Subgroups — C₃×C₄×D₃×Dic₅ #13086526

Order 1440.

462 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₃×Dic₅	C₂	1	2	yes	maximal
3	C₃×C₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₄×Dic₅	—	3	1	no	maximal
7	C₄×D₃×Dic₅	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
11	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×D₃	C₄	1	4	yes	complemented
13	C₂×C₃²×Dic₅	C₂²	1	2	yes	—
14	C₃×D₃×Dic₅	C₄	1	8	yes	direct product factor
15	C₃×C₄²×D₃	—	5	1	no	maximal, Hall
16	C₂²×C₃×C₄×C₅	—	3	1	no	—
17	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	—	3	2	no	—
19	C₂×D₃×Dic₅	C₂×C₃	1	2	yes	—
20	C₄×Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₄×Dic₅	—	2	1	no	—
22	C₃×C₄×Dic₅	—	3	1	no	—
23	C₃×C₄×Dic₅	—	3	2	no	—
24	C₃×C₄×Dic₅	D₃	1	1	yes	characteristic, direct product factor
25	C₃×C₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
26	Dic₃×Dic₅	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×D₃	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
30	C₃²×C₄×C₅	C₂×C₄	1	2	yes	complemented
31	C₃×Dic₁₅	C₂×C₄	1	4	yes	complemented
32	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
33	C₃²×Dic₅	C₂×C₄	1	4	yes	complemented
34	C₂×C₄×Dic₅	—	3	1	no	Hall
35	C₂×C₃×C₄×D₃	—	5	2	no	—
36	C₂×C₃×C₄×D₃	D₅	1	1	yes	characteristic
37	C₃²×C₄²	—	5	1	no	—
38	C₃×C₄×Dic₃	—	5	1	no	—
39	C₃×C₄×Dic₃	—	5	2	no	—
40	C₂³×C₃×C₅	—	3	1	no	—
41	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	1	no	—
43	C₂×C₃×C₄×C₅	—	3	1	no	—
44	C₂×C₃×C₄×C₅	—	3	4	no	—
45	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₂×Dic₁₅	C₂²×C₃	1	2	yes	—
48	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
49	C₄×C₅×D₃	C₃×C₄	1	4	yes	complemented
50	C₂×C₃×Dic₅	—	2	2	no	—
51	C₂×C₃×Dic₅	—	3	2	no	—
52	C₂×C₃×Dic₅	—	3	8	no	—
53	C₂×C₃×Dic₅	C₂×D₃	1	2	yes	—
54	C₂×C₃×Dic₅	C₂²×C₃	1	2	yes	—
55	D₃×Dic₅	C₃×C₄	1	8	yes	direct product factor
56	C₂×C₃×C₄²	—	15	1	no	—
57	C₄²×D₃	—	5	1	no	—
58	C₂×C₃²×C₅	C₂²×C₄	1	2	yes	—
59	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₅×D₃	C₄²	1	4	yes	complemented
61	C₂²×C₄×C₅	—	3	1	no	—
62	C₂²×Dic₅	—	3	2	no	—
63	C₄×Dic₅	—	3	1	no	—
64	C₄×Dic₅	—	3	2	no	—
65	C₄×Dic₅	C₃×D₃	1	1	yes	characteristic, direct product factor
66	C₂²×C₃×D₃	C₂×D₅	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	5	2	no	—
68	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
69	C₂×C₃×Dic₃	—	5	2	no	—
70	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
71	C₃×C₄×D₃	—	5	8	no	—
72	C₃×C₄×D₃	Dic₅	1	4	yes	direct product factor
73	C₂²×C₃×C₅	—	2	1	no	—
74	C₂²×C₃×C₅	—	3	4	no	—
75	C₂²×C₃×C₅	—	3	2	no	—
76	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₅×D₃	C₂×C₃×C₄	1	4	yes	—
79	C₂×C₅×D₃	C₂×C₃×C₄	1	2	yes	—
80	C₃×C₄×C₅	—	2	2	no	—
81	C₃×C₄×C₅	—	3	2	no	—
82	C₃×C₄×C₅	C₄×D₃	1	2	yes	complemented
83	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	complemented
84	Dic₁₅	C₂×C₃×C₄	1	4	yes	complemented
85	C₃×Dic₅	—	2	4	no	—
86	C₃×Dic₅	—	3	4	no	—
87	C₃×Dic₅	C₄×D₃	1	4	yes	direct product factor
