Subgroups — C₃×#10613744 #14478498

Order 1920.

740 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	C₃×#66086	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65197	C₂	1	6	yes	maximal, complemented
4	C₃×#65193	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×D₈₀	C₂	1	3	yes	maximal, complemented
6	#10613744	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4149	—	2	2	no	—
8	C₃×C₅×#131	C₂²	1	1	yes	characteristic
9	C₃×#4054	—	2	6	no	—
10	C₃×#4050	C₂²	1	6	yes	complemented
11	C₂×C₃×D₄₀	C₂²	1	6	yes	complemented
12	C₃×C₅×M₃₂	C₂²	1	3	yes	complemented
13	C₂×C₃×C₅×C₁₆	C₂²	1	3	yes	complemented
14	C₃×Dic₄₀	C₂²	1	1	yes	characteristic, complemented
15	C₃×#3932	C₂²	1	6	yes	complemented
16	C₃×D₈₀	C₂²	1	3	yes	complemented
17	C₃×D₈₀	C₂²	1	6	yes	complemented
18	C₃×#3415	—	5	1	no	maximal, Hall
19	#66086	C₂×C₃	1	2	yes	complemented
20	C₅×#504	C₂×C₃	1	1	yes	characteristic, complemented
21	#65197	C₂×C₃	1	6	yes	complemented
22	#65193	C₂×C₃	1	3	yes	complemented
23	C₂×D₈₀	C₂×C₃	1	3	yes	complemented
24	C₃×C₅×#41	D₄	1	1	yes	characteristic
25	C₃×#713	—	4	2	no	—
26	C₃×D₄×D₅	—	4	6	no	—
27	C₃×#709	—	2	6	no	—
28	C₂×C₃×D₂₀	—	2	6	no	—
29	C₃×C₅×M₁₆	D₄	1	3	yes	—
30	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
31	C₃×Dic₂₀	C₂³	1	2	yes	—
32	C₃×D₄₀	—	2	6	no	—
33	C₃×D₄₀	C₂³	1	6	yes	—
34	C₃×#677	—	2	6	no	—
35	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₁₆	C₂³	1	3	yes	—
37	C₃×#576	—	5	2	no	—
38	C₃×#509	—	5	6	no	—
39	C₃×#508	—	5	3	no	—
40	C₂×C₃×D₁₆	—	5	3	no	—
41	C₃×#504	D₅	1	1	yes	characteristic, complemented
42	#4149	—	2	2	no	—
43	C₅×#131	C₂²×C₃	1	1	yes	characteristic
44	#4054	—	2	6	no	—
45	#4050	C₂²×C₃	1	6	yes	complemented
46	C₂×D₄₀	C₂²×C₃	1	6	yes	complemented
47	C₅×M₃₂	C₂²×C₃	1	3	yes	complemented
48	C₂×C₅×C₁₆	C₂²×C₃	1	3	yes	complemented
49	Dic₄₀	C₂²×C₃	1	1	yes	characteristic, complemented
50	#3932	C₂²×C₃	1	6	yes	complemented
51	D₈₀	C₂²×C₃	1	3	yes	complemented
52	D₈₀	C₂²×C₃	1	6	yes	complemented
53	#3415	—	5	1	no	Sylow
54	C₂²×C₃×D₅	—	4	6	no	—
55	C₃×C₅×Q₈	D₈	1	1	yes	characteristic
56	C₃×C₅×D₄	D₈	1	3	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
58	C₃×C₂²⋊D₅	—	4	6	no	—
59	C₃×D₂₀	—	2	6	no	—
60	C₃×D₂₀	—	4	6	no	—
61	C₃×C₄×D₅	—	4	6	no	—
62	C₃×Dic₁₀	—	2	2	no	—
63	C₃×C₅×C₈	C₂×D₄	1	3	yes	—
64	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
65	C₃×H(5,𝔽₂)	—	10	2	no	—
66	C₃×AGL(1,ℤ/8)	—	10	6	no	—
67	C₃×#135	—	5	6	no	—
68	C₂×C₃×D₈	—	5	6	no	—
69	C₃×#131	C₂×D₅	1	1	yes	characteristic
70	C₃×Q₃₂	—	5	1	no	—
71	C₃×SD₃₂	—	5	6	no	—
72	C₃×D₁₆	—	5	3	no	—
73	C₃×D₁₆	—	5	6	no	—
74	C₃×M₃₂	C₂×D₅	1	3	yes	complemented
75	C₂×C₃×C₁₆	C₂×D₅	1	3	yes	complemented
76	C₅×#41	C₃×D₄	1	1	yes	characteristic
77	#713	—	4	2	no	—
78	D₄×D₅	—	4	6	no	—
79	#709	—	2	6	no	—
80	C₂×D₂₀	—	2	6	no	—
81	C₅×M₁₆	C₃×D₄	1	3	yes	—
82	C₂×C₅×C₈	C₂³×C₃	1	3	yes	—
83	Dic₂₀	C₂³×C₃	1	2	yes	—
84	D₄₀	—	2	6	no	—
85	D₄₀	C₂³×C₃	1	6	yes	—
86	#677	—	2	6	no	—
87	C₅×C₁₆	C₂³×C₃	1	1	yes	characteristic
88	C₅×C₁₆	C₂³×C₃	1	3	yes	—
89	#576	—	5	2	no	—
90	#509	—	5	6	no	—
91	#508	—	5	3	no	—
92	C₂×D₁₆	—	5	3	no	—
93	#504	C₃×D₅	1	1	yes	characteristic, complemented
94	C₂²×C₃×C₅	C₂×D₈	1	3	yes	—
95	C₂×C₃×D₅	—	4	6	no	—
96	C₂×C₃×D₅	—	16	6	no	—
97	C₃×C₄×C₅	C₂×D₈	1	3	yes	—
98	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₃×Dic₅	—	4	2	no	—
100	C₃×#41	—	10	6	no	—
101	C₃×#41	—	20	2	no	—
102	C₃×#41	D₂₀	1	1	yes	characteristic
