Finite Group Databasebeta

← C3×C5×D15×D31 #42124898

Subgroups — C3×C5×D15×D31 #42124898

Order 27900.

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

normal subgroups only

This group has 224 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
1C32×C52×D31C211yesmaximal, characteristic, complemented
2C3×C5×D15×C31C211yesmaximal, characteristic, complemented
3C3×C5×D465C211yesmaximal, characteristic, complemented
4C5×D15×D31C311yesmaximal, characteristic, direct product factor
5C3×C5×D5×D31—31nomaximal
6C32×C52×C31C2211yescharacteristic, complemented, Hall
7C3×D15×D31C511yesmaximal, characteristic, direct product factor
8C3×C5×D3×D31—51nomaximal
9C5×D465C2×C311yescharacteristic, complemented
10C5×D15×C31C2×C311yescharacteristic, complemented
11C3×C5×D155—31no—
12C3×C5×D5×C31—31no—
13C3×C52×D31—21no—
14C3×C52×D31D311yescharacteristic, complemented
15C3×C52×D31C2×C311yescharacteristic, complemented
16C5×D5×D31—31noHall
17C3×D465C2×C511yescharacteristic, complemented
18C3×D15×C31C2×C511yescharacteristic, complemented
19C3×C5×D93—51no—
20C3×C5×D3×C31—51no—
21C32×C5×D31—22no—
22C32×C5×D31D511yescharacteristic, complemented
23C32×C5×D31C2×C511yescharacteristic, complemented
24C3×C52×C31—21no—
25C3×C52×C31C2×D311yescharacteristic, complemented
26C3×C52×C31C22×C311yescharacteristic, complemented
27C2×C3×C5×D31—151no—
28D15×D31C3×C511yescharacteristic, direct product factor
29C5×D3×D31—51no—
30C3×D5×D31—31no—
31C5×D155—31no—
32C5×D5×C31—31no—
33C52×D31C3×D311yescharacteristic, complemented
34C32×C5×C31—22no—
35C32×C5×C31C2×D511yescharacteristic, complemented
36C32×C5×C31C22×C511yescharacteristic, complemented
37C3×D3×D31—51noHall
38C2×C3×C5×C31—151no—
39D465C2×C3×C511yescharacteristic, complemented
40D15×C31C2×C3×C511yescharacteristic, complemented
41C5×D93—51no—
42C5×D3×C31—51no—
43C3×D155—31no—
44C3×D5×C31—31no—
45C3×C5×D31—24no—
46C3×C5×D31—21no—
47C3×C5×D31—21no—
48C3×C5×D31—22no—
49C3×C5×D31—22no—
50C3×C5×D31—151no—
51C3×C5×D31C5×D311yescharacteristic, complemented
52C3×C5×D31C3×D511yescharacteristic, complemented
53C3×C5×D31D1511yescharacteristic, direct product factor
54C3×C5×D31C2×C3×C511yescharacteristic, complemented
55C2×C3×C5×D15—311nomaximal, Hall
56C52×C31C2×C3×D311yescharacteristic, complemented, Hall
57C2×C5×D31—151no—
58D5×D31—31no—
59C3×D93—51no—
60C3×D3×C31—51no—
61C32×D31C5×D511yescharacteristic, complemented
62C3×C5×C31—24no—
63C3×C5×C31—21no—
64C3×C5×C31—21no—
65C3×C5×C31—22no—
66C3×C5×C31—22no—
67C3×C5×C31C2×C3×D511yescharacteristic, complemented
68C3×C5×C31C2×C5×D311yescharacteristic, complemented
69C3×C5×C31C2×D1511yescharacteristic, complemented
70C3×C5×C31C22×C3×C511yescharacteristic, complemented
71C2×C32×C52—311no—
