Finite Group Databasebeta

← C2×C3×C5×D15×C31 #42124987

Subgroups — C2×C3×C5×D15×C31 #42124987

Order 27900.

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×D15×C31C212yesmaximal, direct product factor
2C2×C32×C52×C31C211yesmaximal, characteristic, complemented
3C2×C3×C5×D5×C31—31nomaximal
4C2×C5×D15×C31C311yesmaximal, characteristic, direct product factor
5C32×C52×C31C2211yescharacteristic, complemented, Hall
6C2×C3×C5×D3×C31—51nomaximal
7C2×C3×D15×C31C511yesmaximal, characteristic, direct product factor
8C5×D15×C31C2×C312yesdirect product factor
9C3×C5×D5×C31—32no—
10C2×C3×C52×C31—21no—
11C2×C3×C52×C31D311yescharacteristic, complemented
12C2×C3×C52×C31C2×C311yescharacteristic, complemented
13C2×C5×D5×C31—31noHall
14C3×D15×C31C2×C512yesdirect product factor
15C3×C5×D3×C31—52no—
16C2×C32×C5×C31—22no—
17C2×C32×C5×C31D511yescharacteristic, complemented
18C2×C32×C5×C31C2×C511yescharacteristic, complemented
19C3×C52×C31—21no—
20C3×C52×C31C2×D311yescharacteristic, complemented
21C3×C52×C31C22×C311yescharacteristic, complemented
22C22×C3×C5×C31—151no—
23C2×D15×C31C3×C511yescharacteristic, direct product factor
24C2×C5×D3×C31—51no—
25C2×C3×D5×C31—31no—
26C2×C52×C31C3×D311yescharacteristic, complemented
27C5×D5×C31—32no—
28C32×C5×C31—22no—
29C32×C5×C31C2×D511yescharacteristic, complemented
30C32×C5×C31C22×C511yescharacteristic, complemented
31C2×C3×D3×C31—51noHall
32C2×C3×C5×C31—24no—
33C2×C3×C5×C31—21no—
34C2×C3×C5×C31—21no—
35C2×C3×C5×C31—22no—
36C2×C3×C5×C31—22no—
37C2×C3×C5×C31—152no—
38C2×C3×C5×C31C5×D311yescharacteristic, complemented
39C2×C3×C5×C31C3×D511yescharacteristic, complemented
40C2×C3×C5×C31D1511yescharacteristic, direct product factor
41C2×C3×C5×C31C2×C3×C511yescharacteristic, complemented
42D15×C31C2×C3×C512yesdirect product factor
43C5×D3×C31—52no—
44C3×D5×C31—32no—
45C2×C3×C5×D15C3111yesmaximal, characteristic, direct product factor, Hall
46C52×C31C2×C3×D311yescharacteristic, complemented, Hall
47C22×C5×C31—151no—
48C2×D5×C31—31no—
49C2×C32×C31C5×D511yescharacteristic, complemented
50C3×D3×C31—52no—
51C3×C5×C31—24no—
52C3×C5×C31—21no—
53C3×C5×C31—21no—
54C3×C5×C31—22no—
55C3×C5×C31—22no—
56C3×C5×C31C2×C3×D511yescharacteristic, complemented
57C3×C5×C31C2×C5×D311yescharacteristic, complemented
58C3×C5×C31C2×D1511yescharacteristic, direct product factor
59C3×C5×C31C22×C3×C511yescharacteristic, complemented
60C2×C32×C52C2×C3111yescharacteristic, complemented
61C3×C5×D15C2×C3112yesdirect product factor
62C22×C3×C31—151no—
63C2×D3×C31—51no—
64C2×C5×C31—22no—
65C2×C5×C31—152no—
66C2×C5×C31C3×C5×D311yescharacteristic, complemented
67C2×C5×C31C3×D1511yescharacteristic, direct product factor
68D5×C31—32no—
69C2×C5×D15C3×C3111yescharacteristic, direct product factor
70C2×C3×C5×D5—31no—
71C32×C31C2×C5×D511yescharacteristic, complemented, Hall
72C32×C52C22×C3111yescharacteristic, complemented, Hall
73C2×C3×C31—21no—
74C2×C3×C31—152no—
75C2×C3×C31C3×C5×D511yescharacteristic, complemented
76C2×C3×C31C5×D1511yescharacteristic, direct product factor
77D3×C31—52no—
78C2×C3×D15C5×C3111yescharacteristic, direct product factor
