Subgroups — C₃×#10610173 #14439847

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65804	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#65395	C₂	1	4	yes	maximal
4	C₃×#64972	C₂	1	4	yes	maximal, complemented
5	C₃×#64971	C₂	1	2	yes	maximal, complemented
6	C₃×#64969	C₂	1	2	yes	maximal, complemented
7	#10610173	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4072	C₂²	1	4	yes	—
11	C₃×#4070	—	2	4	no	—
12	C₃×#4016	C₂²	1	4	yes	—
13	C₃×#4015	C₂²	1	4	yes	—
14	C₃×C₄×Dic₁₀	—	2	4	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₃×#3950	C₂²	1	8	yes	—
19	C₃×#3949	C₂²	1	8	yes	—
20	C₃×#3160	—	5	1	no	maximal, Hall
21	#65804	C₂×C₃	1	2	yes	—
22	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
23	#65395	C₂×C₃	1	4	yes	—
24	#64972	C₂×C₃	1	4	yes	complemented
25	#64971	C₂×C₃	1	2	yes	complemented
26	#64969	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×Dic₁₀	—	2	4	no	—
30	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
31	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₂²⋊Dic₅	—	2	4	no	—
37	C₃×C₄⋊Dic₅	—	2	4	no	—
38	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
39	C₃×#683	—	2	8	no	—
40	C₃×C₄×Dic₅	—	2	4	no	—
41	C₃×#533	—	5	2	no	—
42	C₃×#500	—	5	2	no	—
43	C₃×#499	—	5	4	no	—
44	C₃×#498	—	5	2	no	—
45	C₃×#427	—	5	4	no	—
46	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	#4072	C₂²×C₃	1	4	yes	—
50	#4070	—	2	4	no	—
51	#4016	C₂²×C₃	1	4	yes	—
52	#4015	C₂²×C₃	1	4	yes	—
53	C₄×Dic₁₀	—	2	4	no	—
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
57	#3950	C₂²×C₃	1	8	yes	—
58	#3949	C₂²×C₃	1	8	yes	—
59	#3160	—	5	1	no	Sylow
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
66	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
67	C₂×C₃×Dic₅	—	2	8	no	—
68	C₃×Dic₁₀	—	4	8	no	—
69	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
70	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
71	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
72	C₃×C₄⋊Q₈	—	5	4	no	—
73	C₃×#125	—	5	4	no	—
74	C₃×C₂²⋊Q₈	—	10	4	no	—
75	C₃×C₄×Q₈	—	10	4	no	—
76	C₃×#117	—	5	4	no	—
77	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
78	C₃×#107	—	5	8	no	—
79	C₃×#106	—	5	8	no	—
80	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
82	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₂×Dic₁₀	—	2	4	no	—
86	C₂×C₅×C₈	C₃×Q₈	1	8	yes	complemented
87	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
88	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
89	C₄²×C₅	C₃×D₄	1	2	yes	—
90	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₂²⋊Dic₅	—	2	4	no	—
93	C₄⋊Dic₅	—	2	4	no	—
94	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
95	#683	—	2	8	no	—
96	C₄×Dic₅	—	2	4	no	—
97	#533	—	5	2	no	—
98	#500	—	5	2	no	—
99	#499	—	5	4	no	—
100	#498	—	5	2	no	—
101	#427	—	5	4	no	—
102	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
105	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
108	C₃×C₄×C₅	#135	1	4	yes	—
109	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
110	C₃×Dic₅	—	4	8	no	—
111	C₂×C₃×Q₈	—	10	4	no	—
112	C₂²×C₃×C₄	D₂₀	1	2	yes	—
113	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂×C₃×C₈	Dic₁₀	1	8	yes	complemented
115	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
116	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
117	C₃×C₄⋊C₄	—	5	8	no	—
118	C₃×C₄⋊C₄	—	10	8	no	—
119	C₃×C₄⋊C₄	—	10	4	no	—
120	C₃×C₂²⋊C₄	—	10	4	no	—
121	C₃×C₄²	—	10	4	no	—
122	C₃×C₄²	D₂₀	1	2	yes	—
123	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
124	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
125	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
131	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
132	C₂×Dic₅	—	2	8	no	—
133	Dic₁₀	—	4	8	no	—