72C3×C5×D15—311no—
73C3×C5×D15D3111yescharacteristic, direct product factor
74C2×C3×D31—151no—
75D3×D31—51no—
76C2×C5×C31—151no—
77D155—31no—
78C5×D31—22no—
79C5×D31—151no—
80C5×D31C3×C5×D311yescharacteristic, complemented
81C5×D31C3×D1511yescharacteristic, direct product factor
82D5×C31—31no—
83C2×C5×D15—311no—
84C2×C3×C5×D5—931no—
85C32×C31C2×C5×D511yescharacteristic, complemented, Hall
86C32×C52C2×D3111yescharacteristic, complemented, Hall
87C2×C3×C31—151no—
88D93—51no—
89C3×D31—21no—
90C3×D31—151no—
91C3×D31C3×C5×D511yescharacteristic, complemented
92C3×D31C5×D1511yescharacteristic, direct product factor
93D3×C31—51no—
94C2×C3×D15—311no—
95C2×C3×C5×D3—1551no—
96C5×C31—22no—
97C5×C31C2×C3×C5×D311yescharacteristic, complemented
98C5×C31C2×C3×D1511yescharacteristic, complemented
99C2×C3×C52—311no—
100C2×C3×C52—311no—
101C2×C3×C52—621no—
102C5×D15—311no—
103C5×D15C3×D3111yescharacteristic, direct product factor
104C3×C5×D5—31no—
105C3×C5×D5—931no—
106C2×D31—151noHall
107C2×C5×D5—931noHall
108C3×C31—21no—
109C3×C31C2×C3×C5×D511yescharacteristic, complemented
110C3×C31C2×C5×D1511yescharacteristic, complemented
111C2×C32×C5—311no—
112C2×C32×C5—311no—
113C2×C32×C5—622no—
114C3×D15—311no—
115C3×D15C5×D3111yescharacteristic, direct product factor
116C3×C5×D3—51no—
117C3×C5×D3—1551no—
118C3×C52—21no—
119C3×C52D3×D3111yescharacteristic, complemented
120C3×C52C2×C3×D3111yescharacteristic, complemented
121C2×C31—151no—
122D31—151no—
123D31C3×C5×D1511yescharacteristic, direct product factor
124C22×C3×C5—4651no—
125C2×D15—311no—
126C2×C5×D3—1551no—
127C2×C3×D5—931no—
128C2×C52—311no—
129C5×D5—31no—
130C5×D5—931no—
131C32×C5—22no—
132C32×C5D5×D3111yescharacteristic, complemented
133C32×C5C2×C5×D3111yescharacteristic, complemented
134C2×C3×D3—1551noHall
135C31C2×C3×C5×D1511yescharacteristic, complemented, Sylow
136C2×C3×C5—151no—
137C2×C3×C5—311no—
138C2×C3×C5—311no—
139C2×C3×C5—311no—
140C2×C3×C5—311no—
141C2×C3×C5—622no—
142C2×C3×C5—622no—
143C2×C3×C5—621no—
144C2×C3×C5—621no—
145C2×C3×C5—624no—
146C2×C3×C5—4651no—
147D15—311no—
148D15C3×C5×D3111yescharacteristic, direct product factor
149C3×D5—31no—
150C3×D5—931no—
151C5×D3—51no—
152C5×D3—1551no—
153C52C3×D3×D3111yescharacteristic, complemented, Sylow
154C22×C5—4651no—
155C2×D5—931no—
156C2×C32—311no—
157C3×D3—51no—
158C3×D3—1551no—
159C3×C5—24no—
160C3×C5—21no—
161C3×C5—21no—
162C3×C5—22no—
163C3×C5—22no—
164C3×C5C3×D5×D3111yescharacteristic, complemented
165C3×C5C5×D3×D3111yescharacteristic, complemented
166C3×C5D15×D3111yescharacteristic, direct product factor
167C3×C5C2×C3×C5×D3111yescharacteristic, complemented
168C22×C3—4651no—
169C2×D3—1551no—
170C2×C5—151no—
171C2×C5—311no—
172C2×C5—311no—
173C2×C5—622no—
