Finite Group Databasebeta

← C3×D7×D20 #13132642

Subgroups — C3×D7×D20 #13132642

Order 1680.

216 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 72 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×C3×D5×D7C212yesmaximal, complemented
2C3×D140C211yesmaximal, characteristic, complemented
3C3×C7×D20C211yesmaximal, characteristic, complemented
4C3×C4×C5×D7C211yesmaximal, characteristic, complemented
5C3×#63307C212yesmaximal, complemented
6D7×D20C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×D35C2212yescomplemented
8C2×C3×C7×D5C2212yescomplemented
9C2×C3×C5×D7C2211yescharacteristic
10C3×D5×D7—24no—
11C3×C4×C5×C7C2211yescharacteristic, complemented
12C3×C5×Dic7C2211yescharacteristic, complemented
13C3×D4×D7—51nomaximal, Hall
14C2×D5×D7C2×C312yescomplemented
15D140C2×C311yescharacteristic, complemented
16C7×D20C2×C311yescharacteristic, complemented
17C4×C5×D7C2×C311yescharacteristic, complemented
18#63307C2×C312yescomplemented
19C2×C3×D20—71nomaximal, Hall
20C2×C3×C5×C7C2311yescharacteristic
21C3×D35—22no—
22C3×C7×D5—22no—
23C3×C5×D7D412yescomplemented
24C22×C3×D7—52no—
25C3×C7×D4—51no—
26C3×C22⋊D7—52no—
27C3×D28—51no—
28C3×C4×D7D511yescharacteristic, complemented
29C2×D35C22×C312yescomplemented
30C2×C7×D5C22×C312yescomplemented
31C2×C5×D7C22×C311yescharacteristic
32D5×D7—24no—
33C4×C5×C7C22×C311yescharacteristic, complemented
34C5×Dic7C22×C311yescharacteristic, complemented
35C22×C3×D5—72no—
36C2×C3×C4×C5—71no—
37C3×D20—72no—
38C3×D20—71no—
39C3×D20D711yescharacteristic, direct product factor
40D4×D7—51noHall
41C3×C5×C7C2×D411yescharacteristic, complemented, Hall
42C22×C3×C7—52no—
43C2×C3×D7—52no—
44C2×C3×D7—104no—
45C2×C3×D7C2×D511yescharacteristic
46C3×C4×C7C2×D511yescharacteristic, complemented
47C3×Dic7C2×D511yescharacteristic, complemented
48C2×D20—71noHall
49C2×C5×C7C23×C311yescharacteristic
50D35—22no—
51C5×D7C3×D412yescomplemented
52C7×D5—22no—
53C22×C3×C5—71no—
54C2×C3×D5—72no—
55C2×C3×D5—144no—
56C2×C3×D5C2×D712yescomplemented
57C3×C4×C5—71no—
58C3×C4×C5C2×D711yescharacteristic, complemented
59C22×D7—52no—
60C7×D4—51no—
61C22⋊D7—52no—
62D28—51no—
63C4×D7C3×D511yescharacteristic, complemented
64C2×C3×D4—351noHall
65C2×C3×C7—102no—
66C2×C3×C7C22×D511yescharacteristic
67C3×D7—102no—
68C3×D7D2012yesdirect product factor
69C22×D5—72no—
70C2×C4×C5—71no—
71D20—72no—
72D20—71no—
73D20C3×D711yescharacteristic, direct product factor
74C5×C7C2×C3×D411yescharacteristic, complemented, Hall
75C2×C3×C5—72no—
76C2×C3×C5C22×D711yescharacteristic
77C3×D5—22no—
78C3×D5—142no—
79C22×C7—52no—
80C2×D7—52no—
81C2×D7—104no—
82C2×D7C2×C3×D511yescharacteristic
83C4×C7C2×C3×D511yescharacteristic, complemented
84Dic7C2×C3×D511yescharacteristic, complemented
