Subgroups — C₂×C₃×#64972 #14439772

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4016	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3950	C₂	1	2	yes	maximal
4	C₂×C₃×#3949	C₂	1	2	yes	maximal
5	C₃×#64972	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64972	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₅	C₂²	1	4	yes	—
11	C₂×C₃×#683	—	2	4	no	—
12	C₃×#4016	—	2	4	no	—
13	C₃×#4016	C₂²	1	8	yes	—
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₃×#3950	C₂²	1	8	yes	—
16	C₃×#3949	C₂²	1	8	yes	—
17	C₂×C₃×#499	—	5	1	no	maximal, Hall
18	C₂×#4016	C₂×C₃	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×#3950	C₂×C₃	1	2	yes	—
21	C₂×#3949	C₂×C₃	1	2	yes	—
22	#64972	C₂×C₃	1	8	yes	direct product factor
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	—	2	4	no	—
26	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
27	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
28	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂³	1	2	yes	—
32	C₃×C₄⋊Dic₅	—	2	4	no	—
33	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
34	C₃×#683	—	2	16	no	—
35	C₂×C₃×#125	—	5	2	no	—
36	C₃×#499	—	5	8	no	—
37	C₂×C₃×#107	—	5	2	no	—
38	C₂×C₃×#106	—	5	2	no	—
39	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₅	C₂²×C₃	1	4	yes	—
44	C₂×#683	—	2	4	no	—
45	#4016	—	2	4	no	—
46	#4016	C₂²×C₃	1	8	yes	—
47	C₄×C₅×C₈	C₂²×C₃	1	4	yes	—
48	#3950	C₂²×C₃	1	8	yes	—
49	#3949	C₂²×C₃	1	8	yes	—
50	C₂×#499	—	5	1	no	Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₂×C₃×Dic₅	—	2	8	no	—
59	C₂×C₃×Dic₅	—	4	8	no	—
60	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
61	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
62	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
64	C₃×#125	—	5	8	no	—
65	C₃×#125	—	10	4	no	—
66	C₂×C₃×C₄⋊C₄	—	5	4	no	—
67	C₂×C₃×C₄⋊C₄	—	10	4	no	—
68	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
69	C₃×#107	—	5	8	no	—
70	C₃×#106	—	5	8	no	—
71	C₃×C₄×C₈	C₂×D₅	1	4	yes	—
72	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
74	C₂²×Dic₅	—	2	4	no	—
75	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
76	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
77	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
78	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
79	C₄²×C₅	C₃×D₄	1	2	yes	—
80	C₄²×C₅	C₂³×C₃	1	2	yes	—
81	C₄⋊Dic₅	—	2	4	no	—
82	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
83	#683	—	2	16	no	—
84	C₂×#125	—	5	2	no	—
85	#499	—	5	8	no	—
86	C₂×#107	—	5	2	no	—
87	C₂×#106	—	5	2	no	—
88	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
89	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
90	C₂²×C₃×C₅	#135	1	4	yes	—
91	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	4	no	—
93	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
94	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
96	C₃×Dic₅	—	4	8	no	—
97	C₂²×C₃×C₄	—	10	4	no	—
98	C₂²×C₃×C₄	D₂₀	1	2	yes	—
99	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
100	C₂×C₃×C₈	Dic₁₀	1	4	yes	—
101	C₂×C₃×C₈	Dic₁₀	1	4	yes	—
102	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
103	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
104	C₃×C₄⋊C₄	—	5	8	no	—
105	C₃×C₄⋊C₄	—	10	16	no	—
106	C₃×C₄⋊C₄	—	10	4	no	—
107	C₃×C₄²	D₂₀	1	2	yes	—
108	C₃×C₄²	C₂²×D₅	1	2	yes	—
109	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
112	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
115	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
116	C₂×Dic₅	—	2	8	no	—
117	C₂×Dic₅	—	4	8	no	—
118	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
119	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
120	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