79C2×C3×C5×D3—51no—
80C5×C31—22no—
81C5×C31C2×C3×C5×D311yescharacteristic, complemented
82C5×C31C2×C3×D1511yescharacteristic, direct product factor
83C2×C3×C52—21no—
84C2×C3×C52D3×C3111yescharacteristic, complemented
85C2×C3×C52C2×C3×C3111yescharacteristic, complemented
86C5×D15C2×C3×C3112yesdirect product factor
87C3×C5×D5—32no—
88C22×C31—151noHall
89C2×C5×D5—31noHall
90C3×C31—21no—
91C3×C31C2×C3×C5×D511yescharacteristic, complemented
92C3×C31C2×C5×D1511yescharacteristic, direct product factor
93C2×C32×C5—22no—
94C2×C32×C5D5×C3111yescharacteristic, complemented
95C2×C32×C5C2×C5×C3111yescharacteristic, complemented
96C3×D15C2×C5×C3112yesdirect product factor
97C3×C5×D3—52no—
98C3×C52—21no—
99C3×C52C2×D3×C3111yescharacteristic, complemented
100C3×C52C22×C3×C3111yescharacteristic, complemented
101C2×C31—152no—
102C2×C31C3×C5×D1511yescharacteristic, direct product factor
103C22×C3×C5—151no—
104C2×D15C3×C5×C3111yescharacteristic, direct product factor
105C2×C5×D3—51no—
106C2×C3×D5—31no—
107C2×C52C3×D3×C3111yescharacteristic, complemented
108C5×D5—32no—
109C32×C5—22no—
110C32×C5C2×D5×C3111yescharacteristic, complemented
111C32×C5C22×C5×C3111yescharacteristic, complemented
112C2×C3×D3—51noHall
113C31C2×C3×C5×D1511yescharacteristic, direct product factor, Sylow
114C2×C3×C5—24no—
115C2×C3×C5—21no—
116C2×C3×C5—21no—
117C2×C3×C5—22no—
118C2×C3×C5—22no—
119C2×C3×C5—152no—
120C2×C3×C5C3×D5×C3111yescharacteristic, complemented
121C2×C3×C5C5×D3×C3111yescharacteristic, complemented
122C2×C3×C5D15×C3111yescharacteristic, direct product factor
123C2×C3×C5C2×C3×C5×C3111yescharacteristic, complemented
124D15C2×C3×C5×C3112yesdirect product factor
125C3×D5—32no—
126C5×D3—52no—
127C52C2×C3×D3×C3111yescharacteristic, complemented, Sylow
128C22×C5—151no—
129C2×D5—31no—
130C2×C32C5×D5×C3111yescharacteristic, complemented
131C3×D3—52no—
132C3×C5—24no—
133C3×C5—21no—
134C3×C5—21no—
135C3×C5—22no—
136C3×C5—22no—
137C3×C5C2×C3×D5×C3111yescharacteristic, complemented
138C3×C5C2×C5×D3×C3111yescharacteristic, complemented
139C3×C5C2×D15×C3111yescharacteristic, direct product factor
140C3×C5C22×C3×C5×C3111yescharacteristic, complemented
141C22×C3—151no—
142C2×D3—51no—
143C2×C5—22no—
144C2×C5—152no—
145C2×C5C3×C5×D3×C3111yescharacteristic, complemented
146C2×C5C3×D15×C3111yescharacteristic, direct product factor
147D5—32no—
148C32C2×C5×D5×C3111yescharacteristic, complemented, Sylow
149C2×C3—21no—
150C2×C3—152no—
151C2×C3C3×C5×D5×C3111yescharacteristic, complemented
152C2×C3C5×D15×C3111yescharacteristic, direct product factor
153D3—52no—
154C5—22no—
155C5C2×C3×D15×C3111yescharacteristic, direct product factor
156C5C2×C3×C5×D3×C3111yescharacteristic, complemented
157C22—151noSylow
158C3—21no—
159C3C2×C5×D15×C3111yescharacteristic, direct product factor
160C3C2×C3×C5×D5×C3111yescharacteristic, complemented
161C2—152no—
162C2C3×C5×D15×C3111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5×C31D15
commutator, cosocle, hypocenterC3×C5C22×C3×C5×C31
FrattiniC1G
Fitting, socleC2×C32×C52×C31C2
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3×C5×C31

