Finite Group Databasebeta

← C2×C3×C5×C7×D85 #49088044

Subgroups — C2×C3×C5×C7×D85 #49088044

Order 35700.

192 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 88 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×C7×D85C212yesmaximal, direct product factor
2C2×C3×C52×C7×C17C211yesmaximal, characteristic, complemented
3C2×C5×C7×D85C311yesmaximal, characteristic, direct product factor, Hall
4C3×C52×C7×C17C2211yescharacteristic, complemented, Hall
5C2×C3×C5×C7×D17—51nomaximal
6C2×C3×C7×D85C511yesmaximal, characteristic, direct product factor
7C5×C7×D85C2×C312yesdirect product factor
8C2×C52×C7×C17C2×C311yescharacteristic, complemented
9C2×C3×C5×D85C711yesmaximal, characteristic, direct product factor, Hall
10C3×C7×D85C2×C512yesdirect product factor
11C3×C5×C7×D17—52no—
12C2×C3×C5×C7×C17—22no—
13C2×C3×C5×C7×C17D511yescharacteristic, complemented
14C2×C3×C5×C7×C17C2×C511yescharacteristic, complemented
15C52×C7×C17C22×C311yescharacteristic, complemented, Hall
16C3×C5×D85C2×C712yesdirect product factor
17C2×C3×C52×C17C2×C711yescharacteristic, complemented
18C2×C5×C7×D17—51no—
19C2×C7×D85C3×C511yescharacteristic, direct product factor
20C2×C3×C5×C7×D5—171nomaximal, Hall
21C3×C5×C7×C17—22no—
22C3×C5×C7×C17C2×D511yescharacteristic, complemented
23C3×C5×C7×C17C22×C511yescharacteristic, complemented
24C2×C5×D85C3×C711yescharacteristic, direct product factor, Hall
25C2×C3×C7×D17—51noHall
26C3×C52×C17C22×C711yescharacteristic, complemented, Hall
27C2×C5×C7×C17—22no—
28C2×C5×C7×C17C3×D511yescharacteristic, complemented
29C2×C5×C7×C17C2×C3×C511yescharacteristic, complemented
30C7×D85C2×C3×C512yesdirect product factor
31C5×C7×D17—52no—
32C2×C3×C52×C7D1711yescharacteristic, complemented
33C3×C5×C7×D5—172no—
34C2×C3×D85C5×C711yescharacteristic, direct product factor
35C2×C3×C5×D17—51no—
36C2×C52×C17C2×C3×C711yescharacteristic, complemented
37C5×D85C2×C3×C712yesdirect product factor
38C2×C3×C7×C17C5×D511yescharacteristic, complemented
39C3×C7×D17—52no—
40C2×C5×C7×D5—171noHall
41C5×C7×C17—22no—
42C5×C7×C17C2×C3×D511yescharacteristic, complemented
43C5×C7×C17C22×C3×C511yescharacteristic, complemented
44C3×C52×C7C2×D1711yescharacteristic, complemented, Hall
45C2×C3×C5×C17—22no—
46C2×C3×C5×C17C7×D511yescharacteristic, complemented
47C2×C3×C5×C17C2×C5×C711yescharacteristic, complemented
48C3×D85C2×C5×C712yesdirect product factor
49C3×C5×D17—52no—
50C2×C7×D17—51noHall
51C52×C17C22×C3×C711yescharacteristic, complemented, Hall
52C22×C3×C5×C7—851no—
53C2×C3×C7×D5—171no—
54C3×C7×C17C2×C5×D511yescharacteristic, complemented, Hall
55C2×C52×C7C3×D1711yescharacteristic, complemented
56C5×C7×D5—172no—
57C2×D85C3×C5×C711yescharacteristic, direct product factor
58C2×C5×D17—51no—
59C2×C3×C5×D5—171noHall
60C3×C5×C17—22no—
61C3×C5×C17C2×C7×D511yescharacteristic, complemented
62C3×C5×C17C22×C5×C711yescharacteristic, complemented
63C2×C7×C17C3×C5×D511yescharacteristic, complemented
64C7×D17—52no—
65C2×C3×C5×C7—22no—
66C2×C3×C5×C7—852no—
