Subgroups — C₃×#10613746 #14478626

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66087	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65198	C₂	1	6	yes	maximal, complemented
4	C₂×C₃×Dic₄₀	C₂	1	3	yes	maximal, complemented
5	C₃×#65193	C₂	1	3	yes	maximal, complemented
6	#10613746	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4150	—	2	2	no	—
8	C₃×C₅×#131	C₂²	1	1	yes	characteristic
9	C₃×#4055	—	2	6	no	—
10	C₂×C₃×Dic₂₀	C₂²	1	6	yes	—
11	C₃×#4050	C₂²	1	6	yes	—
12	C₃×C₅×M₃₂	C₂²	1	3	yes	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	3	yes	—
14	C₃×Dic₄₀	C₂²	1	3	yes	—
15	C₃×Dic₄₀	C₂²	1	6	yes	—
16	C₃×#3932	C₂²	1	6	yes	—
17	C₃×D₈₀	C₂²	1	1	yes	characteristic
18	C₃×#3417	—	5	1	no	maximal, Hall
19	#66087	C₂×C₃	1	2	yes	complemented
20	C₅×#504	C₂×C₃	1	1	yes	characteristic, complemented
21	#65198	C₂×C₃	1	6	yes	complemented
22	C₂×Dic₄₀	C₂×C₃	1	3	yes	complemented
23	#65193	C₂×C₃	1	3	yes	complemented
24	C₃×C₅×#41	D₄	1	1	yes	characteristic
25	C₃×Q₈×D₅	—	4	2	no	—
26	C₃×#711	—	4	6	no	—
27	C₃×#709	—	2	6	no	—
28	C₂×C₃×Dic₁₀	—	2	6	no	—
29	C₃×C₅×M₁₆	D₄	1	3	yes	—
30	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
31	C₃×Dic₂₀	—	2	6	no	—
32	C₃×Dic₂₀	C₂³	1	6	yes	—
33	C₃×D₄₀	C₂³	1	2	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×#578	—	5	2	no	—
38	C₃×#510	—	5	6	no	—
39	C₃×#508	—	5	3	no	—
40	C₂×C₃×Q₃₂	—	5	3	no	—
41	C₃×#504	D₅	1	1	yes	characteristic, complemented
42	#4150	—	2	2	no	—
43	C₅×#131	C₂²×C₃	1	1	yes	characteristic
44	#4055	—	2	6	no	—
45	C₂×Dic₂₀	C₂²×C₃	1	6	yes	—
46	#4050	C₂²×C₃	1	6	yes	—
47	C₅×M₃₂	C₂²×C₃	1	3	yes	—
48	C₂×C₅×C₁₆	C₂²×C₃	1	3	yes	—
49	Dic₄₀	C₂²×C₃	1	3	yes	—
50	Dic₄₀	C₂²×C₃	1	6	yes	—
51	#3932	C₂²×C₃	1	6	yes	—
52	D₈₀	C₂²×C₃	1	1	yes	characteristic
53	#3417	—	5	1	no	Sylow
54	C₃×C₅×Q₈	D₈	1	1	yes	characteristic
55	C₃×C₅×D₄	D₈	1	3	yes	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
57	C₃×C₂²⋊D₅	—	4	6	no	—
58	C₂×C₃×Dic₅	—	4	6	no	—
59	C₃×D₂₀	—	2	2	no	—
60	C₃×C₄×D₅	—	4	6	no	—
61	C₃×Dic₁₀	—	2	6	no	—
62	C₃×Dic₁₀	—	4	6	no	—
63	C₃×C₅×C₈	C₂×D₄	1	3	yes	—
64	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
65	C₃×2₋¹⁺⁴	—	10	2	no	—
66	C₃×#137	—	10	6	no	—
67	C₃×#135	—	5	6	no	—
68	C₂×C₃×Q₁₆	—	5	6	no	—
69	C₃×#131	C₂×D₅	1	1	yes	characteristic
70	C₃×Q₃₂	—	5	3	no	—
71	C₃×Q₃₂	—	5	6	no	—
72	C₃×SD₃₂	—	5	6	no	—
73	C₃×D₁₆	—	5	1	no	—
74	C₃×M₃₂	C₂×D₅	1	3	yes	—
75	C₂×C₃×C₁₆	C₂×D₅	1	3	yes	—
76	C₅×#41	C₃×D₄	1	1	yes	characteristic
77	Q₈×D₅	—	4	2	no	—
78	#711	—	4	6	no	—
79	#709	—	2	6	no	—
80	C₂×Dic₁₀	—	2	6	no	—
81	C₅×M₁₆	C₃×D₄	1	3	yes	—
82	C₂×C₅×C₈	C₂³×C₃	1	3	yes	—
83	Dic₂₀	—	2	6	no	—
84	Dic₂₀	C₂³×C₃	1	6	yes	—
85	D₄₀	C₂³×C₃	1	2	yes	—
86	#677	—	2	6	no	—
87	C₅×C₁₆	C₂³×C₃	1	1	yes	characteristic
88	C₅×C₁₆	C₂³×C₃	1	3	yes	—
89	#578	—	5	2	no	—
90	#510	—	5	6	no	—
91	#508	—	5	3	no	—
92	C₂×Q₃₂	—	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	2	no	—
96	C₃×C₄×C₅	C₂×D₈	1	3	yes	—
97	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₃×Dic₅	—	4	6	no	—
99	C₃×#41	—	10	6	no	—
100	C₃×#41	—	20	6	no	—
101	C₃×#41	D₂₀	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	10	6	no	—
103	C₂×C₃×Q₈	—	20	2	no	—
104	C₃×Q₁₆	—	5	6	no	—
105	C₃×Q₁₆	—	10	6	no	—
106	C₃×SD₁₆	—	10	6	no	—
107	C₃×D₈	—	5	2	no	—
108	C₃×M₁₆	D₂₀	1	3	yes	—
109	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
110	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
111	C₃×C₁₆	C₂²×D₅	1	3	yes	—
