Finite Group Databasebeta

← C2×C3×C7×C23×D15 #44394673

Subgroups — C2×C3×C7×C23×D15 #44394673

Order 28980.

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

normal subgroups only

This group has 176 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×C7×C23×D15C212yesmaximal, direct product factor
2C2×C32×C5×C7×C23C211yesmaximal, characteristic, complemented
3C2×C3×C7×D5×C23—31nomaximal
4C2×C7×C23×D15C311yesmaximal, characteristic, direct product factor
5C32×C5×C7×C23C2211yescharacteristic, complemented, Hall
6C2×C3×D3×C7×C23—51nomaximal, Hall
7C7×C23×D15C2×C312yesdirect product factor
8C3×C7×D5×C23—32no—
9C2×C3×C5×C7×C23—21no—
10C2×C3×C5×C7×C23D311yescharacteristic, complemented
11C2×C3×C5×C7×C23C2×C311yescharacteristic, complemented
12C2×C3×C23×D15C711yesmaximal, characteristic, direct product factor, Hall
13C2×C7×D5×C23—31noHall
14C3×D3×C7×C23—52no—
15C2×C32×C7×C23D511yescharacteristic, complemented
16C3×C5×C7×C23—21no—
17C3×C5×C7×C23C2×D311yescharacteristic, complemented
18C3×C5×C7×C23C22×C311yescharacteristic, complemented
19C3×C23×D15C2×C712yesdirect product factor
20C2×C32×C5×C23C2×C711yescharacteristic, complemented
21C22×C3×C7×C23—151no—
22C2×D3×C7×C23—51no—
23C2×C5×C7×C23C3×D311yescharacteristic, complemented
24C7×D5×C23—32no—
25C32×C7×C23C2×D511yescharacteristic, complemented, Hall
26C2×C23×D15C3×C711yescharacteristic, direct product factor
27C2×C3×D5×C23—31no—
28C2×C3×C7×D15C2311yesmaximal, characteristic, direct product factor, Hall
29C32×C5×C23C22×C711yescharacteristic, complemented, Hall
30C2×C3×C7×C23—21no—
31C2×C3×C7×C23—152no—
32C2×C3×C7×C23C3×D511yescharacteristic, complemented
33C2×C3×C7×C23D1511yescharacteristic, direct product factor
34D3×C7×C23—52no—
35C2×C3×D3×C23—51noHall
36C5×C7×C23C2×C3×D311yescharacteristic, complemented, Hall
37C2×C3×C5×C23—21no—
38C2×C3×C5×C23D3×C711yescharacteristic, complemented
39C2×C3×C5×C23C2×C3×C711yescharacteristic, complemented
40C23×D15C2×C3×C712yesdirect product factor
41C3×D5×C23—32no—
42C22×C7×C23—151noHall
43C2×C32×C5×C7C2×C2311yescharacteristic, complemented
44C3×C7×D15C2×C2312yesdirect product factor
45C3×C7×C23—21no—
46C3×C7×C23C2×C3×D511yescharacteristic, complemented
47C3×C7×C23C2×D1511yescharacteristic, direct product factor
48C2×D5×C23—31noHall
49C2×C7×D15C3×C2311yescharacteristic, direct product factor
50C2×C3×C7×D5—31no—
51C2×C32×C23C7×D511yescharacteristic, complemented
52C3×D3×C23—52no—
53C3×C5×C23—21no—
54C3×C5×C23C2×D3×C711yescharacteristic, complemented
55C3×C5×C23C22×C3×C711yescharacteristic, complemented
56C2×C7×C23—152no—
57C2×C7×C23C3×D1511yescharacteristic, direct product factor
58C32×C5×C7C22×C2311yescharacteristic, complemented, Hall
59C22×C3×C23—151no—
60C2×D3×C23—51no—
61C2×C3×D3×C7—51noHall
62C2×C5×C23C3×D3×C711yescharacteristic, complemented
63D5×C23—32no—
64C2×C3×C5×C7—21no—
65C2×C3×C5×C7D3×C2311yescharacteristic, complemented
66C2×C3×C5×C7C2×C3×C2311yescharacteristic, complemented
67C7×D15C2×C3×C2312yesdirect product factor
68C3×C7×D5—32no—