67C2×C3×C5×C7C5×D1711yescharacteristic, complemented
68C2×C3×C5×C7D8511yescharacteristic, direct product factor
69C3×C7×D5—172no—
70C2×C3×D17—51noHall
71C52×C7C2×C3×D1711yescharacteristic, complemented, Hall
72C2×C5×C17—22no—
73C2×C5×C17C3×C7×D511yescharacteristic, complemented
74C2×C5×C17C2×C3×C5×C711yescharacteristic, complemented
75D85C2×C3×C5×C712yesdirect product factor
76C5×D17—52no—
77C2×C3×C52C7×D1711yescharacteristic, complemented
78C3×C5×D5—172no—
79C22×C5×C7—851no—
80C2×C7×D5—171no—
81C7×C17C2×C3×C5×D511yescharacteristic, complemented, Hall
82C3×C5×C7—22no—
83C3×C5×C7C2×C5×D1711yescharacteristic, complemented
84C3×C5×C7C2×D8511yescharacteristic, direct product factor
85C2×C3×C17C5×C7×D511yescharacteristic, complemented
86C3×D17—52no—
87C2×C5×D5—171noHall
88C5×C17—22no—
89C5×C17C2×C3×C7×D511yescharacteristic, complemented
90C5×C17C22×C3×C5×C711yescharacteristic, complemented
91C22×C3×C7—851noHall
92C3×C52C2×C7×D1711yescharacteristic, complemented, Hall
93C2×C5×C7—22no—
94C2×C5×C7—852no—
95C2×C5×C7C3×C5×D1711yescharacteristic, complemented
96C2×C5×C7C3×D8511yescharacteristic, direct product factor
97C7×D5—172no—
98C2×D17—51noHall
99C22×C3×C5—851no—
100C2×C3×D5—171no—
101C3×C17C2×C5×C7×D511yescharacteristic, complemented, Hall
102C2×C52C3×C7×D1711yescharacteristic, complemented
103C5×D5—172no—
104C2×C3×C7—852no—
105C2×C3×C7C5×D8511yescharacteristic, direct product factor
106C5×C7—22no—
107C5×C7C2×C3×C5×D1711yescharacteristic, complemented
108C5×C7C2×C3×D8511yescharacteristic, direct product factor
109C2×C17C3×C5×C7×D511yescharacteristic, complemented
110D17—52no—
111C2×C3×C5—22no—
112C2×C3×C5—852no—
113C2×C3×C5C5×C7×D1711yescharacteristic, complemented
114C2×C3×C5C7×D8511yescharacteristic, direct product factor
115C3×D5—172no—
116C22×C7—851noHall
117C52C2×C3×C7×D1711yescharacteristic, complemented, Sylow
118C3×C7C2×C5×D8511yescharacteristic, direct product factor, Hall
119C22×C5—851no—
120C2×D5—171no—
121C17C2×C3×C5×C7×D511yescharacteristic, complemented, Sylow
122C3×C5—22no—
123C3×C5C2×C7×D8511yescharacteristic, direct product factor
124C3×C5C2×C5×C7×D1711yescharacteristic, complemented
125C2×C7—852no—
126C2×C7C3×C5×D8511yescharacteristic, direct product factor
127C22×C3—851noHall
128C2×C5—22no—
129C2×C5—852no—
130C2×C5C3×C5×C7×D1711yescharacteristic, complemented
131C2×C5C3×C7×D8511yescharacteristic, direct product factor
132D5—172no—
133C7C2×C3×C5×D8511yescharacteristic, direct product factor, Sylow
134C2×C3—852no—
135C2×C3C5×C7×D8511yescharacteristic, direct product factor
136C5—22no—
137C5C2×C3×C7×D8511yescharacteristic, direct product factor
138C5C2×C3×C5×C7×D1711yescharacteristic, complemented
139C22—851noSylow
140C3C2×C5×C7×D8511yescharacteristic, direct product factor, Sylow
141C2—852no—
142C2C3×C5×C7×D8511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5×C7D85
commutator, cosocle, hypocenterC5×C17C22×C3×C5×C7
FrattiniC1G
Fitting, socleC2×C3×C52×C7×C17C2
LCSG ▷ C5×C17
UCSC1 ◁ C2×C3×C5×C7

