Finite Group Databasebeta

← C2×C3×C5×C7×D15 #20263165

Subgroups — C2×C3×C5×C7×D15 #20263165

Order 6300.

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

normal subgroups only

This group has 224 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×D15C212yesmaximal, direct product factor
2C2×C32×C52×C7C211yesmaximal, characteristic, complemented
3C2×C5×C7×D15C311yesmaximal, characteristic, direct product factor
4C2×C3×C5×C7×D5—31nomaximal
5C32×C52×C7C2211yescharacteristic, complemented, Hall
6C2×C3×C7×D15C511yesmaximal, characteristic, direct product factor
7C2×C3×C5×D3×C7—51nomaximal
8C2×C3×C52×C7—21no—
9C2×C3×C52×C7D311yescharacteristic, complemented
10C2×C3×C52×C7C2×C311yescharacteristic, complemented
11C5×C7×D15C2×C312yesdirect product factor
12C3×C5×C7×D5—32no—
13C2×C3×C5×D15C711yesmaximal, characteristic, direct product factor, Hall
14C2×C5×C7×D5—31noHall
15C2×C32×C5×C7—22no—
16C2×C32×C5×C7D511yescharacteristic, complemented
17C2×C32×C5×C7C2×C511yescharacteristic, complemented
18C3×C7×D15C2×C512yesdirect product factor
19C3×C5×D3×C7—52no—
20C3×C52×C7—21no—
21C3×C52×C7C2×D311yescharacteristic, complemented
22C3×C52×C7C22×C311yescharacteristic, complemented
23C2×C32×C52C2×C711yescharacteristic, complemented
24C3×C5×D15C2×C712yesdirect product factor
25C22×C3×C5×C7—151no—
26C2×C7×D15C3×C511yescharacteristic, direct product factor
27C2×C5×D3×C7—51no—
28C2×C3×C7×D5—31no—
29C2×C52×C7C3×D311yescharacteristic, complemented
30C5×C7×D5—32no—
31C32×C5×C7—22no—
32C32×C5×C7C2×D511yescharacteristic, complemented
33C32×C5×C7C22×C511yescharacteristic, complemented
34C2×C5×D15C3×C711yescharacteristic, direct product factor
35C2×C3×C5×D5—31no—
36C2×C3×D3×C7—51noHall
37C32×C52C22×C711yescharacteristic, complemented, Hall
38C2×C3×C5×C7—24no—
39C2×C3×C5×C7—22no—
40C2×C3×C5×C7—22no—
41C2×C3×C5×C7—21no—
42C2×C3×C5×C7—21no—
43C2×C3×C5×C7—152no—
44C2×C3×C5×C7C5×D311yescharacteristic, complemented
45C2×C3×C5×C7C3×D511yescharacteristic, complemented
46C2×C3×C5×C7D1511yescharacteristic, direct product factor
47C2×C3×C5×C7C2×C3×C511yescharacteristic, complemented
48C7×D15C2×C3×C512yesdirect product factor
49C5×D3×C7—52no—
50C3×C7×D5—32no—
51C2×C3×D15C5×C711yescharacteristic, direct product factor
52C2×C3×C5×D3—51no—
53C52×C7C2×C3×D311yescharacteristic, complemented, Hall
54C2×C3×C52—21no—
55C2×C3×C52D3×C711yescharacteristic, complemented
56C2×C3×C52C2×C3×C711yescharacteristic, complemented
57C5×D15C2×C3×C712yesdirect product factor
58C3×C5×D5—32no—
59C22×C5×C7—151no—
60C2×C7×D5—31no—
61C2×C32×C7C5×D511yescharacteristic, complemented
62C3×D3×C7—52no—
63C3×C5×C7—24no—
64C3×C5×C7—21no—
65C3×C5×C7—21no—
66C3×C5×C7—22no—
67C3×C5×C7—22no—
68C3×C5×C7C2×C3×D511yescharacteristic, complemented
69C3×C5×C7C2×C5×D311yescharacteristic, complemented
70C3×C5×C7C2×D1511yescharacteristic, direct product factor
71C3×C5×C7C22×C3×C511yescharacteristic, complemented
72C2×C5×D5—31noHall
73C2×C32×C5—22no—
74C2×C32×C5C7×D511yescharacteristic, complemented
75C2×C32×C5C2×C5×C711yescharacteristic, complemented
76C3×D15C2×C5×C712yesdirect product factor
