Finite Group Databasebeta

← C5×D3×D15×C31 #42124926

Subgroups — C5×D3×D15×C31 #42124926

Order 27900.

212 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 64 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×C52×D3×C31C211yesmaximal, characteristic, complemented
2C3×C5×D15×C31C211yesmaximal, characteristic, complemented
3C5×C31×C3⋊D15C211yesmaximal, characteristic, complemented
4C2×C5×D15×C31—31nomaximal
5C5×D3×D5×C31—31nomaximal
6C32×C52×C31C2211yescharacteristic, complemented, Hall
7D3×D15×C31C511yesmaximal, characteristic, direct product factor
8C5×D32×C31—51nomaximal
9C5×D15×C31—31no—
10C5×D15×C31—31no—
11C5×D15×C31—61no—
12C5×D15×C31D311yescharacteristic, direct product factor
13C52×D3×C31D311yescharacteristic, complemented
14C3×C5×D5×C31—31no—
15C2×C3×C52×C31—31no—
16C2×C5×D5×C31—91noHall
17C31×C3⋊D15C2×C511yescharacteristic, complemented
18C5×C3⋊D3×C31—51no—
19C3×D15×C31C2×C511yescharacteristic, complemented
20C3×C5×D3×C31—22no—
21C3×C5×D3×C31—51no—
22C3×C5×D3×C31D511yescharacteristic, complemented
23C3×C5×D3×C31C2×C511yescharacteristic, complemented
24C3×C52×C31—21no—
25C3×C52×C31C2×D311yescharacteristic, complemented
26C3×C52×C31C2×D311yescharacteristic, complemented
27C2×D15×C31—31no—
28C2×C5×D3×C31—151no—
29C2×C5×D3×C31—151no—
30D3×D5×C31—31no—
31C2×C52×C31—31no—
32C5×D5×C31—31no—
33C5×D5×C31—91no—
34C32×C5×C31—22no—
35C32×C5×C31C2×D511yescharacteristic, complemented
36C32×C5×C31C22×C511yescharacteristic, complemented
37D32×C31—51noHall
38C2×C3×C5×C31—31no—
39C2×C3×C5×C31—31no—
40C2×C3×C5×C31—62no—
41C2×C3×C5×C31—151no—
42D15×C31—31no—
43D15×C31—31no—
44D15×C31—61no—
45D15×C31C5×D311yescharacteristic, direct product factor
46C5×D3×C31—22no—
47C5×D3×C31—51no—
48C5×D3×C31—151no—
49C5×D3×C31—151no—
50C5×D3×C31—301no—
51C5×D3×C31C5×D311yescharacteristic, complemented
52C5×D3×C31D1511yescharacteristic, direct product factor
53C3×D5×C31—31no—
54C5×D3×D15C3111yesmaximal, characteristic, direct product factor, Hall
55C52×C31D3211yescharacteristic, complemented, Hall
56C22×C5×C31—451no—
57C2×D5×C31—91no—
58C3⋊D3×C31—51no—
59C3×D3×C31—51no—
60C3×D3×C31C5×D511yescharacteristic, complemented
61C3×C5×C31—21no—
62C3×C5×C31—21no—
63C3×C5×C31—22no—
64C3×C5×C31—22no—
65C3×C5×C31—42no—
66C3×C5×C31D3×D511yescharacteristic, complemented
67C3×C5×C31C2×C5×D311yescharacteristic, complemented
68C3×C5×C31C2×C5×D311yescharacteristic, complemented
69C3×C5×C31C2×D1511yescharacteristic, complemented
70C5×C3⋊D15C2×C3111yescharacteristic, complemented
71C3×C5×D15C2×C3111yescharacteristic, complemented
72C3×C52×D3C2×C3111yescharacteristic, complemented
73C2×D3×C31—151no—
74C2×D3×C31—151no—
75C2×C5×C31—31no—
76C2×C5×C31—31no—
77C2×C5×C31—62no—
78C2×C5×C31—151no—
79C2×C5×C31—451no—
80D5×C31—31no—
81D5×C31—91no—
82C2×C5×D15—31no—
83C5×D3×D5—31no—
84C32×C31C2×C5×D511yescharacteristic, complemented, Hall
85C32×C52C22×C3111yescharacteristic, complemented, Hall
86C2×C3×C31—31no—
87C2×C3×C31—151no—
88D3×C31—51no—
89D3×C31—151no—
90D3×C31—151no—