69C32×C23C2×C7×D511yescharacteristic, complemented, Hall
70C2×C3×D15C7×C2311yescharacteristic, direct product factor, Hall
71C7×C23C2×C3×D1511yescharacteristic, direct product factor, Hall
72C2×C7×D5—31noHall
73C2×C3×C23—21no—
74C2×C3×C23—152no—
75C2×C3×C23C3×C7×D511yescharacteristic, complemented
76C2×C3×C23C7×D1511yescharacteristic, direct product factor
77D3×C23—52no—
78C2×C32×C7D5×C2311yescharacteristic, complemented
79C3×D3×C7—52no—
80C5×C23C2×C3×D3×C711yescharacteristic, complemented, Hall
81C3×C5×C7—21no—
82C3×C5×C7C2×D3×C2311yescharacteristic, complemented
83C3×C5×C7C22×C3×C2311yescharacteristic, complemented
84C22×C23—151noHall
85C2×C32×C5C2×C7×C2311yescharacteristic, complemented
86C3×D15C2×C7×C2312yesdirect product factor
87C22×C3×C7—151no—
88C2×D3×C7—51no—
89C2×C5×C7C3×D3×C2311yescharacteristic, complemented
90C7×D5—32no—
91C3×C23—21no—
92C3×C23C2×C3×C7×D511yescharacteristic, complemented
93C3×C23C2×C7×D1511yescharacteristic, direct product factor
94C32×C7C2×D5×C2311yescharacteristic, complemented, Hall
95C2×D15C3×C7×C2311yescharacteristic, direct product factor
96C2×C3×D5—31no—
97C2×C23—152no—
98C2×C23C3×C7×D1511yescharacteristic, direct product factor
99C32×C5C22×C7×C2311yescharacteristic, complemented, Hall
100C2×C3×C7—21no—
101C2×C3×C7—152no—
102C2×C3×C7C3×D5×C2311yescharacteristic, complemented
103C2×C3×C7C23×D1511yescharacteristic, direct product factor
104D3×C7—52no—
105C2×C3×D3—51noHall
106C5×C7C2×C3×D3×C2311yescharacteristic, complemented, Hall
107C2×C3×C5—21no—
108C2×C3×C5D3×C7×C2311yescharacteristic, complemented
109C2×C3×C5C2×C3×C7×C2311yescharacteristic, complemented
110D15C2×C3×C7×C2312yesdirect product factor
111C3×D5—32no—
112C22×C7—151noHall
113C23C2×C3×C7×D1511yescharacteristic, direct product factor, Sylow
114C3×C7—21no—
115C3×C7C2×C3×D5×C2311yescharacteristic, complemented
116C3×C7C2×C23×D1511yescharacteristic, direct product factor
117C2×D5—31noHall
118C2×C32C7×D5×C2311yescharacteristic, complemented
119C3×D3—52no—
120C3×C5—21no—
121C3×C5C2×D3×C7×C2311yescharacteristic, complemented
122C3×C5C22×C3×C7×C2311yescharacteristic, complemented
123C2×C7—152no—
124C2×C7C3×C23×D1511yescharacteristic, direct product factor
125C22×C3—151no—
126C2×D3—51no—
127C2×C5C3×D3×C7×C2311yescharacteristic, complemented
128D5—32no—
129C32C2×C7×D5×C2311yescharacteristic, complemented, Sylow
130C7C2×C3×C23×D1511yescharacteristic, direct product factor, Sylow
131C2×C3—21no—
132C2×C3—152no—
133C2×C3C3×C7×D5×C2311yescharacteristic, complemented
134C2×C3C7×C23×D1511yescharacteristic, direct product factor
135D3—52no—
136C5C2×C3×D3×C7×C2311yescharacteristic, complemented, Sylow
137C22—151noSylow
138C3—21no—
139C3C2×C7×C23×D1511yescharacteristic, direct product factor
140C3C2×C3×C7×D5×C2311yescharacteristic, complemented
141C2—152no—
142C2C3×C7×C23×D1511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C7×C23D15
commutator, cosocle, hypocenterC3×C5C22×C3×C7×C23
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C23C2
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3×C7×C23