77C3×C5×D3—52no—
78C22×C3×C7—151no—
79C2×D3×C7—51no—
80C3×C52—21no—
81C3×C52C2×D3×C711yescharacteristic, complemented
82C3×C52C22×C3×C711yescharacteristic, complemented
83C2×C5×C7—22no—
84C2×C5×C7—152no—
85C2×C5×C7C3×C5×D311yescharacteristic, complemented
86C2×C5×C7C3×D1511yescharacteristic, direct product factor
87C7×D5—32no—
88C32×C7C2×C5×D511yescharacteristic, complemented, Hall
89C22×C3×C5—151no—
90C2×D15C3×C5×C711yescharacteristic, direct product factor
91C2×C5×D3—51no—
92C2×C3×D5—31no—
93C2×C52C3×D3×C711yescharacteristic, complemented
94C5×D5—32no—
95C32×C5—22no—
96C32×C5C2×C7×D511yescharacteristic, complemented
97C32×C5C22×C5×C711yescharacteristic, complemented
98C2×C3×C7—21no—
99C2×C3×C7—152no—
100C2×C3×C7C3×C5×D511yescharacteristic, complemented
101C2×C3×C7C5×D1511yescharacteristic, direct product factor
102D3×C7—52no—
103C2×C3×D3—51noHall
104C5×C7—22no—
105C5×C7C2×C3×C5×D311yescharacteristic, complemented
106C5×C7C2×C3×D1511yescharacteristic, direct product factor
107C2×C3×C5—24no—
108C2×C3×C5—21no—
109C2×C3×C5—21no—
110C2×C3×C5—22no—
111C2×C3×C5—22no—
112C2×C3×C5—152no—
113C2×C3×C5C3×C7×D511yescharacteristic, complemented
114C2×C3×C5C5×D3×C711yescharacteristic, complemented
115C2×C3×C5C7×D1511yescharacteristic, direct product factor
116C2×C3×C5C2×C3×C5×C711yescharacteristic, complemented
117D15C2×C3×C5×C712yesdirect product factor
118C3×D5—32no—
119C5×D3—52no—
120C22×C7—151noHall
121C52C2×C3×D3×C711yescharacteristic, complemented, Sylow
122C3×C7—21no—
123C3×C7C2×C3×C5×D511yescharacteristic, complemented
124C3×C7C2×C5×D1511yescharacteristic, direct product factor
125C22×C5—151no—
126C2×D5—31no—
127C2×C32C5×C7×D511yescharacteristic, complemented
128C3×D3—52no—
129C3×C5—24no—
130C3×C5—21no—
131C3×C5—21no—
132C3×C5—22no—
133C3×C5—22no—
134C3×C5C2×C3×C7×D511yescharacteristic, complemented
135C3×C5C2×C5×D3×C711yescharacteristic, complemented
136C3×C5C2×C7×D1511yescharacteristic, direct product factor
137C3×C5C22×C3×C5×C711yescharacteristic, complemented
138C2×C7—152no—
139C2×C7C3×C5×D1511yescharacteristic, direct product factor
140C22×C3—151no—
141C2×D3—51no—
142C2×C5—22no—
143C2×C5—152no—
144C2×C5C3×C5×D3×C711yescharacteristic, complemented
145C2×C5C3×C7×D1511yescharacteristic, direct product factor
146D5—32no—
147C32C2×C5×C7×D511yescharacteristic, complemented, Sylow
148C7C2×C3×C5×D1511yescharacteristic, direct product factor, Sylow
149C2×C3—21no—
150C2×C3—152no—
151C2×C3C3×C5×C7×D511yescharacteristic, complemented
152C2×C3C5×C7×D1511yescharacteristic, direct product factor
153D3—52no—
154C5—22no—
155C5C2×C3×C5×D3×C711yescharacteristic, complemented
156C5C2×C3×C7×D1511yescharacteristic, direct product factor
157C22—151noSylow
158C3—21no—
159C3C2×C3×C5×C7×D511yescharacteristic, complemented
160C3C2×C5×C7×D1511yescharacteristic, direct product factor
161C2—152no—
162C2C3×C5×C7×D1511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5×C7D15
commutator, hypocenterC3×C5C22×C3×C5×C7
FrattiniC1G
Fitting, socleC2×C32×C52×C7C2
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3×C5×C7