91D3×C31—301no—
92D3×C31C5×D1511yescharacteristic, direct product factor
93D3×D15C5×C3111yescharacteristic, direct product factor
94C5×D32—51no—
95C5×C31—22no—
96C5×C31C5×D3211yescharacteristic, complemented
97C5×C31D3×D1511yescharacteristic, direct product factor
98C2×C3×C52—31no—
99C5×D15—31no—
100C5×D15—31no—
101C5×D15—61no—
102C5×D15D3×C3111yescharacteristic, direct product factor
103C52×D3D3×C3111yescharacteristic, complemented
104C3×C5×D5—31no—
105C22×C31—451noHall
106C2×C5×D5—91noHall
107C3×C31—21no—
108C3×C31C5×D3×D511yescharacteristic, complemented
109C3×C31C2×C5×D1511yescharacteristic, complemented
110C3⋊D15C2×C5×C3111yescharacteristic, complemented
111C5×C3⋊D3—51no—
112C3×D15C2×C5×C3111yescharacteristic, complemented
113C3×C5×D3—22no—
114C3×C5×D3—51no—
115C3×C5×D3D5×C3111yescharacteristic, complemented
116C3×C5×D3C2×C5×C3111yescharacteristic, complemented
117C3×C52—21no—
118C3×C52C2×D3×C3111yescharacteristic, complemented
119C3×C52C2×D3×C3111yescharacteristic, complemented
120C2×C31—31no—
121C2×C31—151no—
122C2×C31—451no—
123C2×D15—31no—
124C2×C5×D3—151no—
125C2×C5×D3—151no—
126D3×D5—31no—
127C2×C52—31no—
128C5×D5—31no—
129C5×D5—91no—
130C32×C5—22no—
131C32×C5C2×D5×C3111yescharacteristic, complemented
132C32×C5C22×C5×C3111yescharacteristic, complemented
133D32—51noHall
134C31C5×D3×D1511yescharacteristic, direct product factor, Sylow
135C2×C3×C5—31no—
136C2×C3×C5—31no—
137C2×C3×C5—62no—
138C2×C3×C5—151no—
139D15—31no—
140D15—31no—
141D15—61no—
142D15C5×D3×C3111yescharacteristic, direct product factor
143C3×D5—31no—
144C5×D3—22no—
145C5×D3—51no—
146C5×D3—151no—
147C5×D3—151no—
148C5×D3—301no—
149C5×D3C5×D3×C3111yescharacteristic, complemented
150C5×D3D15×C3111yescharacteristic, direct product factor
151C52D32×C3111yescharacteristic, complemented, Sylow
152C22×C5—451no—
153C2×D5—91no—
154C3⋊D3—51no—
155C3×D3—51no—
156C3×D3C5×D5×C3111yescharacteristic, complemented
157C3×C5—21no—
158C3×C5—21no—
159C3×C5—22no—
160C3×C5—22no—
161C3×C5—42no—
162C3×C5D3×D5×C3111yescharacteristic, complemented
163C3×C5C2×C5×D3×C3111yescharacteristic, complemented
164C3×C5C2×C5×D3×C3111yescharacteristic, complemented
165C3×C5C2×D15×C3111yescharacteristic, complemented
166C2×D3—151no—
167C2×D3—151no—
168C2×C5—31no—
169C2×C5—31no—
170C2×C5—62no—
171C2×C5—151no—
172C2×C5—451no—
173D5—31no—
174D5—91no—
175C32C2×C5×D5×C3111yescharacteristic, complemented, Sylow
176C2×C3—31no—
177C2×C3—151no—
178D3—51no—
179D3—151no—
180D3—151no—
181D3—301no—
182D3C5×D15×C3111yescharacteristic, direct product factor
183C5—22no—
184C5C5×D32×C3111yescharacteristic, complemented
185C5D3×D15×C3111yescharacteristic, direct product factor
186C22—451noSylow
187C3—21no—
188C3C5×D3×D5×C3111yescharacteristic, complemented
189C3C2×C5×D15×C3111yescharacteristic, complemented
190C2—31no—
191C2—151no—
192C2—451no—
subgroupG/H
center, hypercenterC5×C31D3×D15
commutator, cosocle, hypocenterC32×C5C22×C5×C31
FrattiniC1G
Fitting, socleC32×C52×C31C22
LCSG ▷ C32×C5
UCSC1 ◁ C5×C31

