Finite Group Databasebeta

← C3×C5×D3×D5×C31 #42124899

Subgroups — C3×C5×D3×D5×C31 #42124899

Order 27900.

212 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C32×C5×D5×C31C211yesmaximal, characteristic, complemented
2C3×C52×D3×C31C211yesmaximal, characteristic, complemented
3C3×C5×D15×C31C211yesmaximal, characteristic, complemented
4C2×C3×C5×D5×C31—31nomaximal
5C5×D3×D5×C31C311yesmaximal, characteristic, direct product factor
6C32×C52×C31C2211yescharacteristic, complemented, Hall
7C2×C3×C5×D3×C31—51nomaximal
8C3×D3×D5×C31C511yesmaximal, characteristic, direct product factor
9C5×D15×C31C2×C311yescharacteristic, complemented
10C52×D3×C31C2×C311yescharacteristic, complemented
11C3×C5×D5×C31—21no—
12C3×C5×D5×C31—31no—
13C3×C5×D5×C31D311yescharacteristic, direct product factor
14C3×C5×D5×C31C2×C311yescharacteristic, complemented
15C2×C3×C52×C31—31no—
16C2×C5×D5×C31—31noHall
17C3×D15×C31C2×C511yescharacteristic, complemented
18C3×C5×D3×C31—22no—
19C3×C5×D3×C31—51no—
20C3×C5×D3×C31D511yescharacteristic, direct product factor
21C3×C5×D3×C31C2×C511yescharacteristic, complemented
22C32×D5×C31C2×C511yescharacteristic, complemented
23C2×C32×C5×C31—51no—
24C3×C52×C31—21no—
25C3×C52×C31C2×D311yescharacteristic, complemented
26C3×C52×C31C22×C311yescharacteristic, complemented
27C22×C3×C5×C31—151no—
28C2×C5×D3×C31—51no—
29C2×C3×D5×C31—31no—
30D3×D5×C31C3×C511yescharacteristic, direct product factor
31C2×C52×C31—31no—
32C5×D5×C31—31no—
33C5×D5×C31C3×D311yescharacteristic, direct product factor
34C32×C5×C31—22no—
35C32×C5×C31C2×D511yescharacteristic, complemented
36C32×C5×C31C22×C511yescharacteristic, complemented
37C2×C3×D3×C31—51noHall
38C2×C3×C5×C31—31no—
39C2×C3×C5×C31—31no—
40C2×C3×C5×C31—51no—
41C2×C3×C5×C31—51no—
42C2×C3×C5×C31—62no—
43C2×C3×C5×C31—101no—
44C2×C3×C5×C31—151no—
45D15×C31C2×C3×C511yescharacteristic, complemented
46C5×D3×C31—22no—
47C5×D3×C31—51no—
48C5×D3×C31C3×D511yescharacteristic, direct product factor
49C5×D3×C31C2×C3×C511yescharacteristic, complemented
50C3×D5×C31—21no—
51C3×D5×C31—31no—
52C3×D5×C31C5×D311yescharacteristic, direct product factor
53C3×D5×C31C2×C3×C511yescharacteristic, complemented
54C3×C5×D3×D5C3111yesmaximal, characteristic, direct product factor, Hall
55C52×C31C2×C3×D311yescharacteristic, complemented, Hall
56C22×C5×C31—151no—
57C2×D5×C31—31no—
58C2×C32×C31—51no—
59C3×D3×C31—51no—
60C3×D3×C31C5×D511yescharacteristic, direct product factor
61C3×C5×C31—21no—
62C3×C5×C31—21no—
63C3×C5×C31—22no—
64C3×C5×C31—22no—
65C3×C5×C31—42no—
66C3×C5×C31D3×D511yescharacteristic, direct product factor
67C3×C5×C31C2×C3×D511yescharacteristic, complemented
68C3×C5×C31C2×C5×D311yescharacteristic, complemented
69C3×C5×C31C22×C3×C511yescharacteristic, complemented
70C3×C5×D15C2×C3111yescharacteristic, complemented
71C3×C52×D3C2×C3111yescharacteristic, complemented
72C32×C5×D5C2×C3111yescharacteristic, complemented
73C22×C3×C31—151no—
74C2×D3×C31—51no—
75C2×C5×C31—31no—
76C2×C5×C31—31no—
77C2×C5×C31—51no—
78C2×C5×C31—62no—
79C2×C5×C31—151no—
80D5×C31—31no—
81D5×C31C3×C5×D311yescharacteristic, direct product factor
82C2×C3×C5×D5—31no—
83C5×D3×D5C3×C3111yescharacteristic, direct product factor
84C32×C31C2×C5×D511yescharacteristic, complemented, Hall
85C32×C52C22×C3111yescharacteristic, complemented, Hall
86C2×C3×C31—31no—
87C2×C3×C31—51no—
88C2×C3×C31—51no—
89C2×C3×C31—101no—
90C2×C3×C31—151no—
91D3×C31—51no—