Complements

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

NH ≅ G/Ntype
1. C3×C7×C23×D15C2direct (N × H)
2. C2×C32×C5×C7×C23C2semidirect
4. C2×C7×C23×D15C3direct (N × H)
5. C32×C5×C7×C23C22semidirect
7. C7×C23×D15C2×C3direct (N × H)
10. C2×C3×C5×C7×C23D3semidirect
11. C2×C3×C5×C7×C23C2×C3semidirect
12. C2×C3×C23×D15C7direct (N × H)
15. C2×C32×C7×C23D5semidirect
17. C3×C5×C7×C23C2×D3semidirect
18. C3×C5×C7×C23C22×C3semidirect
19. C3×C23×D15C2×C7direct (N × H)
20. C2×C32×C5×C23C2×C7semidirect
23. C2×C5×C7×C23C3×D3semidirect
25. C32×C7×C23C2×D5semidirect
26. C2×C23×D15C3×C7direct (N × H)
28. C2×C3×C7×D15C23direct (N × H)
29. C32×C5×C23C22×C7semidirect
32. C2×C3×C7×C23C3×D5semidirect
33. C2×C3×C7×C23D15direct (N × H)
36. C5×C7×C23C2×C3×D3semidirect
38. C2×C3×C5×C23D3×C7semidirect
39. C2×C3×C5×C23C2×C3×C7semidirect
40. C23×D15C2×C3×C7direct (N × H)
43. C2×C32×C5×C7C2×C23semidirect
44. C3×C7×D15C2×C23direct (N × H)
46. C3×C7×C23C2×C3×D5semidirect
47. C3×C7×C23C2×D15direct (N × H)
49. C2×C7×D15C3×C23direct (N × H)
51. C2×C32×C23C7×D5semidirect
54. C3×C5×C23C2×D3×C7semidirect
55. C3×C5×C23C22×C3×C7semidirect
57. C2×C7×C23C3×D15direct (N × H)
58. C32×C5×C7C22×C23semidirect
62. C2×C5×C23C3×D3×C7semidirect
65. C2×C3×C5×C7D3×C23semidirect
66. C2×C3×C5×C7C2×C3×C23semidirect
67. C7×D15C2×C3×C23direct (N × H)
69. C32×C23C2×C7×D5semidirect
70. C2×C3×D15C7×C23direct (N × H)
71. C7×C23C2×C3×D15direct (N × H)
75. C2×C3×C23C3×C7×D5semidirect
76. C2×C3×C23C7×D15direct (N × H)
78. C2×C32×C7D5×C23semidirect
80. C5×C23C2×C3×D3×C7semidirect
82. C3×C5×C7C2×D3×C23semidirect
83. C3×C5×C7C22×C3×C23semidirect
85. C2×C32×C5C2×C7×C23semidirect
86. C3×D15C2×C7×C23direct (N × H)
89. C2×C5×C7C3×D3×C23semidirect
92. C3×C23C2×C3×C7×D5semidirect
93. C3×C23C2×C7×D15direct (N × H)
94. C32×C7C2×D5×C23semidirect
95. C2×D15C3×C7×C23direct (N × H)
98. C2×C23C3×C7×D15direct (N × H)
99. C32×C5C22×C7×C23semidirect
102. C2×C3×C7C3×D5×C23semidirect
103. C2×C3×C7C23×D15direct (N × H)
106. C5×C7C2×C3×D3×C23semidirect
108. C2×C3×C5D3×C7×C23semidirect
109. C2×C3×C5C2×C3×C7×C23semidirect
110. D15C2×C3×C7×C23direct (N × H)
113. C23C2×C3×C7×D15direct (N × H)
115. C3×C7C2×C3×D5×C23semidirect
116. C3×C7C2×C23×D15direct (N × H)
118. C2×C32C7×D5×C23semidirect
121. C3×C5C2×D3×C7×C23semidirect
122. C3×C5C22×C3×C7×C23semidirect
124. C2×C7C3×C23×D15direct (N × H)
127. C2×C5C3×D3×C7×C23semidirect
129. C32C2×C7×D5×C23semidirect
130. C7C2×C3×C23×D15direct (N × H)
133. C2×C3C3×C7×D5×C23semidirect
134. C2×C3C7×C23×D15direct (N × H)
136. C5C2×C3×D3×C7×C23semidirect
139. C3C2×C7×C23×D15direct (N × H)
140. C3C2×C3×C7×D5×C23semidirect
142. C2C3×C7×C23×D15direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C7×C23×D152
C2×C32×C5×C7×C232
C2×C3×C7×D5×C233
C2×C7×C23×D153
C2×C3×D3×C7×C235
C2×C3×C23×D157
C2×C3×C7×D1523