103	C₂×C₃×D₄	—	10	6	no	—
104	C₂×C₃×D₄	—	20	6	no	—
105	C₃×Q₁₆	—	5	2	no	—
106	C₃×SD₁₆	—	10	6	no	—
107	C₃×D₈	—	5	6	no	—
108	C₃×D₈	—	10	6	no	—
109	C₃×M₁₆	D₂₀	1	3	yes	—
110	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
111	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
112	C₃×C₁₆	C₂²×D₅	1	3	yes	—
113	C₂²×D₅	—	4	6	no	—
114	C₅×Q₈	C₃×D₈	1	1	yes	characteristic
115	C₅×D₄	C₃×D₈	1	3	yes	—
116	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
117	C₂²⋊D₅	—	4	6	no	—
118	D₂₀	—	2	6	no	—
119	D₂₀	—	4	6	no	—
120	C₄×D₅	—	4	6	no	—
121	Dic₁₀	—	2	2	no	—
122	C₅×C₈	C₂×C₃×D₄	1	3	yes	—
123	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
124	H(5,𝔽₂)	—	10	2	no	—
125	AGL(1,ℤ/8)	—	10	6	no	—
126	#135	—	5	6	no	—
127	C₂×D₈	—	5	6	no	—
128	#131	C₂×C₃×D₅	1	1	yes	characteristic
129	Q₃₂	—	5	1	no	—
130	SD₃₂	—	5	6	no	—
131	D₁₆	—	5	3	no	—
132	D₁₆	—	5	6	no	—
133	M₃₂	C₂×C₃×D₅	1	3	yes	complemented
134	C₂×C₁₆	C₂×C₃×D₅	1	3	yes	complemented
135	C₂×C₃×C₅	—	2	3	no	—
136	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
137	C₃×D₅	—	8	6	no	—
138	C₂³×C₃	—	20	6	no	—
139	C₃×Q₈	—	10	2	no	—
140	C₃×Q₈	D₄₀	1	1	yes	characteristic
141	C₃×D₄	—	10	6	no	—
142	C₃×D₄	—	20	6	no	—
143	C₃×D₄	—	20	6	no	—
144	C₃×D₄	D₄₀	1	3	yes	—
145	C₂×C₃×C₄	—	20	6	no	—
146	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
147	C₃×C₈	C₂×D₂₀	1	3	yes	—
148	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
149	C₂²×C₅	C₂×C₃×D₈	1	3	yes	—
150	C₂×D₅	—	4	6	no	—
151	C₂×D₅	—	16	6	no	—
152	C₄×C₅	C₂×C₃×D₈	1	3	yes	—
153	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
154	Dic₅	—	4	2	no	—
155	#41	—	10	6	no	—
156	#41	—	20	2	no	—
157	#41	C₃×D₂₀	1	1	yes	characteristic
158	C₂×D₄	—	10	6	no	—
159	C₂×D₄	—	20	6	no	—
160	Q₁₆	—	5	2	no	—
161	SD₁₆	—	10	6	no	—
162	D₈	—	5	6	no	—
163	D₈	—	10	6	no	—
164	M₁₆	C₃×D₂₀	1	3	yes	—
165	C₂×C₈	C₂²×C₃×D₅	1	3	yes	—
166	C₁₆	C₂²×C₃×D₅	1	1	yes	characteristic
167	C₁₆	C₂²×C₃×D₅	1	3	yes	—
168	C₃×C₅	#3415	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	—	20	6	no	—
170	C₂²×C₃	—	80	6	no	—
171	C₂²×C₃	C₂×D₄₀	1	3	yes	—
172	C₃×C₄	—	20	2	no	—
173	C₃×C₄	C₂×D₄₀	1	3	yes	—
174	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
175	C₂×C₅	—	2	3	no	—
176	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
177	D₅	—	8	6	no	—
178	C₂³	—	20	6	no	—
179	Q₈	—	10	2	no	—
180	Q₈	C₃×D₄₀	1	1	yes	characteristic
181	D₄	—	10	6	no	—
182	D₄	—	20	6	no	—
183	D₄	—	20	6	no	—
184	D₄	C₃×D₄₀	1	3	yes	—
185	C₂×C₄	—	20	6	no	—
186	C₂×C₄	C₂×C₃×D₂₀	1	3	yes	—
187	C₈	C₂×C₃×D₂₀	1	3	yes	—
188	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
189	C₂×C₃	—	2	3	no	—
190	C₂×C₃	—	40	6	no	—
191	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
192	C₅	C₃×#3415	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	20	6	no	—
194	C₂²	—	80	6	no	—
195	C₂²	C₂×C₃×D₄₀	1	3	yes	—
196	C₄	—	20	2	no	—
197	C₄	C₂×C₃×D₄₀	1	3	yes	—
198	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
199	C₃	#10613744	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	2	3	no	—
201	C₂	—	40	6	no	—
202	C₂	C₂²×C₃×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₄₀
commutator, cosocle	C₅×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₅
Fitting	C₃×C₅×#504	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₃×#504	D₅
hypocenter	C₅	C₃×#3415
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×#131 ◁ C₃×#504