92D3×C31C3×C5×D511yescharacteristic, direct product factor
93C2×C3×C5×D3—51no—
94C3×D3×D5C5×C3111yescharacteristic, direct product factor
95C5×C31—22no—
96C5×C31C3×D3×D511yescharacteristic, direct product factor
97C5×C31C2×C3×C5×D311yescharacteristic, complemented
98C2×C3×C52—31no—
99C5×D15C2×C3×C3111yescharacteristic, complemented
100C52×D3C2×C3×C3111yescharacteristic, complemented
101C3×C5×D5—21no—
102C3×C5×D5—31no—
103C3×C5×D5D3×C3111yescharacteristic, direct product factor
104C3×C5×D5C2×C3×C3111yescharacteristic, complemented
105C22×C31—151noHall
106C2×C5×D5—31noHall
107C3×C31—21no—
108C3×C31C5×D3×D511yescharacteristic, direct product factor
109C3×C31C2×C3×C5×D511yescharacteristic, complemented
110C2×C32×C5—51no—
111C3×D15C2×C5×C3111yescharacteristic, complemented
112C3×C5×D3—22no—
113C3×C5×D3—51no—
114C3×C5×D3D5×C3111yescharacteristic, direct product factor
115C3×C5×D3C2×C5×C3111yescharacteristic, complemented
116C32×D5C2×C5×C3111yescharacteristic, complemented
117C3×C52—21no—
118C3×C52C2×D3×C3111yescharacteristic, complemented
119C3×C52C22×C3×C3111yescharacteristic, complemented
120C2×C31—31no—
121C2×C31—51no—
122C2×C31—151no—
123C22×C3×C5—151no—
124C2×C5×D3—51no—
125C2×C3×D5—31no—
126D3×D5C3×C5×C3111yescharacteristic, direct product factor
127C2×C52—31no—
128C5×D5—31no—
129C5×D5C3×D3×C3111yescharacteristic, direct product factor
130C32×C5—22no—
131C32×C5C2×D5×C3111yescharacteristic, complemented
132C32×C5C22×C5×C3111yescharacteristic, complemented
133C2×C3×D3—51noHall
134C31C3×C5×D3×D511yescharacteristic, direct product factor, Sylow
135C2×C3×C5—31no—
136C2×C3×C5—31no—
137C2×C3×C5—51no—
138C2×C3×C5—51no—
139C2×C3×C5—62no—
140C2×C3×C5—101no—
141C2×C3×C5—151no—
142D15C2×C3×C5×C3111yescharacteristic, complemented
143C3×D5—21no—
144C3×D5—31no—
145C3×D5C5×D3×C3111yescharacteristic, direct product factor
146C3×D5C2×C3×C5×C3111yescharacteristic, complemented
147C5×D3—22no—
148C5×D3—51no—
149C5×D3C3×D5×C3111yescharacteristic, direct product factor
150C5×D3C2×C3×C5×C3111yescharacteristic, complemented
151C52C2×C3×D3×C3111yescharacteristic, complemented, Sylow
152C22×C5—151no—
153C2×D5—31no—
154C2×C32—51no—
155C3×D3—51no—
156C3×D3C5×D5×C3111yescharacteristic, direct product factor
157C3×C5—21no—
158C3×C5—21no—
159C3×C5—22no—
160C3×C5—22no—
161C3×C5—42no—
162C3×C5D3×D5×C3111yescharacteristic, direct product factor
163C3×C5C2×C3×D5×C3111yescharacteristic, complemented
164C3×C5C2×C5×D3×C3111yescharacteristic, complemented
165C3×C5C22×C3×C5×C3111yescharacteristic, complemented
166C22×C3—151no—
167C2×D3—51no—
168C2×C5—31no—
169C2×C5—31no—
170C2×C5—51no—
171C2×C5—62no—
172C2×C5—151no—
173D5—31no—
174D5C3×C5×D3×C3111yescharacteristic, direct product factor
175C32C2×C5×D5×C3111yescharacteristic, complemented, Sylow
176C2×C3—31no—
177C2×C3—51no—
178C2×C3—51no—
179C2×C3—101no—
180C2×C3—151no—
181D3—51no—
182D3C3×C5×D5×C3111yescharacteristic, direct product factor
183C5—22no—
184C5C3×D3×D5×C3111yescharacteristic, direct product factor
185C5C2×C3×C5×D3×C3111yescharacteristic, complemented
186C22—151noSylow
187C3—21no—
188C3C5×D3×D5×C3111yescharacteristic, direct product factor
189C3C2×C3×C5×D5×C3111yescharacteristic, complemented
190C2—31no—
191C2—51no—
192C2—151no—
subgroupG/H
center, hypercenterC3×C5×C31D3×D5
commutator, cosocle, hypocenterC3×C5C22×C3×C5×C31
FrattiniC1G
Fitting, socleC32×C52×C31C22
LCSG ▷ C3×C5
UCSC1 ◁ C3×C5×C31