88	C₃×Dic₅	C₂×C₃×C₄	1	4	yes	complemented
89	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	complemented
90	C₂²×C₃×C₄	—	3	1	no	—
91	C₂²×C₃×C₄	—	15	2	no	—
92	C₂×C₄×D₃	—	5	2	no	—
93	C₂×C₄×D₃	C₃×D₅	1	1	yes	characteristic
94	C₃×C₄²	—	5	1	no	—
95	C₃×C₄²	—	5	1	no	—
96	C₃×C₄²	—	10	1	no	—
97	C₃×C₄²	—	15	2	no	—
98	C₃×C₄²	—	15	1	no	—
99	C₄×Dic₃	—	5	1	no	—
100	C₄×Dic₃	—	5	2	no	—
101	C₃²×C₅	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
102	C₂³×C₅	—	3	1	no	—
103	C₂×C₄×C₅	—	3	1	no	—
104	C₂×C₄×C₅	—	3	4	no	—
105	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
106	C₂×Dic₅	—	3	2	no	—
107	C₂×Dic₅	—	3	8	no	—
108	C₂×Dic₅	C₂×C₃×D₃	1	2	yes	—
109	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
110	C₂×C₃×D₃	C₄×D₅	1	4	yes	—
111	C₂×C₃×D₃	C₂×Dic₅	1	2	yes	—
112	C₃²×C₄	—	5	4	no	—
113	C₃²×C₄	C₂×Dic₅	1	2	yes	complemented
114	C₃×Dic₃	—	5	4	no	—
115	C₃×Dic₃	C₂×Dic₅	1	2	yes	complemented
116	C₂×C₄²	—	15	1	no	Sylow
117	C₂×C₃×C₅	—	2	2	no	—
118	C₂×C₃×C₅	—	2	1	no	—
119	C₂×C₃×C₅	—	3	4	no	—
120	C₂×C₃×C₅	C₂×C₄×D₃	1	2	yes	—
121	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂²×C₃×C₄	1	2	yes	—
123	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₅×D₃	C₃×C₄²	1	4	yes	complemented
125	C₂³×C₃	—	3	1	no	—
126	C₂²×D₃	C₂×C₃×D₅	1	1	yes	characteristic
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	3	1	no	—
129	C₂×C₃×C₄	—	3	4	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	5	2	no	—
132	C₂×C₃×C₄	—	10	2	no	—
133	C₂×C₃×C₄	—	15	8	no	—
134	C₂×C₃×C₄	—	15	2	no	—
135	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
137	C₂×Dic₃	—	5	2	no	—
138	C₂×Dic₃	C₂×C₃×D₅	1	1	yes	characteristic
139	C₄×D₃	—	5	8	no	—
140	C₄×D₃	C₃×Dic₅	1	4	yes	direct product factor
141	C₂²×C₅	—	3	4	no	—
142	C₂²×C₅	—	3	2	no	—
143	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
144	C₄×C₅	—	3	2	no	—
145	C₄×C₅	C₃×C₄×D₃	1	2	yes	complemented
146	Dic₅	—	3	4	no	—
147	Dic₅	C₃×C₄×D₃	1	4	yes	direct product factor
148	C₂×C₃²	C₂×C₄×D₅	1	2	yes	—
149	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
150	C₃×D₃	C₄×Dic₅	1	4	yes	direct product factor
151	C₂²×C₄	—	3	1	no	—
152	C₂²×C₄	—	15	2	no	—
153	C₄²	—	5	1	no	—
154	C₄²	—	15	2	no	—
155	C₄²	—	15	1	no	—
156	C₃×C₅	—	2	1	no	—
157	C₃×C₅	C₄²×D₃	1	1	yes	characteristic, complemented
158	C₃×C₅	C₂×C₃×C₄²	1	1	yes	characteristic, complemented
159	C₂²×C₃	—	2	1	no	—
160	C₂²×C₃	—	3	4	no	—
161	C₂²×C₃	—	3	2	no	—
162	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
163	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
164	C₂×D₃	C₃×C₄×D₅	1	4	yes	—
165	C₂×D₃	C₂×C₃×Dic₅	1	2	yes	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	3	2	no	—
168	C₃×C₄	—	5	4	no	—
169	C₃×C₄	—	5	4	no	—
170	C₃×C₄	—	10	4	no	—
171	C₃×C₄	—	15	4	no	—
172	C₃×C₄	D₃×Dic₅	1	2	yes	direct product factor
173	C₃×C₄	C₂×C₃×Dic₅	1	2	yes	complemented
174	Dic₃	—	5	4	no	—
175	Dic₃	C₂×C₃×Dic₅	1	2	yes	complemented
176	C₂×C₅	—	3	4	no	—
177	C₂×C₅	C₂×C₃×C₄×D₃	1	2	yes	—