134	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
135	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
136	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
137	C₄⋊Q₈	—	5	4	no	—
138	#125	—	5	4	no	—
139	C₂²⋊Q₈	—	10	4	no	—
140	C₄×Q₈	—	10	4	no	—
141	#117	—	5	4	no	—
142	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
143	#107	—	5	8	no	—
144	#106	—	5	8	no	—
145	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
146	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
147	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
148	C₂×C₃×C₅	—	2	2	no	—
149	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×#135	1	2	yes	—
151	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
152	C₃×Q₈	—	20	8	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	10	8	no	—
155	C₂×C₃×C₄	C₂×Dic₁₀	1	4	yes	—
156	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
157	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
158	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
160	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
161	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
164	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
167	C₄×C₅	C₃×#135	1	4	yes	—
168	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
169	Dic₅	—	4	8	no	—
170	C₂×Q₈	—	10	4	no	—
171	C₂²×C₄	C₃×D₂₀	1	2	yes	—
172	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
173	C₂×C₈	C₃×Dic₁₀	1	8	yes	complemented
174	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
175	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
176	C₄⋊C₄	—	5	8	no	—
177	C₄⋊C₄	—	10	8	no	—
178	C₄⋊C₄	—	10	4	no	—
179	C₂²⋊C₄	—	10	4	no	—
180	C₄²	—	10	4	no	—
181	C₄²	C₃×D₂₀	1	2	yes	—
182	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
183	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
184	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	#4015	1	2	yes	—
187	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	20	8	no	—
190	C₃×C₄	#4015	1	2	yes	—
191	C₃×C₄	#4050	1	4	yes	—
192	C₃×C₄	C₂²×Dic₁₀	1	2	yes	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×C₅	C₂×C₃×#135	1	2	yes	—
196	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
197	Q₈	—	20	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	10	8	no	—
200	C₂×C₄	C₂×C₃×Dic₁₀	1	4	yes	—
201	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
202	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
204	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
205	C₈	C₂×C₃×Dic₁₀	1	4	yes	—
206	C₈	C₂×C₃×Dic₁₀	1	4	yes	—
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂×#4015	1	1	yes	characteristic
209	C₂×C₃	C₂×#4050	1	2	yes	—
210	C₅	C₃×#3160	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	C₃×#4015	1	2	yes	—
213	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	20	8	no	—
216	C₄	C₃×#4015	1	2	yes	—
217	C₄	C₃×#4050	1	4	yes	—
218	C₄	C₂²×C₃×Dic₁₀	1	2	yes	—
219	C₃	#10610173	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	2	2	no	—
221	C₂	C₂×C₃×#4015	1	1	yes	characteristic
222	C₂	C₂×C₃×#4050	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#64972	C₂	semidirect
5. C₃×#64971	C₂	semidirect
6. C₃×#64969	C₂	semidirect
7. #10610173	C₃	direct (N × H)
24. #64972	C₂×C₃	semidirect
25. #64971	C₂×C₃	semidirect
26. #64969	C₂×C₃	semidirect
30. C₂×C₃×C₅×C₈	Q₈	semidirect
86. C₂×C₅×C₈	C₃×Q₈	semidirect
114. C₂×C₃×C₈	Dic₁₀	semidirect
173. C₂×C₈	C₃×Dic₁₀	semidirect
184. C₃×C₅	#3160	semidirect
210. C₅	C₃×#3160	semidirect
219. C₃	#10610173	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#65804	2
C₂×C₃×C₄×C₅×C₈	2
C₃×#65395	2
C₃×#64972	2
C₃×#64971	2
C₃×#64969	2
#10610173	3
C₃×#3160	5

Subgroups — C3×#10610173 #14439847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610173 #14439847