Complements

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

NH ≅ G/Ntype
1. C3×C5×D15×C31C2direct (N × H)
2. C2×C32×C52×C31C2semidirect
4. C2×C5×D15×C31C3direct (N × H)
5. C32×C52×C31C22semidirect
7. C2×C3×D15×C31C5direct (N × H)
8. C5×D15×C31C2×C3direct (N × H)
11. C2×C3×C52×C31D3semidirect
12. C2×C3×C52×C31C2×C3semidirect
14. C3×D15×C31C2×C5direct (N × H)
17. C2×C32×C5×C31D5semidirect
18. C2×C32×C5×C31C2×C5semidirect
20. C3×C52×C31C2×D3semidirect
21. C3×C52×C31C22×C3semidirect
23. C2×D15×C31C3×C5direct (N × H)
26. C2×C52×C31C3×D3semidirect
29. C32×C5×C31C2×D5semidirect
30. C32×C5×C31C22×C5semidirect
38. C2×C3×C5×C31C5×D3semidirect
39. C2×C3×C5×C31C3×D5semidirect
40. C2×C3×C5×C31D15direct (N × H)
41. C2×C3×C5×C31C2×C3×C5semidirect
42. D15×C31C2×C3×C5direct (N × H)
45. C2×C3×C5×D15C31direct (N × H)
46. C52×C31C2×C3×D3semidirect
49. C2×C32×C31C5×D5semidirect
56. C3×C5×C31C2×C3×D5semidirect
57. C3×C5×C31C2×C5×D3semidirect
58. C3×C5×C31C2×D15direct (N × H)
59. C3×C5×C31C22×C3×C5semidirect
60. C2×C32×C52C2×C31semidirect
61. C3×C5×D15C2×C31direct (N × H)
66. C2×C5×C31C3×C5×D3semidirect
67. C2×C5×C31C3×D15direct (N × H)
69. C2×C5×D15C3×C31direct (N × H)
71. C32×C31C2×C5×D5semidirect
72. C32×C52C22×C31semidirect
75. C2×C3×C31C3×C5×D5semidirect
76. C2×C3×C31C5×D15direct (N × H)
78. C2×C3×D15C5×C31direct (N × H)
81. C5×C31C2×C3×C5×D3semidirect
82. C5×C31C2×C3×D15direct (N × H)
84. C2×C3×C52D3×C31semidirect
85. C2×C3×C52C2×C3×C31semidirect
86. C5×D15C2×C3×C31direct (N × H)
91. C3×C31C2×C3×C5×D5semidirect
92. C3×C31C2×C5×D15direct (N × H)
94. C2×C32×C5D5×C31semidirect
95. C2×C32×C5C2×C5×C31semidirect
96. C3×D15C2×C5×C31direct (N × H)
99. C3×C52C2×D3×C31semidirect
100. C3×C52C22×C3×C31semidirect
102. C2×C31C3×C5×D15direct (N × H)
104. C2×D15C3×C5×C31direct (N × H)
107. C2×C52C3×D3×C31semidirect
110. C32×C5C2×D5×C31semidirect
111. C32×C5C22×C5×C31semidirect
113. C31C2×C3×C5×D15direct (N × H)
120. C2×C3×C5C3×D5×C31semidirect
121. C2×C3×C5C5×D3×C31semidirect
122. C2×C3×C5D15×C31direct (N × H)
123. C2×C3×C5C2×C3×C5×C31semidirect
124. D15C2×C3×C5×C31direct (N × H)
127. C52C2×C3×D3×C31semidirect
130. C2×C32C5×D5×C31semidirect
137. C3×C5C2×C3×D5×C31semidirect
138. C3×C5C2×C5×D3×C31semidirect
139. C3×C5C2×D15×C31direct (N × H)
140. C3×C5C22×C3×C5×C31semidirect
145. C2×C5C3×C5×D3×C31semidirect
146. C2×C5C3×D15×C31direct (N × H)
148. C32C2×C5×D5×C31semidirect
151. C2×C3C3×C5×D5×C31semidirect
152. C2×C3C5×D15×C31direct (N × H)
155. C5C2×C3×D15×C31direct (N × H)
156. C5C2×C3×C5×D3×C31semidirect
159. C3C2×C5×D15×C31direct (N × H)
160. C3C2×C3×C5×D5×C31semidirect
162. C2C3×C5×D15×C31direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D15×C312
C2×C32×C52×C312
C2×C3×C5×D5×C313
C2×C5×D15×C313
C2×C3×C5×D3×C315
C2×C3×D15×C315
C2×C3×C5×D1531