178	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
179	C₃²	C₂×C₄×Dic₅	1	1	yes	characteristic, complemented, Sylow
180	C₂³	—	3	1	no	—
181	C₂×C₄	—	3	1	no	—
182	C₂×C₄	—	3	4	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	15	8	no	—
185	C₂×C₄	—	15	2	no	—
186	C₂×C₄	C₃×D₃×D₅	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	—	3	4	no	—
190	C₂×C₃	C₄×D₃×D₅	1	2	yes	—
191	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
192	C₂×C₃	C₂×C₃×C₄×D₅	1	2	yes	—
193	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
194	D₃	C₃×C₄×Dic₅	1	4	yes	direct product factor
195	C₅	C₃×C₄²×D₃	1	1	yes	characteristic, complemented, Sylow
196	C₂²	—	3	4	no	—
197	C₂²	—	3	2	no	—
198	C₂²	C₂×C₃×D₃×D₅	1	1	yes	characteristic
199	C₄	—	3	2	no	—
200	C₄	—	5	4	no	—
201	C₄	—	15	4	no	—
202	C₄	C₃×D₃×Dic₅	1	2	yes	direct product factor
203	C₃	—	2	1	no	—
204	C₃	C₄×D₃×Dic₅	1	1	yes	characteristic, direct product factor
205	C₃	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic, complemented
206	C₂	—	3	4	no	—
207	C₂	C₃×C₄×D₃×D₅	1	2	yes	—
208	C₂	C₂×C₃×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂×C₃×C₄²
Frattini	C₂²	C₂×C₃×D₃×D₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
cosocle	C₂²×C₃×C₅	C₂³×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×Dic₁₅	C₂	semidirect
4. C₃²×C₄×Dic₅	C₂	semidirect
5. C₃×Dic₃×Dic₅	C₂	semidirect
7. C₄×D₃×Dic₅	C₃	direct (N × H)
12. C₃×C₄×C₅×D₃	C₄	semidirect
14. C₃×D₃×Dic₅	C₄	direct (N × H)
20. C₄×Dic₁₅	C₂×C₃	semidirect
24. C₃×C₄×Dic₅	D₃	direct (N × H)
25. C₃×C₄×Dic₅	C₂×C₃	semidirect
26. Dic₃×Dic₅	C₂×C₃	semidirect
30. C₃²×C₄×C₅	C₂×C₄	semidirect
31. C₃×Dic₁₅	C₂×C₄	semidirect
32. C₃×C₅×Dic₃	C₂×C₄	semidirect
33. C₃²×Dic₅	C₂×C₄	semidirect
49. C₄×C₅×D₃	C₃×C₄	semidirect
55. D₃×Dic₅	C₃×C₄	direct (N × H)
60. C₃×C₅×D₃	C₄²	semidirect
65. C₄×Dic₅	C₃×D₃	direct (N × H)
72. C₃×C₄×D₃	Dic₅	direct (N × H)
82. C₃×C₄×C₅	C₄×D₃	semidirect
83. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
84. Dic₁₅	C₂×C₃×C₄	semidirect
87. C₃×Dic₅	C₄×D₃	direct (N × H)
88. C₃×Dic₅	C₂×C₃×C₄	semidirect
89. C₅×Dic₃	C₂×C₃×C₄	semidirect
101. C₃²×C₅	C₂×C₄²	semidirect
113. C₃²×C₄	C₂×Dic₅	semidirect
115. C₃×Dic₃	C₂×Dic₅	semidirect
124. C₅×D₃	C₃×C₄²	semidirect
140. C₄×D₃	C₃×Dic₅	direct (N × H)
145. C₄×C₅	C₃×C₄×D₃	semidirect
147. Dic₅	C₃×C₄×D₃	direct (N × H)
150. C₃×D₃	C₄×Dic₅	direct (N × H)
157. C₃×C₅	C₄²×D₃	semidirect
158. C₃×C₅	C₂×C₃×C₄²	semidirect
172. C₃×C₄	D₃×Dic₅	direct (N × H)
173. C₃×C₄	C₂×C₃×Dic₅	semidirect
175. Dic₃	C₂×C₃×Dic₅	semidirect
179. C₃²	C₂×C₄×Dic₅	semidirect
194. D₃	C₃×C₄×Dic₅	direct (N × H)
195. C₅	C₃×C₄²×D₃	semidirect
202. C₄	C₃×D₃×Dic₅	direct (N × H)
204. C₃	C₄×D₃×Dic₅	direct (N × H)
205. C₃	C₂×C₃×C₄×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₃	2
C₂×C₃×D₃×Dic₅	2
C₃×C₄×Dic₁₅	2
C₃²×C₄×Dic₅	2
C₃×Dic₃×Dic₅	2
C₂×C₃×C₄×Dic₅	3
C₄×D₃×Dic₅	3
C₃×C₄²×D₃	5

Subgroups — C3×C4×D3×Dic5 #13086526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₃×Dic₅ #13086526