Complements

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

NH ≅ G/Ntype
1. C3×C5×C7×D15C2direct (N × H)
2. C2×C32×C52×C7C2semidirect
3. C2×C5×C7×D15C3direct (N × H)
5. C32×C52×C7C22semidirect
6. C2×C3×C7×D15C5direct (N × H)
9. C2×C3×C52×C7D3semidirect
10. C2×C3×C52×C7C2×C3semidirect
11. C5×C7×D15C2×C3direct (N × H)
13. C2×C3×C5×D15C7direct (N × H)
16. C2×C32×C5×C7D5semidirect
17. C2×C32×C5×C7C2×C5semidirect
18. C3×C7×D15C2×C5direct (N × H)
21. C3×C52×C7C2×D3semidirect
22. C3×C52×C7C22×C3semidirect
23. C2×C32×C52C2×C7semidirect
24. C3×C5×D15C2×C7direct (N × H)
26. C2×C7×D15C3×C5direct (N × H)
29. C2×C52×C7C3×D3semidirect
32. C32×C5×C7C2×D5semidirect
33. C32×C5×C7C22×C5semidirect
34. C2×C5×D15C3×C7direct (N × H)
37. C32×C52C22×C7semidirect
44. C2×C3×C5×C7C5×D3semidirect
45. C2×C3×C5×C7C3×D5semidirect
46. C2×C3×C5×C7D15direct (N × H)
47. C2×C3×C5×C7C2×C3×C5semidirect
48. C7×D15C2×C3×C5direct (N × H)
51. C2×C3×D15C5×C7direct (N × H)
53. C52×C7C2×C3×D3semidirect
55. C2×C3×C52D3×C7semidirect
56. C2×C3×C52C2×C3×C7semidirect
57. C5×D15C2×C3×C7direct (N × H)
61. C2×C32×C7C5×D5semidirect
68. C3×C5×C7C2×C3×D5semidirect
69. C3×C5×C7C2×C5×D3semidirect
70. C3×C5×C7C2×D15direct (N × H)
71. C3×C5×C7C22×C3×C5semidirect
74. C2×C32×C5C7×D5semidirect
75. C2×C32×C5C2×C5×C7semidirect
76. C3×D15C2×C5×C7direct (N × H)
81. C3×C52C2×D3×C7semidirect
82. C3×C52C22×C3×C7semidirect
85. C2×C5×C7C3×C5×D3semidirect
86. C2×C5×C7C3×D15direct (N × H)
88. C32×C7C2×C5×D5semidirect
90. C2×D15C3×C5×C7direct (N × H)
93. C2×C52C3×D3×C7semidirect
96. C32×C5C2×C7×D5semidirect
97. C32×C5C22×C5×C7semidirect
100. C2×C3×C7C3×C5×D5semidirect
101. C2×C3×C7C5×D15direct (N × H)
105. C5×C7C2×C3×C5×D3semidirect
106. C5×C7C2×C3×D15direct (N × H)
113. C2×C3×C5C3×C7×D5semidirect
114. C2×C3×C5C5×D3×C7semidirect
115. C2×C3×C5C7×D15direct (N × H)
116. C2×C3×C5C2×C3×C5×C7semidirect
117. D15C2×C3×C5×C7direct (N × H)
121. C52C2×C3×D3×C7semidirect
123. C3×C7C2×C3×C5×D5semidirect
124. C3×C7C2×C5×D15direct (N × H)
127. C2×C32C5×C7×D5semidirect
134. C3×C5C2×C3×C7×D5semidirect
135. C3×C5C2×C5×D3×C7semidirect
136. C3×C5C2×C7×D15direct (N × H)
137. C3×C5C22×C3×C5×C7semidirect
139. C2×C7C3×C5×D15direct (N × H)
144. C2×C5C3×C5×D3×C7semidirect
145. C2×C5C3×C7×D15direct (N × H)
147. C32C2×C5×C7×D5semidirect
148. C7C2×C3×C5×D15direct (N × H)
151. C2×C3C3×C5×C7×D5semidirect
152. C2×C3C5×C7×D15direct (N × H)
155. C5C2×C3×C5×D3×C7semidirect
156. C5C2×C3×C7×D15direct (N × H)
159. C3C2×C3×C5×C7×D5semidirect
160. C3C2×C5×C7×D15direct (N × H)
162. C2C3×C5×C7×D15direct (N × H)