Subgroups — C₃×C₄×D₆₀ #13087970

Order 1440.

406 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₁₅	C₂	1	2	yes	maximal, complemented
3	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6714	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₆₀	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×C₄×D₂₀	—	3	1	no	maximal
8	C₂²×C₃×D₁₅	C₂²	1	2	yes	—
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
13	C₃×D₆₀	C₄	1	4	yes	direct product factor
14	C₃×C₄×D₁₅	—	2	4	no	—
15	C₃×C₄×D₁₂	—	5	1	no	maximal, Hall
16	C₂×D₆₀	C₂×C₃	1	1	yes	characteristic
17	C₂×C₄×D₁₅	C₂×C₃	1	2	yes	complemented
18	C₂×C₃×D₂₀	—	3	1	no	—
19	C₂×C₃×C₄×D₅	—	3	2	no	—
20	C₃×C₄²×C₅	—	2	1	no	—
21	C₃×C₄²×C₅	D₃	1	1	yes	characteristic, complemented
22	C₃×C₄²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
23	#6714	C₂×C₃	1	2	yes	complemented
24	C₄⋊Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#685	—	3	2	no	—
26	C₃×C₄⋊Dic₅	—	3	1	no	—
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×D₁₅	—	2	2	no	—
29	C₂×C₃×D₁₅	—	2	2	no	—
30	C₂×C₃×D₁₅	C₂×C₄	1	4	yes	complemented
31	C₃²×C₄×C₅	—	2	1	no	—
32	C₃²×C₄×C₅	C₂×C₄	1	2	yes	complemented
33	C₃²×C₄×C₅	D₄	1	2	yes	complemented
34	C₃×Dic₁₅	—	2	2	no	—
35	C₄×D₂₀	—	3	1	no	Hall
36	C₂×C₃×D₁₂	—	5	1	no	—
37	C₂×C₃×C₄×D₃	—	5	2	no	—
38	C₃²×C₄²	D₅	1	1	yes	characteristic, complemented
39	C₃×(C₂×D₃)⋊C₄	—	5	2	no	—
40	C₃×C₄⋊Dic₃	—	5	1	no	—
41	C₂²×D₁₅	C₂²×C₃	1	2	yes	—
42	C₂²×C₃×D₅	—	3	2	no	—
43	C₂×C₃×C₄×C₅	—	2	1	no	—
44	C₂×C₃×C₄×C₅	—	2	1	no	—
45	C₂×C₃×C₄×C₅	—	2	1	no	—
46	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂²×C₃	1	2	yes	—
53	D₆₀	C₃×C₄	1	4	yes	direct product factor
54	C₄×D₁₅	—	2	4	no	—
55	C₂×C₃×Dic₅	—	3	2	no	—
56	C₃×D₂₀	—	3	4	no	—
57	C₃×C₄×D₅	—	6	4	no	—
58	C₃×C₄×D₄	—	15	1	no	—
59	C₄×D₁₂	—	5	1	no	—
60	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₅	#41	1	1	yes	characteristic
63	C₃×D₁₅	—	2	4	no	—
64	C₂×D₂₀	—	3	1	no	—
65	C₂×C₄×D₅	—	3	2	no	—
66	C₄²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
67	#685	—	3	2	no	—
68	C₄⋊Dic₅	—	3	1	no	—
69	C₂²×C₃×D₃	—	5	2	no	—
70	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	5	2	no	—
74	C₃×D₁₂	—	5	4	no	—
75	C₃×C₄×D₃	—	10	4	no	—
76	C₂²×C₃×C₅	—	2	1	no	—
77	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
79	C₂×D₁₅	—	2	2	no	—
80	C₂×D₁₅	—	2	2	no	—
81	C₂×D₁₅	C₂×C₃×C₄	1	4	yes	complemented
82	C₂×C₃×D₅	—	3	4	no	—
83	C₂×C₃×D₅	—	6	2	no	—
84	C₂×C₃×D₅	—	6	2	no	—
85	C₃×C₄×C₅	—	2	2	no	—
86	C₃×C₄×C₅	—	2	1	no	—
87	C₃×C₄×C₅	—	2	1	no	—
88	C₃×C₄×C₅	—	2	2	no	—
89	C₃×C₄×C₅	—	2	2	no	—
90	C₃×C₄×C₅	C₄×D₃	1	2	yes	complemented
91	C₃×C₄×C₅	D₁₂	1	2	yes	complemented
92	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	complemented
93	C₃×C₄×C₅	C₃×D₄	1	2	yes	complemented
94	Dic₁₅	—	2	2	no	—
95	C₃×Dic₅	—	6	2	no	—
96	C₂×C₃×D₄	—	15	1	no	—
97	C₂²×C₃×C₄	—	15	2	no	—
