Finite Group Databasebeta

← C2×C3×C5×D357 #36339581

Subgroups — C2×C3×C5×D357 #36339581

Order 21420.

176 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C3×C5×D357C212yesmaximal, direct product factor
2C2×C32×C5×C7×C17C211yesmaximal, characteristic, complemented
3C2×C3×C5×D119—31nomaximal
4C2×C5×D357C311yesmaximal, characteristic, direct product factor
5C32×C5×C7×C17C2211yescharacteristic, complemented, Hall
6C2×C3×D357C511yesmaximal, characteristic, direct product factor, Hall
7C5×D357C2×C312yesdirect product factor
8C3×C5×D119—32no—
9C2×C3×C5×C7×C17—21no—
10C2×C3×C5×C7×C17D311yescharacteristic, complemented
11C2×C3×C5×C7×C17C2×C311yescharacteristic, complemented
12C2×C3×C5×D51—71nomaximal, Hall
13C2×C5×D119—31noHall
14C3×D357C2×C512yesdirect product factor
15C2×C32×C7×C17C2×C511yescharacteristic, complemented
16C3×C5×C7×C17—21no—
17C3×C5×C7×C17C2×D311yescharacteristic, complemented
18C3×C5×C7×C17C22×C311yescharacteristic, complemented
19C2×C32×C5×C17D711yescharacteristic, complemented
20C3×C5×D51—72no—
21C2×D357C3×C511yescharacteristic, direct product factor
22C2×C3×D119—31no—
23C2×C3×C5×D21—171nomaximal, Hall
24C2×C5×C7×C17C3×D311yescharacteristic, complemented
25C5×D119—32no—
26C32×C7×C17C22×C511yescharacteristic, complemented, Hall
27C2×C5×D51—71no—
28C2×C3×C5×D17—211no—
29C32×C5×C17C2×D711yescharacteristic, complemented, Hall
30C2×C3×C7×C17—21no—
31C2×C3×C7×C17C5×D311yescharacteristic, complemented
32C2×C3×C7×C17C2×C3×C511yescharacteristic, complemented
33D357C2×C3×C512yesdirect product factor
34C3×D119—32no—
35C2×C32×C5×C7D1711yescharacteristic, complemented
36C3×C5×D21—172no—
37C2×C3×D51—71noHall
38C5×C7×C17C2×C3×D311yescharacteristic, complemented, Hall
39C2×C3×C5×C17—21no—
40C2×C3×C5×C17C3×D711yescharacteristic, complemented
41C2×C3×C5×C17D2111yescharacteristic, complemented
42C5×D51—72no—
43C3×C5×D17—212no—
44C2×D119—31noHall
45C2×C5×D21—171no—
46C2×C3×C5×D7—511no—
47C3×C7×C17—21no—
48C3×C7×C17C2×C5×D311yescharacteristic, complemented
49C3×C7×C17C22×C3×C511yescharacteristic, complemented
50C2×C5×D17—211noHall
51C32×C5×C7C2×D1711yescharacteristic, complemented, Hall
52C2×C32×C17C5×D711yescharacteristic, complemented
53C3×D51—72no—
54C3×C5×C17—21no—
55C3×C5×C17C2×C3×D711yescharacteristic, complemented
56C3×C5×C17C2×D2111yescharacteristic, complemented
57C2×C3×D21—171noHall
58C2×C7×C17C3×C5×D311yescharacteristic, complemented
59D119—32no—
60C2×C3×C5×C7—21no—
61C2×C3×C5×C7C3×D1711yescharacteristic, complemented
62C2×C3×C5×C7D5111yescharacteristic, complemented
63C5×D21—172no—
64C3×C5×D7—512no—
65C2×D51—71no—
66C2×C3×D17—211no—
67C2×C3×C5×D3—1191noHall
68C2×C5×C17C3×D2111yescharacteristic, complemented
69C5×D17—212no—
70C32×C17C2×C5×D711yescharacteristic, complemented, Hall
71C2×C5×D7—511noHall
72C2×C32×C7C5×D1711yescharacteristic, complemented
73C3×D21—172no—
74C7×C17C2×C3×C5×D311yescharacteristic, complemented, Hall
75C3×C5×C7—21no—
76C3×C5×C7C2×C3×D1711yescharacteristic, complemented
77C3×C5×C7C2×D5111yescharacteristic, complemented
78C2×C3×C17—21no—
79C2×C3×C17C3×C5×D711yescharacteristic, complemented
80C2×C3×C17C5×D2111yescharacteristic, complemented
81D51—72no—
82C3×D17—212no—
83C2×C32×C5D11911yescharacteristic, complemented
84C3×C5×D3—1192no—
85C5×C17C2×C3×D2111yescharacteristic, complemented, Hall
86C2×D21—171no—
87C2×C3×D7—511no—
88C2×C5×C7C3×D5111yescharacteristic, complemented
89C5×D7—512no—
90C2×D17—211noHall
91C32×C7C2×C5×D1711yescharacteristic, complemented, Hall
92C22×C3×C5—3571no—
93C2×C5×D3—1191no—
94C3×C17—21no—
95C3×C17C2×C3×C5×D711yescharacteristic, complemented
96C3×C17C2×C5×D2111yescharacteristic, complemented
97C32×C5C2×D11911yescharacteristic, complemented, Hall
98C2×C3×C7—21no—
99C2×C3×C7C3×C5×D1711yescharacteristic, complemented
100C2×C3×C7C5×D5111yescharacteristic, complemented
101D21—172no—
102C3×D7—512no—
103C2×C3×D3—1191noHall
104C5×C7C2×C3×D5111yescharacteristic, complemented, Hall