Complements

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

NH ≅ G/Ntype
1. C3×C52×D3×C31C2semidirect
2. C3×C5×D15×C31C2semidirect
3. C5×C31×C3⋊D15C2semidirect
6. C32×C52×C31C22semidirect
7. D3×D15×C31C5direct (N × H)
12. C5×D15×C31D3direct (N × H)
13. C52×D3×C31D3semidirect
17. C31×C3⋊D15C2×C5semidirect
19. C3×D15×C31C2×C5semidirect
22. C3×C5×D3×C31D5semidirect
23. C3×C5×D3×C31C2×C5semidirect
25. C3×C52×C31C2×D3semidirect
26. C3×C52×C31C2×D3semidirect
35. C32×C5×C31C2×D5semidirect
36. C32×C5×C31C22×C5semidirect
45. D15×C31C5×D3direct (N × H)
51. C5×D3×C31C5×D3semidirect
52. C5×D3×C31D15direct (N × H)
54. C5×D3×D15C31direct (N × H)
55. C52×C31D32semidirect
60. C3×D3×C31C5×D5semidirect
66. C3×C5×C31D3×D5semidirect
67. C3×C5×C31C2×C5×D3semidirect
68. C3×C5×C31C2×C5×D3semidirect
69. C3×C5×C31C2×D15semidirect
70. C5×C3⋊D15C2×C31semidirect
71. C3×C5×D15C2×C31semidirect
72. C3×C52×D3C2×C31semidirect
84. C32×C31C2×C5×D5semidirect
85. C32×C52C22×C31semidirect
92. D3×C31C5×D15direct (N × H)
93. D3×D15C5×C31direct (N × H)
96. C5×C31C5×D32semidirect
97. C5×C31D3×D15direct (N × H)
102. C5×D15D3×C31direct (N × H)
103. C52×D3D3×C31semidirect
108. C3×C31C5×D3×D5semidirect
109. C3×C31C2×C5×D15semidirect
110. C3⋊D15C2×C5×C31semidirect
112. C3×D15C2×C5×C31semidirect
115. C3×C5×D3D5×C31semidirect
116. C3×C5×D3C2×C5×C31semidirect
118. C3×C52C2×D3×C31semidirect
119. C3×C52C2×D3×C31semidirect
131. C32×C5C2×D5×C31semidirect
132. C32×C5C22×C5×C31semidirect
134. C31C5×D3×D15direct (N × H)
142. D15C5×D3×C31direct (N × H)
149. C5×D3C5×D3×C31semidirect
150. C5×D3D15×C31direct (N × H)
151. C52D32×C31semidirect
156. C3×D3C5×D5×C31semidirect
162. C3×C5D3×D5×C31semidirect
163. C3×C5C2×C5×D3×C31semidirect
164. C3×C5C2×C5×D3×C31semidirect
165. C3×C5C2×D15×C31semidirect
175. C32C2×C5×D5×C31semidirect
182. D3C5×D15×C31direct (N × H)
184. C5C5×D32×C31semidirect
185. C5D3×D15×C31direct (N × H)
188. C3C5×D3×D5×C31semidirect
189. C3C2×C5×D15×C31semidirect

Maximal subgroups

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

subgroupindex
C3×C52×D3×C312
C3×C5×D15×C312
C5×C31×C3⋊D152
C2×C5×D15×C313
C5×D3×D5×C313
D3×D15×C315
C5×D32×C315
C5×D3×D1531