Subgroups — C₈×Dic₆₀ #14286231

Order 1920.

328 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₀	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₀	C₂	1	2	yes	maximal
3	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
4	#11746279	C₂	1	1	yes	maximal, characteristic
5	#11746277	C₂	1	2	yes	maximal
6	C₈×Dic₂₀	—	3	1	no	maximal, Hall
7	C₂×Dic₆₀	C₄	1	1	yes	characteristic
8	C₄×Dic₃₀	C₂²	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91102	C₄	1	1	yes	characteristic
13	#91100	C₄	1	2	yes	—
14	#91098	—	2	2	no	—
15	C₈×Dic₁₅	—	2	2	no	—
16	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
17	C₈×Dic₁₂	—	5	1	no	maximal, Hall
18	C₄×Dic₂₀	—	3	1	no	—
19	C₈×Dic₁₀	—	3	2	no	—
20	C₅×C₈²	D₃	1	1	yes	characteristic
21	#64678	—	3	1	no	—
22	#64676	—	3	2	no	—
23	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	—	2	1	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
31	#6712	—	2	2	no	—
32	C₄×Dic₁₅	—	2	2	no	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	Dic₆₀	C₈	1	4	yes	direct product factor
35	C₄×Dic₁₂	—	5	1	no	—
36	C₈×Dic₆	—	5	2	no	—
37	C₃×C₈²	D₅	1	1	yes	characteristic
38	#4458	—	5	1	no	—
39	#4456	—	5	2	no	—
40	C₂×Dic₂₀	—	3	1	no	—
41	C₄×Dic₁₀	—	3	2	no	—
42	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	#3950	—	3	1	no	—
46	#3948	—	3	2	no	—
47	#3946	—	6	2	no	—
48	C₈×Dic₅	—	6	2	no	—
49	C₄⋊#168	—	3	2	no	—
50	C₈×Q₁₆	—	15	1	no	Sylow
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
54	C₂×Dic₁₅	—	2	2	no	—
55	C₂×Dic₁₅	—	2	2	no	—
56	Dic₃₀	—	2	2	no	—
57	Dic₃₀	C₂×C₈	1	4	yes	—
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
62	C₃×C₅×C₈	Q₁₆	1	2	yes	complemented
63	C₃⋊#168	—	4	2	no	—
64	C₂×Dic₁₂	—	5	1	no	—
65	C₄×Dic₆	—	5	2	no	—
66	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	#818	—	5	1	no	—
70	#816	—	5	2	no	—
71	Dic₃⋊C₈	—	10	2	no	—
72	C₈×Dic₃	—	10	2	no	—
73	C₄⋊(C₃⋊C₈)	—	5	2	no	—
74	C₂×Dic₁₀	—	3	2	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
77	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
78	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
79	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
80	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₄⋊Dic₅	—	3	2	no	—
82	#683	—	6	2	no	—
83	C₄×Dic₅	—	6	2	no	—
84	C₂×#168	—	6	2	no	—
85	Dic₂₀	—	3	4	no	—
86	C₈×Q₈	—	15	2	no	—
87	C₄×Q₁₆	—	15	1	no	—
88	#335	—	15	1	no	—
89	Q₈⋊C₈	—	15	2	no	—
90	C₈²	D₁₅	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₃×C₄×C₅	#131	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₃×C₄×C₅	#135	1	1	yes	characteristic
97	Dic₁₅	—	2	4	no	—
98	Dic₁₅	—	4	2	no	—
99	C₂×Dic₆	—	5	2	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
103	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
104	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
105	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
106	C₄⋊Dic₃	—	5	2	no	—
107	Dic₃⋊C₄	—	10	2	no	—
108	C₄×Dic₃	—	10	2	no	—
109	C₂×C₃⋊C₈	—	10	2	no	—
110	Dic₁₂	—	5	4	no	—
111	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
113	C₂×C₄×C₅	#235	1	1	yes	characteristic
114	C₂×Dic₅	—	6	2	no	—
115	C₂×Dic₅	—	6	2	no	—
116	Dic₁₀	—	3	4	no	—
117	Dic₁₀	—	6	2	no	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	D₃×C₈	1	2	yes	—
122	C₅×C₈	Dic₁₂	1	2	yes	complemented
123	#168	—	12	2	no	—
124	C₂×Q₁₆	—	15	1	no	—
125	C₄×Q₈	—	15	2	no	—
126	#107	—	15	1	no	—
127	C₄⋊C₈	—	15	2	no	—
128	C₄⋊C₈	—	30	2	no	—