105C2×C17C3×C5×D2111yescharacteristic, complemented
106D17—212no—
107C2×C3×C5—21no—
108C2×C3×C5—3572no—
109C2×C3×C5C3×D11911yescharacteristic, complemented
110C2×C3×C5D35711yescharacteristic, direct product factor
111C5×D3—1192no—
112C2×D7—511noHall
113C3×C7—21no—
114C3×C7C2×C3×C5×D1711yescharacteristic, complemented
115C3×C7C2×C5×D5111yescharacteristic, complemented
116C22×C5—3571noHall
117C2×C32C5×D11911yescharacteristic, complemented
118C3×D3—1192no—
119C17C2×C3×C5×D2111yescharacteristic, complemented, Sylow
120C3×C5—21no—
121C3×C5C2×C3×D11911yescharacteristic, complemented
122C3×C5C2×D35711yescharacteristic, direct product factor
123C2×C7C3×C5×D5111yescharacteristic, complemented
124D7—512no—
125C22×C3—3571no—
126C2×D3—1191no—
127C2×C5—3572no—
128C2×C5C3×D35711yescharacteristic, direct product factor
129C32C2×C5×D11911yescharacteristic, complemented, Sylow
130C7C2×C3×C5×D5111yescharacteristic, complemented, Sylow
131C2×C3—21no—
132C2×C3—3572no—
133C2×C3C3×C5×D11911yescharacteristic, complemented
134C2×C3C5×D35711yescharacteristic, direct product factor
135D3—1192no—
136C5C2×C3×D35711yescharacteristic, direct product factor, Sylow
137C22—3571noSylow
138C3—21no—
139C3C2×C5×D35711yescharacteristic, direct product factor
140C3C2×C3×C5×D11911yescharacteristic, complemented
141C2—3572no—
142C2C3×C5×D35711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5D357
commutator, cosocle, hypocenterC3×C7×C17C22×C3×C5
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C17C2
LCSG ▷ C3×C7×C17
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×D357C2direct (N × H)
2. C2×C32×C5×C7×C17C2semidirect
4. C2×C5×D357C3direct (N × H)
5. C32×C5×C7×C17C22semidirect
6. C2×C3×D357C5direct (N × H)
7. C5×D357C2×C3direct (N × H)
10. C2×C3×C5×C7×C17D3semidirect
11. C2×C3×C5×C7×C17C2×C3semidirect
14. C3×D357C2×C5direct (N × H)
15. C2×C32×C7×C17C2×C5semidirect
17. C3×C5×C7×C17C2×D3semidirect
18. C3×C5×C7×C17C22×C3semidirect
19. C2×C32×C5×C17D7semidirect
21. C2×D357C3×C5direct (N × H)
24. C2×C5×C7×C17C3×D3semidirect
26. C32×C7×C17C22×C5semidirect
29. C32×C5×C17C2×D7semidirect
31. C2×C3×C7×C17C5×D3semidirect
32. C2×C3×C7×C17C2×C3×C5semidirect
33. D357C2×C3×C5direct (N × H)
35. C2×C32×C5×C7D17semidirect
38. C5×C7×C17C2×C3×D3semidirect
40. C2×C3×C5×C17C3×D7semidirect
41. C2×C3×C5×C17D21semidirect
48. C3×C7×C17C2×C5×D3semidirect
49. C3×C7×C17C22×C3×C5semidirect
51. C32×C5×C7C2×D17semidirect
52. C2×C32×C17C5×D7semidirect
55. C3×C5×C17C2×C3×D7semidirect
56. C3×C5×C17C2×D21semidirect
58. C2×C7×C17C3×C5×D3semidirect
61. C2×C3×C5×C7C3×D17semidirect
62. C2×C3×C5×C7D51semidirect
68. C2×C5×C17C3×D21semidirect
70. C32×C17C2×C5×D7semidirect
72. C2×C32×C7C5×D17semidirect
74. C7×C17C2×C3×C5×D3semidirect
76. C3×C5×C7C2×C3×D17semidirect
77. C3×C5×C7C2×D51semidirect
79. C2×C3×C17C3×C5×D7semidirect
80. C2×C3×C17C5×D21semidirect
83. C2×C32×C5D119semidirect
85. C5×C17C2×C3×D21semidirect
88. C2×C5×C7C3×D51semidirect
91. C32×C7C2×C5×D17semidirect
95. C3×C17C2×C3×C5×D7semidirect
96. C3×C17C2×C5×D21semidirect
97. C32×C5C2×D119semidirect
99. C2×C3×C7C3×C5×D17semidirect
100. C2×C3×C7C5×D51semidirect
104. C5×C7C2×C3×D51semidirect
105. C2×C17C3×C5×D21semidirect
109. C2×C3×C5C3×D119semidirect
110. C2×C3×C5D357direct (N × H)
114. C3×C7C2×C3×C5×D17semidirect
115. C3×C7C2×C5×D51semidirect
117. C2×C32C5×D119semidirect
119. C17C2×C3×C5×D21semidirect
121. C3×C5C2×C3×D119semidirect
122. C3×C5C2×D357direct (N × H)
123. C2×C7C3×C5×D51semidirect
128. C2×C5C3×D357direct (N × H)
129. C32C2×C5×D119semidirect
130. C7C2×C3×C5×D51semidirect
133. C2×C3C3×C5×D119semidirect
134. C2×C3C5×D357direct (N × H)
136. C5C2×C3×D357direct (N × H)
139. C3C2×C5×D357direct (N × H)
140. C3C2×C3×C5×D119semidirect
142. C2C3×C5×D357direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D3572
C2×C32×C5×C7×C172
C2×C3×C5×D1193
C2×C5×D3573
C2×C3×D3575
C2×C3×C5×D517
C2×C3×C5×D2117