121	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
122	#125	—	5	8	no	—
123	#125	—	10	4	no	—
124	C₂×C₄⋊C₄	—	5	4	no	—
125	C₂×C₄⋊C₄	—	10	4	no	—
126	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
127	#107	—	5	8	no	—
128	#106	—	5	8	no	—
129	C₄×C₈	C₂×C₃×D₅	1	4	yes	—
130	C₂×C₃×C₅	#499	1	4	yes	complemented
131	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#135	1	2	yes	—
133	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	10	8	no	—
136	C₂×C₃×C₄	—	20	8	no	—
137	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
138	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
139	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
140	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
142	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
143	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
144	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
145	C₂²×C₅	C₃×#135	1	4	yes	—
146	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
149	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
150	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
151	Dic₅	—	4	8	no	—
152	C₂²×C₄	—	10	4	no	—
153	C₂²×C₄	C₃×D₂₀	1	2	yes	—
154	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
155	C₂×C₈	C₃×Dic₁₀	1	4	yes	—
156	C₂×C₈	C₃×Dic₁₀	1	4	yes	—
157	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
158	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
159	C₄⋊C₄	—	5	8	no	—
160	C₄⋊C₄	—	10	4	no	—
161	C₄⋊C₄	—	10	16	no	—
162	C₄²	C₃×D₂₀	1	2	yes	—
163	C₄²	C₂²×C₃×D₅	1	2	yes	—
164	C₃×C₅	C₂×#499	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	#4015	1	2	yes	—
166	C₂²×C₃	#4050	1	4	yes	—
167	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	—	20	8	no	—
170	C₃×C₄	#4015	1	2	yes	—
171	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
172	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
173	C₂×C₅	C₃×#499	1	4	yes	complemented
174	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₅	C₂×C₃×#135	1	2	yes	—
176	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	10	8	no	—
179	C₂×C₄	—	20	8	no	—
180	C₂×C₄	C₂×C₃×Dic₁₀	1	2	yes	—
181	C₂×C₄	C₂×C₃×Dic₁₀	1	2	yes	—
182	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
183	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
184	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
185	C₈	C₂×C₃×Dic₁₀	1	4	yes	—
186	C₈	C₂×C₃×Dic₁₀	1	4	yes	—
187	C₂×C₃	#64972	1	4	yes	direct product factor
188	C₂×C₃	C₂×#4015	1	1	yes	characteristic
189	C₂×C₃	C₂×#4050	1	2	yes	—
190	C₅	C₂×C₃×#499	1	1	yes	characteristic, complemented, Sylow
191	C₂²	C₃×#4015	1	2	yes	—
192	C₂²	C₃×#4050	1	4	yes	—
193	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	20	8	no	—
196	C₄	C₃×#4015	1	2	yes	—
197	C₄	C₂²×C₃×Dic₁₀	1	1	yes	characteristic
198	C₄	C₂²×C₃×Dic₁₀	1	1	yes	characteristic
199	C₃	C₂×#64972	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	C₃×#64972	1	4	yes	direct product factor
201	C₂	C₂×C₃×#4015	1	1	yes	characteristic
202	C₂	C₂×C₃×#4050	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₂×C₃×#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. C₃×#64972	C₂	direct (N × H)
6. C₂×#64972	C₃	direct (N × H)
22. #64972	C₂×C₃	direct (N × H)
130. C₂×C₃×C₅	#499	semidirect
164. C₃×C₅	C₂×#499	semidirect
173. C₂×C₅	C₃×#499	semidirect
187. C₂×C₃	#64972	direct (N × H)
190. C₅	C₂×C₃×#499	semidirect
199. C₃	C₂×#64972	direct (N × H)
200. C₂	C₃×#64972	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×#4016	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₃×#3950	2
C₂×C₃×#3949	2
C₃×#64972	2
C₂×#64972	3
C₂×C₃×#499	5

Subgroups — C2×C3×#64972 #14439772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64972 #14439772