Complements

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

NH ≅ G/Ntype
1. C32×C5×D5×C31C2semidirect
2. C3×C52×D3×C31C2semidirect
3. C3×C5×D15×C31C2semidirect
5. C5×D3×D5×C31C3direct (N × H)
6. C32×C52×C31C22semidirect
8. C3×D3×D5×C31C5direct (N × H)
9. C5×D15×C31C2×C3semidirect
10. C52×D3×C31C2×C3semidirect
13. C3×C5×D5×C31D3direct (N × H)
14. C3×C5×D5×C31C2×C3semidirect
17. C3×D15×C31C2×C5semidirect
20. C3×C5×D3×C31D5direct (N × H)
21. C3×C5×D3×C31C2×C5semidirect
22. C32×D5×C31C2×C5semidirect
25. C3×C52×C31C2×D3semidirect
26. C3×C52×C31C22×C3semidirect
30. D3×D5×C31C3×C5direct (N × H)
33. C5×D5×C31C3×D3direct (N × H)
35. C32×C5×C31C2×D5semidirect
36. C32×C5×C31C22×C5semidirect
45. D15×C31C2×C3×C5semidirect
48. C5×D3×C31C3×D5direct (N × H)
49. C5×D3×C31C2×C3×C5semidirect
52. C3×D5×C31C5×D3direct (N × H)
53. C3×D5×C31C2×C3×C5semidirect
54. C3×C5×D3×D5C31direct (N × H)
55. C52×C31C2×C3×D3semidirect
60. C3×D3×C31C5×D5direct (N × H)
66. C3×C5×C31D3×D5direct (N × H)
67. C3×C5×C31C2×C3×D5semidirect
68. C3×C5×C31C2×C5×D3semidirect
69. C3×C5×C31C22×C3×C5semidirect
70. C3×C5×D15C2×C31semidirect
71. C3×C52×D3C2×C31semidirect
72. C32×C5×D5C2×C31semidirect
81. D5×C31C3×C5×D3direct (N × H)
83. C5×D3×D5C3×C31direct (N × H)
84. C32×C31C2×C5×D5semidirect
85. C32×C52C22×C31semidirect
92. D3×C31C3×C5×D5direct (N × H)
94. C3×D3×D5C5×C31direct (N × H)
96. C5×C31C3×D3×D5direct (N × H)
97. C5×C31C2×C3×C5×D3semidirect
99. C5×D15C2×C3×C31semidirect
100. C52×D3C2×C3×C31semidirect
103. C3×C5×D5D3×C31direct (N × H)
104. C3×C5×D5C2×C3×C31semidirect
108. C3×C31C5×D3×D5direct (N × H)
109. C3×C31C2×C3×C5×D5semidirect
111. C3×D15C2×C5×C31semidirect
114. C3×C5×D3D5×C31direct (N × H)
115. C3×C5×D3C2×C5×C31semidirect
116. C32×D5C2×C5×C31semidirect
118. C3×C52C2×D3×C31semidirect
119. C3×C52C22×C3×C31semidirect
126. D3×D5C3×C5×C31direct (N × H)
129. C5×D5C3×D3×C31direct (N × H)
131. C32×C5C2×D5×C31semidirect
132. C32×C5C22×C5×C31semidirect
134. C31C3×C5×D3×D5direct (N × H)
142. D15C2×C3×C5×C31semidirect
145. C3×D5C5×D3×C31direct (N × H)
146. C3×D5C2×C3×C5×C31semidirect
149. C5×D3C3×D5×C31direct (N × H)
150. C5×D3C2×C3×C5×C31semidirect
151. C52C2×C3×D3×C31semidirect
156. C3×D3C5×D5×C31direct (N × H)
162. C3×C5D3×D5×C31direct (N × H)
163. C3×C5C2×C3×D5×C31semidirect
164. C3×C5C2×C5×D3×C31semidirect
165. C3×C5C22×C3×C5×C31semidirect
174. D5C3×C5×D3×C31direct (N × H)
175. C32C2×C5×D5×C31semidirect
182. D3C3×C5×D5×C31direct (N × H)
184. C5C3×D3×D5×C31direct (N × H)
185. C5C2×C3×C5×D3×C31semidirect
188. C3C5×D3×D5×C31direct (N × H)
189. C3C2×C3×C5×D5×C31semidirect

Maximal subgroups

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

subgroupindex
C32×C5×D5×C312
C3×C52×D3×C312
C3×C5×D15×C312
C2×C3×C5×D5×C313
C5×D3×D5×C313
C2×C3×C5×D3×C315
C3×D3×D5×C315
C3×C5×D3×D531