98	C₂×D₁₂	—	5	1	no	—
99	C₂×C₄×D₃	—	5	2	no	—
100	C₃×C₄⋊C₄	—	15	1	no	—
101	C₃×C₂²⋊C₄	—	15	2	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	C₃×D₅	1	1	yes	characteristic, complemented
104	C₃×C₄²	D₁₅	1	1	yes	characteristic, complemented
105	(C₂×D₃)⋊C₄	—	5	2	no	—
106	C₄⋊Dic₃	—	5	1	no	—
107	C₃²×C₅	C₄×D₄	1	1	yes	characteristic, complemented, Hall
108	C₂²×D₅	—	3	2	no	—
109	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
112	C₂×Dic₅	—	3	2	no	—
113	D₂₀	—	3	4	no	—
114	C₄×D₅	—	6	4	no	—
115	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
116	C₂×C₃×D₃	—	5	4	no	—
117	C₂×C₃×D₃	—	10	2	no	—
118	C₂×C₃×D₃	—	10	2	no	—
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	C₄×D₅	1	2	yes	complemented
121	C₃²×C₄	D₂₀	1	2	yes	complemented
122	C₃×Dic₃	—	10	2	no	—
123	C₄×D₄	—	15	1	no	Sylow
124	C₂×C₃×C₅	—	2	1	no	—
125	C₂×C₃×C₅	—	2	1	no	—
126	C₂×C₃×C₅	—	2	1	no	—
127	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
129	C₂×C₃×C₅	#235	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
132	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
133	D₁₅	—	2	4	no	—
134	C₃×D₅	—	6	4	no	—
135	C₂³×C₃	—	15	2	no	—
136	C₂²×D₃	—	5	2	no	—
137	C₃×D₄	—	15	4	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	15	2	no	—
142	C₂×C₃×C₄	—	30	4	no	—
143	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
149	C₂×Dic₃	—	5	2	no	—
150	D₁₂	—	5	4	no	—
151	C₄×D₃	—	10	4	no	—
152	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
153	C₂×D₅	—	3	4	no	—
154	C₂×D₅	—	6	2	no	—
155	C₂×D₅	—	6	2	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	C₃×C₄×D₃	1	2	yes	complemented
158	C₄×C₅	C₃×D₁₂	1	2	yes	complemented
159	Dic₅	—	6	2	no	—
160	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
161	C₂×C₃²	C₂×D₂₀	1	1	yes	characteristic
162	C₂×C₃²	#709	1	1	yes	characteristic
163	C₃×D₃	—	10	4	no	—
164	C₂×D₄	—	15	1	no	—
165	C₂²×C₄	—	15	2	no	—
166	C₄⋊C₄	—	15	1	no	—
167	C₂²⋊C₄	—	15	2	no	—
168	C₄²	C₃×D₁₅	1	1	yes	characteristic, complemented
169	C₃×C₅	—	2	1	no	—
170	C₃×C₅	C₄×D₁₂	1	1	yes	characteristic, complemented
171	C₃×C₅	C₃×C₄×D₄	1	1	yes	characteristic, complemented
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	15	4	no	—
174	C₂²×C₃	—	30	2	no	—
175	C₂²×C₃	—	30	2	no	—
176	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
177	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
178	C₂×D₃	—	5	4	no	—
179	C₂×D₃	—	10	2	no	—
180	C₂×D₃	—	10	2	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	30	2	no	—
187	C₃×C₄	C₃×C₄×D₅	1	2	yes	complemented
188	C₃×C₄	C₃×D₂₀	1	2	yes	complemented
189	C₃×C₄	C₄×D₁₅	1	2	yes	complemented
190	C₃×C₄	D₆₀	1	2	yes	direct product factor
191	Dic₃	—	10	2	no	—
192	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
193	C₂×C₅	C₂×C₃×D₁₂	1	1	yes	characteristic
194	C₂×C₅	C₃×#235	1	1	yes	characteristic
195	D₅	—	6	4	no	—
