7C22×C511yescharacteristic, complemented
47Dic21C22×C511yescharacteristic, complemented
48C2×C5×D4—211noHall
49C2×C5×C7—62no—
50C2×C5×C7C22×D311yescharacteristic
51C5×D7—22no—
52C5×D7—32no—
53C22×C3×C5—72no—
54C2×C5×D3—71no—
55C2×C5×D3—144no—
56C2×C5×D3C2×D712yescomplemented
57C3×C4×C5C2×D711yescharacteristic, complemented
58C5×Dic3—71no—
59C22×D7—32no—
60C7×D4—31no—
61C22⋊D7—32no—
62D28C5×D311yescharacteristic, complemented
63C4×D7—31no—
64D3×D4—71noHall
65C2×C3×C7C23×C511yescharacteristic
66D21C5×D412yescomplemented
67C3×D7—22no—
68D3×C7—22no—
69C23×C5—212no—
70C5×D4—31no—
71C5×D4—71no—
72C5×D4—212no—
73C2×C4×C5—211no—
74C5×C7D3×D411yescharacteristic, complemented, Hall
75C2×C3×C5—142no—
76C2×C3×C5C22×D711yescharacteristic
77C5×D3—22no—
78C5×D3—72no—
79C22×C7—32no—
80C2×D7—31no—
81C2×D7—64no—
82C2×D7C2×C5×D312yescomplemented
83C4×C7C2×C5×D311yescharacteristic, complemented
84Dic7—31no—
85C22×D3—72no—
86C3×D4—71no—
87#67—72no—
88D12C5×D711yescharacteristic, complemented
89C4×D3—71no—
90C3×C7C2×C5×D411yescharacteristic, complemented, Hall
91C22×C5—32no—
92C22×C5—72no—
93C22×C5—211no—
94C22×C5—424no—
95C4×C5—211no—
96C4×C5D3×D711yescharacteristic, complemented
97C2×D4—211noSylow
98C3×C5D4×D711yescharacteristic, complemented, Hall
99C2×C7—62no—
100C2×C7C22×C5×D311yescharacteristic
101D7—22no—
102D7—32no—
103C22×C3—72no—
104C2×D3—71no—
105C2×D3—144no—
106C2×D3C2×C5×D712yescomplemented
107C3×C4C2×C5×D711yescharacteristic, complemented
108Dic3—71no—
109C2×C5—62no—
110C2×C5—142no—
111C2×C5—212no—
112C2×C5C2×D3×D711yescharacteristic
113C23—212no—
114D4—31no—
115D4—71no—
116D4—212no—
117C2×C4—211no—
118C7C5×D3×D411yescharacteristic, complemented, Sylow
119C2×C3—142no—
120C2×C3C22×C5×D711yescharacteristic
121D3—22no—
122D3—72no—
123C5#6668111yescharacteristic, direct product factor, Sylow
124C22—32no—
125C22—72no—
126C22—211no—
127C22—424no—
128C4—211no—
129C4C5×D3×D711yescharacteristic, complemented
130C3C5×D4×D711yescharacteristic, complemented, Sylow
131C2—62no—
132C2—142no—
133C2—212no—
134C2C2×C5×D3×D711yescharacteristic
subgroupG/H
centerC2×C5C2×D3×D7
commutator, cosocleC2×C3×C7C23×C5
FrattiniC2C2×C5×D3×D7
FittingC3×C4×C5×C7C22
socleC2×C3×C5×C7C23
hypercenterC4×C5D3×D7
hypocenterC3×C7C2×C5×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C5 ◁ C4×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×D3×D7C2semidirect
2. C4×C5×D21C2semidirect
3. C5×C7×D12C2semidirect
4. C3×C5×D28C2semidirect
5. C5×#4245C2semidirect
8. C2×C5×D3×C7C22semidirect
9. C2×C3×C5×D7C22semidirect
11. C3×C4×C5×C7C22semidirect
12. C5×Dic21C22semidirect
13. #66681C5direct (N × H)
17. C5×D28D3semidirect
21. C5×D21D4semidirect
24. C2×D3×D7C2×C5semidirect
25. C4×D21C2×C5semidirect
26. C7×D12C2×C5semidirect
27. C3×D28C2×C5semidirect
28. #4245C2×C5semidirect
32. C2×C5×D7C2×D3semidirect
33. C4×C5×C7C2×D3semidirect
38. C5×D12D7semidirect
41. C3×C5×C7C2×D4semidirect
43. C2×D3×C7C22×C5semidirect
44. C2×C3×D7C22×C5semidirect
46. C3×C4×C7C22×C5semidirect
47. Dic21C22×C5semidirect
56. C2×C5×D3C2×D7semidirect
57. C3×C4×C5C2×D7semidirect
62. D28C5×D3semidirect
66. D21C5×D4semidirect
74. C5×C7D3×D4semidirect
82. C2×D7C2×C5×D3semidirect
83. C4×C7C2×C5×D3semidirect
88. D12C5×D7semidirect
90. C3×C7C2×C5×D4semidirect
96. C4×C5D3×D7semidirect
98. C3×C5D4×D7semidirect
106. C2×D3C2×C5×D7semidirect
107. C3×C4C2×C5×D7semidirect
118. C7C5×D3×D4semidirect
123. C5#66681direct (N × H)
129. C4C5×D3×D7semidirect
130. C3C5×D4×D7semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D3×D72
C4×C5×D212
C5×C7×D122
C3×C5×D282
C5×#42452
C5×D4×D73
#666815
C5×D3×D47