174C2×C5—4651no—
175D5—31no—
176D5—931no—
177C32C5×D5×D3111yescharacteristic, complemented, Sylow
178C2×C3—151no—
179C2×C3—311no—
180C2×C3—311no—
181C2×C3—621no—
182C2×C3—4651no—
183D3—51no—
184D3—1551no—
185C5—22no—
186C5C3×C5×D3×D3111yescharacteristic, complemented
187C5C3×D15×D3111yescharacteristic, direct product factor
188C22—4651noSylow
189C3—21no—
190C3C3×C5×D5×D3111yescharacteristic, complemented
191C3C5×D15×D3111yescharacteristic, direct product factor
192C2—151no—
193C2—311no—
194C2—4651no—
subgroupG/H
center, hypercenterC3×C5D15×D31
commutator, cosocle, hypocenterC3×C5×C31C22×C3×C5
FrattiniC1G
Fitting, socleC32×C52×C31C22
LCSG ▷ C3×C5×C31
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C32×C52×D31C2semidirect
2. C3×C5×D15×C31C2semidirect
3. C3×C5×D465C2semidirect
4. C5×D15×D31C3direct (N × H)
6. C32×C52×C31C22semidirect
7. C3×D15×D31C5direct (N × H)
9. C5×D465C2×C3semidirect
10. C5×D15×C31C2×C3semidirect
14. C3×C52×D31D3semidirect
15. C3×C52×D31C2×C3semidirect
17. C3×D465C2×C5semidirect
18. C3×D15×C31C2×C5semidirect
22. C32×C5×D31D5semidirect
23. C32×C5×D31C2×C5semidirect
25. C3×C52×C31C2×D3semidirect
26. C3×C52×C31C22×C3semidirect
28. D15×D31C3×C5direct (N × H)
33. C52×D31C3×D3semidirect
35. C32×C5×C31C2×D5semidirect
36. C32×C5×C31C22×C5semidirect
39. D465C2×C3×C5semidirect
40. D15×C31C2×C3×C5semidirect
51. C3×C5×D31C5×D3semidirect
52. C3×C5×D31C3×D5semidirect
53. C3×C5×D31D15direct (N × H)
54. C3×C5×D31C2×C3×C5semidirect
56. C52×C31C2×C3×D3semidirect
61. C32×D31C5×D5semidirect
67. C3×C5×C31C2×C3×D5semidirect
68. C3×C5×C31C2×C5×D3semidirect
69. C3×C5×C31C2×D15semidirect
70. C3×C5×C31C22×C3×C5semidirect
73. C3×C5×D15D31direct (N × H)
80. C5×D31C3×C5×D3semidirect
81. C5×D31C3×D15direct (N × H)
85. C32×C31C2×C5×D5semidirect
86. C32×C52C2×D31semidirect
91. C3×D31C3×C5×D5semidirect
92. C3×D31C5×D15direct (N × H)
97. C5×C31C2×C3×C5×D3semidirect
98. C5×C31C2×C3×D15semidirect
103. C5×D15C3×D31direct (N × H)
109. C3×C31C2×C3×C5×D5semidirect
110. C3×C31C2×C5×D15semidirect
115. C3×D15C5×D31direct (N × H)
119. C3×C52D3×D31semidirect
120. C3×C52C2×C3×D31semidirect
123. D31C3×C5×D15direct (N × H)
132. C32×C5D5×D31semidirect
133. C32×C5C2×C5×D31semidirect
135. C31C2×C3×C5×D15semidirect
148. D15C3×C5×D31direct (N × H)
153. C52C3×D3×D31semidirect
164. C3×C5C3×D5×D31semidirect
165. C3×C5C5×D3×D31semidirect
166. C3×C5D15×D31direct (N × H)
167. C3×C5C2×C3×C5×D31semidirect
177. C32C5×D5×D31semidirect
186. C5C3×C5×D3×D31semidirect
187. C5C3×D15×D31direct (N × H)
190. C3C3×C5×D5×D31semidirect
191. C3C5×D15×D31direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C52×D312
C3×C5×D15×C312
C3×C5×D4652
C5×D15×D313
C3×C5×D5×D313
C3×D15×D315
C3×C5×D3×D315
C2×C3×C5×D1531