196	C₃²	C₄×D₂₀	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	15	2	no	—
198	D₄	—	15	4	no	—
199	C₂×C₄	—	15	2	no	—
200	C₂×C₄	—	30	4	no	—
201	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	2	1	no	—
207	C₂×C₃	—	30	4	no	—
208	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
209	C₂×C₃	C₂×C₃×D₂₀	1	1	yes	characteristic
210	C₂×C₃	C₃×#709	1	1	yes	characteristic
211	C₂×C₃	C₂×C₄×D₁₅	1	1	yes	characteristic
212	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
213	C₂×C₃	#6817	1	1	yes	characteristic
214	D₃	—	10	4	no	—
215	C₅	C₃×C₄×D₁₂	1	1	yes	characteristic, complemented, Sylow
216	C₂²	—	15	4	no	—
217	C₂²	—	30	2	no	—
218	C₂²	—	30	2	no	—
219	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	30	2	no	—
222	C₄	C₃×C₄×D₁₅	1	2	yes	complemented
223	C₄	C₃×D₆₀	1	2	yes	direct product factor
224	C₃	—	2	1	no	—
225	C₃	C₃×C₄×D₂₀	1	1	yes	characteristic, complemented
226	C₃	C₄×D₆₀	1	1	yes	characteristic, direct product factor
227	C₂	—	30	4	no	—
228	C₂	C₂×C₃×C₄×D₁₅	1	1	yes	characteristic
229	C₂	C₂×C₃×D₆₀	1	1	yes	characteristic
230	C₂	C₃×#6817	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₁₅
Fitting	C₃²×C₄²×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂×C₃×D₆₀	C₂
cosocle	C₂²×C₃×C₅	C₂³×C₃
hypercenter	C₃×C₄²	D₁₅
hypocenter	C₃×C₅	C₃×C₄×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×D₁₅	C₂	semidirect
3. C₃²×C₄²×C₅	C₂	semidirect
4. C₃×#6714	C₂	semidirect
5. C₃×C₄⋊Dic₁₅	C₂	semidirect
6. C₄×D₆₀	C₃	direct (N × H)
13. C₃×D₆₀	C₄	direct (N × H)
17. C₂×C₄×D₁₅	C₂×C₃	semidirect
21. C₃×C₄²×C₅	D₃	semidirect
22. C₃×C₄²×C₅	C₂×C₃	semidirect
23. #6714	C₂×C₃	semidirect
24. C₄⋊Dic₁₅	C₂×C₃	semidirect
30. C₂×C₃×D₁₅	C₂×C₄	semidirect
32. C₃²×C₄×C₅	C₂×C₄	semidirect
33. C₃²×C₄×C₅	D₄	semidirect
38. C₃²×C₄²	D₅	semidirect
53. D₆₀	C₃×C₄	direct (N × H)
66. C₄²×C₅	C₃×D₃	semidirect
81. C₂×D₁₅	C₂×C₃×C₄	semidirect
90. C₃×C₄×C₅	C₄×D₃	semidirect
91. C₃×C₄×C₅	D₁₂	semidirect
92. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
93. C₃×C₄×C₅	C₃×D₄	semidirect
103. C₃×C₄²	C₃×D₅	semidirect
104. C₃×C₄²	D₁₅	semidirect
107. C₃²×C₅	C₄×D₄	semidirect
120. C₃²×C₄	C₄×D₅	semidirect
121. C₃²×C₄	D₂₀	semidirect
157. C₄×C₅	C₃×C₄×D₃	semidirect
158. C₄×C₅	C₃×D₁₂	semidirect
168. C₄²	C₃×D₁₅	semidirect
170. C₃×C₅	C₄×D₁₂	semidirect
171. C₃×C₅	C₃×C₄×D₄	semidirect
187. C₃×C₄	C₃×C₄×D₅	semidirect
188. C₃×C₄	C₃×D₂₀	semidirect
189. C₃×C₄	C₄×D₁₅	semidirect
190. C₃×C₄	D₆₀	direct (N × H)
196. C₃²	C₄×D₂₀	semidirect
215. C₅	C₃×C₄×D₁₂	semidirect
222. C₄	C₃×C₄×D₁₅	semidirect
223. C₄	C₃×D₆₀	direct (N × H)
225. C₃	C₃×C₄×D₂₀	semidirect
226. C₃	C₄×D₆₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₆₀	2
C₂×C₃×C₄×D₁₅	2
C₃²×C₄²×C₅	2
C₃×#6714	2
C₃×C₄⋊Dic₁₅	2
C₄×D₆₀	3
C₃×C₄×D₂₀	3
C₃×C₄×D₁₂	5

Subgroups — C3×C4×D60 #13087970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₆₀ #13087970