Complements

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

NH ≅ G/Ntype
1. C3×C5×C7×D85C2direct (N × H)
2. C2×C3×C52×C7×C17C2semidirect
3. C2×C5×C7×D85C3direct (N × H)
4. C3×C52×C7×C17C22semidirect
6. C2×C3×C7×D85C5direct (N × H)
7. C5×C7×D85C2×C3direct (N × H)
8. C2×C52×C7×C17C2×C3semidirect
9. C2×C3×C5×D85C7direct (N × H)
10. C3×C7×D85C2×C5direct (N × H)
13. C2×C3×C5×C7×C17D5semidirect
14. C2×C3×C5×C7×C17C2×C5semidirect
15. C52×C7×C17C22×C3semidirect
16. C3×C5×D85C2×C7direct (N × H)
17. C2×C3×C52×C17C2×C7semidirect
19. C2×C7×D85C3×C5direct (N × H)
22. C3×C5×C7×C17C2×D5semidirect
23. C3×C5×C7×C17C22×C5semidirect
24. C2×C5×D85C3×C7direct (N × H)
26. C3×C52×C17C22×C7semidirect
28. C2×C5×C7×C17C3×D5semidirect
29. C2×C5×C7×C17C2×C3×C5semidirect
30. C7×D85C2×C3×C5direct (N × H)
32. C2×C3×C52×C7D17semidirect
34. C2×C3×D85C5×C7direct (N × H)
36. C2×C52×C17C2×C3×C7semidirect
37. C5×D85C2×C3×C7direct (N × H)
38. C2×C3×C7×C17C5×D5semidirect
42. C5×C7×C17C2×C3×D5semidirect
43. C5×C7×C17C22×C3×C5semidirect
44. C3×C52×C7C2×D17semidirect
46. C2×C3×C5×C17C7×D5semidirect
47. C2×C3×C5×C17C2×C5×C7semidirect
48. C3×D85C2×C5×C7direct (N × H)
51. C52×C17C22×C3×C7semidirect
54. C3×C7×C17C2×C5×D5semidirect
55. C2×C52×C7C3×D17semidirect
57. C2×D85C3×C5×C7direct (N × H)
61. C3×C5×C17C2×C7×D5semidirect
62. C3×C5×C17C22×C5×C7semidirect
63. C2×C7×C17C3×C5×D5semidirect
67. C2×C3×C5×C7C5×D17semidirect
68. C2×C3×C5×C7D85direct (N × H)
71. C52×C7C2×C3×D17semidirect
73. C2×C5×C17C3×C7×D5semidirect
74. C2×C5×C17C2×C3×C5×C7semidirect
75. D85C2×C3×C5×C7direct (N × H)
77. C2×C3×C52C7×D17semidirect
81. C7×C17C2×C3×C5×D5semidirect
83. C3×C5×C7C2×C5×D17semidirect
84. C3×C5×C7C2×D85direct (N × H)
85. C2×C3×C17C5×C7×D5semidirect
89. C5×C17C2×C3×C7×D5semidirect
90. C5×C17C22×C3×C5×C7semidirect
92. C3×C52C2×C7×D17semidirect
95. C2×C5×C7C3×C5×D17semidirect
96. C2×C5×C7C3×D85direct (N × H)
101. C3×C17C2×C5×C7×D5semidirect
102. C2×C52C3×C7×D17semidirect
105. C2×C3×C7C5×D85direct (N × H)
107. C5×C7C2×C3×C5×D17semidirect
108. C5×C7C2×C3×D85direct (N × H)
109. C2×C17C3×C5×C7×D5semidirect
113. C2×C3×C5C5×C7×D17semidirect
114. C2×C3×C5C7×D85direct (N × H)
117. C52C2×C3×C7×D17semidirect
118. C3×C7C2×C5×D85direct (N × H)
121. C17C2×C3×C5×C7×D5semidirect
123. C3×C5C2×C7×D85direct (N × H)
124. C3×C5C2×C5×C7×D17semidirect
126. C2×C7C3×C5×D85direct (N × H)
130. C2×C5C3×C5×C7×D17semidirect
131. C2×C5C3×C7×D85direct (N × H)
133. C7C2×C3×C5×D85direct (N × H)
135. C2×C3C5×C7×D85direct (N × H)
137. C5C2×C3×C7×D85direct (N × H)
138. C5C2×C3×C5×C7×D17semidirect
140. C3C2×C5×C7×D85direct (N × H)
142. C2C3×C5×C7×D85direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×C7×D852
C2×C3×C52×C7×C172
C2×C5×C7×D853
C2×C3×C5×C7×D175
C2×C3×C7×D855
C2×C3×C5×D857
C2×C3×C5×C7×D517