85C23×C3—352no—
86C3×D4—51no—
87C3×D4—352no—
88C3×D4—351no—
89C2×C3×C4—71no—
90C3×C7C2×D2011yescharacteristic, complemented, Hall
91C22×C5—71no—
92C2×D5—72no—
93C2×D5—144no—
94C2×D5C2×C3×D712yescomplemented
95C4×C5—71no—
96C4×C5C2×C3×D711yescharacteristic, complemented
97C2×D4—351noSylow
98C3×C5D4×D711yescharacteristic, complemented, Hall
99C2×C7—102no—
100C2×C7C22×C3×D511yescharacteristic
101D7—102no—
102D7C3×D2012yesdirect product factor
103C22×C3—52no—
104C22×C3—71no—
105C22×C3—352no—
106C22×C3—704no—
107C3×C4—71no—
108C3×C4D5×D711yescharacteristic, complemented
109C2×C5—72no—
110C2×C5C22×C3×D711yescharacteristic
111D5—22no—
112D5—142no—
113C23—352no—
114D4—51no—
115D4—352no—
116D4—351no—
117C2×C4—71no—
118C7C2×C3×D2011yescharacteristic, complemented, Sylow
119C2×C3—72no—
120C2×C3—102no—
121C2×C3—702no—
122C2×C3C2×D5×D711yescharacteristic
123C5C3×D4×D711yescharacteristic, complemented, Sylow
124C22—52no—
125C22—71no—
126C22—352no—
127C22—704no—
128C4—71no—
129C4C3×D5×D711yescharacteristic, complemented
130C3D7×D2011yescharacteristic, direct product factor, Sylow
131C2—72no—
132C2—102no—
133C2—702no—
134C2C2×C3×D5×D711yescharacteristic
subgroupG/H
centerC2×C3C2×D5×D7
commutator, cosocleC2×C5×C7C23×C3
FrattiniC2C2×C3×D5×D7
FittingC3×C4×C5×C7C22
socleC2×C3×C5×C7C23
hypercenterC3×C4D5×D7
hypocenterC5×C7C2×C3×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D5×D7C2semidirect
2. C3×D140C2semidirect
3. C3×C7×D20C2semidirect
4. C3×C4×C5×D7C2semidirect
5. C3×#63307C2semidirect
6. D7×D20C3direct (N × H)
7. C2×C3×D35C22semidirect
8. C2×C3×C7×D5C22semidirect
11. C3×C4×C5×C7C22semidirect
12. C3×C5×Dic7C22semidirect
14. C2×D5×D7C2×C3semidirect
15. D140C2×C3semidirect
16. C7×D20C2×C3semidirect
17. C4×C5×D7C2×C3semidirect
18. #63307C2×C3semidirect
23. C3×C5×D7D4semidirect
28. C3×C4×D7D5semidirect
29. C2×D35C22×C3semidirect
30. C2×C7×D5C22×C3semidirect
33. C4×C5×C7C22×C3semidirect
34. C5×Dic7C22×C3semidirect
39. C3×D20D7direct (N × H)
41. C3×C5×C7C2×D4semidirect
46. C3×C4×C7C2×D5semidirect
47. C3×Dic7C2×D5semidirect
51. C5×D7C3×D4semidirect
56. C2×C3×D5C2×D7semidirect
58. C3×C4×C5C2×D7semidirect
63. C4×D7C3×D5semidirect
68. C3×D7D20direct (N × H)
73. D20C3×D7direct (N × H)
74. C5×C7C2×C3×D4semidirect
83. C4×C7C2×C3×D5semidirect
84. Dic7C2×C3×D5semidirect
90. C3×C7C2×D20semidirect
94. C2×D5C2×C3×D7semidirect
96. C4×C5C2×C3×D7semidirect
98. C3×C5D4×D7semidirect
102. D7C3×D20direct (N × H)
108. C3×C4D5×D7semidirect
118. C7C2×C3×D20semidirect
123. C5C3×D4×D7semidirect
129. C4C3×D5×D7semidirect
130. C3D7×D20direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×D72
C3×D1402
C3×C7×D202
C3×C4×C5×D72
C3×#633072
D7×D203
C3×D4×D75
C2×C3×D207