Complements

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

N	H ≅ G/N	type
1. C₃×#66086	C₂	semidirect
2. C₃×C₅×#504	C₂	semidirect
3. C₃×#65197	C₂	semidirect
4. C₃×#65193	C₂	semidirect
5. C₂×C₃×D₈₀	C₂	semidirect
6. #10613744	C₃	direct (N × H)
10. C₃×#4050	C₂²	semidirect
11. C₂×C₃×D₄₀	C₂²	semidirect
12. C₃×C₅×M₃₂	C₂²	semidirect
13. C₂×C₃×C₅×C₁₆	C₂²	semidirect
14. C₃×Dic₄₀	C₂²	semidirect
15. C₃×#3932	C₂²	semidirect
16. C₃×D₈₀	C₂²	semidirect
17. C₃×D₈₀	C₂²	semidirect
19. #66086	C₂×C₃	semidirect
20. C₅×#504	C₂×C₃	semidirect
21. #65197	C₂×C₃	semidirect
22. #65193	C₂×C₃	semidirect
23. C₂×D₈₀	C₂×C₃	semidirect
41. C₃×#504	D₅	semidirect
45. #4050	C₂²×C₃	semidirect
46. C₂×D₄₀	C₂²×C₃	semidirect
47. C₅×M₃₂	C₂²×C₃	semidirect
48. C₂×C₅×C₁₆	C₂²×C₃	semidirect
49. Dic₄₀	C₂²×C₃	semidirect
50. #3932	C₂²×C₃	semidirect
51. D₈₀	C₂²×C₃	semidirect
52. D₈₀	C₂²×C₃	semidirect
74. C₃×M₃₂	C₂×D₅	semidirect
75. C₂×C₃×C₁₆	C₂×D₅	semidirect
93. #504	C₃×D₅	semidirect
133. M₃₂	C₂×C₃×D₅	semidirect
134. C₂×C₁₆	C₂×C₃×D₅	semidirect
168. C₃×C₅	#3415	semidirect
192. C₅	C₃×#3415	semidirect
199. C₃	#10613744	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×#66086	2
C₃×C₅×#504	2
C₃×#65197	2
C₃×#65193	2
C₂×C₃×D₈₀	2
#10613744	3
C₃×#3415	5

Subgroups — C3×#10613744 #14478498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613744 #14478498