129	#103	—	15	2	no	—
130	C₄×C₈	—	30	2	no	—
131	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
132	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
133	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
134	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
135	C₂×C₃×C₅	C₄×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₅	#443	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
139	C₂×C₃×C₄	#709	1	1	yes	characteristic
140	C₂×Dic₃	—	10	2	no	—
141	C₂×Dic₃	—	10	2	no	—
142	Dic₆	—	5	4	no	—
143	Dic₆	—	10	2	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	C₈×D₅	1	2	yes	—
148	C₃×C₈	Dic₂₀	1	2	yes	complemented
149	C₃⋊C₈	—	20	2	no	—
150	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
153	C₄×C₅	#901	1	1	yes	characteristic
154	C₄×C₅	#904	1	1	yes	characteristic
155	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
156	Dic₅	—	6	4	no	—
157	Dic₅	—	12	2	no	—
158	C₂×Q₈	—	15	2	no	—
159	Q₁₆	—	15	4	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	30	2	no	—
162	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
163	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
164	C₂×C₈	D₆₀	1	1	yes	characteristic
165	C₂×C₈	D₆₀	1	1	yes	characteristic
166	C₄⋊C₄	—	15	2	no	—
167	C₄⋊C₄	—	30	2	no	—
168	C₄²	—	30	2	no	—
169	C₄²	C₂²×D₁₅	1	1	yes	characteristic
170	C₃×C₅	C₈×Q₁₆	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
174	C₃×C₄	#4047	1	1	yes	characteristic
175	C₃×C₄	#4050	1	1	yes	characteristic
176	C₃×C₄	C₂×Dic₂₀	1	1	yes	characteristic
177	Dic₃	—	10	4	no	—
178	Dic₃	—	20	2	no	—
179	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
180	C₂×C₅	C₄×Dic₁₂	1	1	yes	characteristic
181	C₂×C₅	#4700	1	1	yes	characteristic
182	Q₈	—	15	4	no	—
183	Q₈	—	30	2	no	—
184	C₂×C₄	—	30	2	no	—
185	C₂×C₄	—	30	2	no	—
186	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
187	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
188	C₂×C₄	#6817	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	60	2	no	—
193	C₈	C₈×D₁₅	1	2	yes	—
194	C₈	Dic₆₀	1	2	yes	direct product factor
195	C₂×C₃	C₈×D₂₀	1	1	yes	characteristic
196	C₂×C₃	C₄×Dic₂₀	1	1	yes	characteristic
197	C₂×C₃	#64989	1	1	yes	characteristic
198	C₅	C₈×Dic₁₂	1	1	yes	characteristic, complemented, Sylow
199	C₂²	C₄×D₆₀	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	30	4	no	—
202	C₄	—	60	2	no	—
203	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
204	C₄	#91790	1	1	yes	characteristic
205	C₄	#91793	1	1	yes	characteristic
206	C₄	C₂×Dic₆₀	1	1	yes	characteristic
207	C₃	C₈×Dic₂₀	1	1	yes	characteristic, complemented, Sylow
208	C₂	C₈×D₆₀	1	1	yes	characteristic
209	C₂	C₄×Dic₆₀	1	1	yes	characteristic
210	C₂	#11750801	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₆₀
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×C₈²	C₂
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₈²	D₁₅
hypocenter	C₃×C₅	C₈×Q₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
34. Dic₆₀	C₈	direct (N × H)
62. C₃×C₅×C₈	Q₁₆	semidirect
122. C₅×C₈	Dic₁₂	semidirect
148. C₃×C₈	Dic₂₀	semidirect
170. C₃×C₅	C₈×Q₁₆	semidirect
194. C₈	Dic₆₀	direct (N × H)
198. C₅	C₈×Dic₁₂	semidirect
207. C₃	C₈×Dic₂₀	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₀	2
C₈×Dic₃₀	2
C₃×C₅×C₈²	2
#11746279	2
#11746277	2
C₈×Dic₂₀	3
C₈×Dic₁₂	5

Subgroups — C8×Dic60 #14286231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Dic₆₀ #14286231