112	C₅×Q₈	C₃×D₈	1	1	yes	characteristic
113	C₅×D₄	C₃×D₈	1	3	yes	—
114	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
115	C₂²⋊D₅	—	4	6	no	—
116	C₂×Dic₅	—	4	6	no	—
117	D₂₀	—	2	2	no	—
118	C₄×D₅	—	4	6	no	—
119	Dic₁₀	—	2	6	no	—
120	Dic₁₀	—	4	6	no	—
121	C₅×C₈	C₂×C₃×D₄	1	3	yes	—
122	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
123	2₋¹⁺⁴	—	10	2	no	—
124	#137	—	10	6	no	—
125	#135	—	5	6	no	—
126	C₂×Q₁₆	—	5	6	no	—
127	#131	C₂×C₃×D₅	1	1	yes	characteristic
128	Q₃₂	—	5	3	no	—
129	Q₃₂	—	5	6	no	—
130	SD₃₂	—	5	6	no	—
131	D₁₆	—	5	1	no	—
132	M₃₂	C₂×C₃×D₅	1	3	yes	—
133	C₂×C₁₆	C₂×C₃×D₅	1	3	yes	—
134	C₂×C₃×C₅	—	2	3	no	—
135	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
136	C₃×D₅	—	8	2	no	—
137	C₃×Q₈	—	10	6	no	—
138	C₃×Q₈	—	20	6	no	—
139	C₃×Q₈	D₄₀	1	1	yes	characteristic
140	C₃×D₄	—	10	2	no	—
141	C₃×D₄	—	20	6	no	—
142	C₃×D₄	D₄₀	1	3	yes	—
143	C₂×C₃×C₄	—	20	6	no	—
144	C₂×C₃×C₄	—	20	6	no	—
145	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
146	C₃×C₈	C₂×D₂₀	1	3	yes	—
147	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₃×D₈	1	3	yes	—
149	C₂×D₅	—	4	2	no	—
150	C₄×C₅	C₂×C₃×D₈	1	3	yes	—
151	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
152	Dic₅	—	4	6	no	—
153	#41	—	10	6	no	—
154	#41	—	20	6	no	—
155	#41	C₃×D₂₀	1	1	yes	characteristic
156	C₂×Q₈	—	10	6	no	—
157	C₂×Q₈	—	20	2	no	—
158	Q₁₆	—	5	6	no	—
159	Q₁₆	—	10	6	no	—
160	SD₁₆	—	10	6	no	—
161	D₈	—	5	2	no	—
162	M₁₆	C₃×D₂₀	1	3	yes	—
163	C₂×C₈	C₂²×C₃×D₅	1	3	yes	—
164	C₁₆	C₂²×C₃×D₅	1	1	yes	characteristic
165	C₁₆	C₂²×C₃×D₅	1	3	yes	—
166	C₃×C₅	#3417	1	1	yes	characteristic, complemented, Hall
167	C₂²×C₃	—	20	2	no	—
168	C₂²×C₃	C₂×D₄₀	1	3	yes	—
169	C₃×C₄	—	20	6	no	—
170	C₃×C₄	C₂×D₄₀	1	3	yes	—
171	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
172	C₂×C₅	—	2	3	no	—
173	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
174	D₅	—	8	2	no	—
175	Q₈	—	10	6	no	—
176	Q₈	—	20	6	no	—
177	Q₈	C₃×D₄₀	1	1	yes	characteristic
178	D₄	—	10	2	no	—
179	D₄	—	20	6	no	—
180	D₄	C₃×D₄₀	1	3	yes	—
181	C₂×C₄	—	20	6	no	—
182	C₂×C₄	—	20	6	no	—
183	C₂×C₄	C₂×C₃×D₂₀	1	3	yes	—
184	C₈	C₂×C₃×D₂₀	1	3	yes	—
185	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
186	C₂×C₃	—	2	3	no	—
187	C₂×C₃	—	40	2	no	—
188	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
189	C₅	C₃×#3417	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	20	2	no	—
191	C₂²	C₂×C₃×D₄₀	1	3	yes	—
192	C₄	—	20	6	no	—
193	C₄	C₂×C₃×D₄₀	1	3	yes	—
194	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
195	C₃	#10613746	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	2	3	no	—
197	C₂	—	40	2	no	—
198	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×#3417
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×#66087	C₂	semidirect
2. C₃×C₅×#504	C₂	semidirect
3. C₃×#65198	C₂	semidirect
4. C₂×C₃×Dic₄₀	C₂	semidirect
5. C₃×#65193	C₂	semidirect
6. #10613746	C₃	direct (N × H)
19. #66087	C₂×C₃	semidirect
20. C₅×#504	C₂×C₃	semidirect
21. #65198	C₂×C₃	semidirect
22. C₂×Dic₄₀	C₂×C₃	semidirect
23. #65193	C₂×C₃	semidirect
41. C₃×#504	D₅	semidirect
93. #504	C₃×D₅	semidirect
166. C₃×C₅	#3417	semidirect
189. C₅	C₃×#3417	semidirect
195. C₃	#10613746	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×#66087	2
C₃×C₅×#504	2
C₃×#65198	2
C₂×C₃×Dic₄₀	2
C₃×#65193	2
#10613746	3
C₃×#3417	5

Subgroups — C3×#10613746 #14478626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613746 #14478626