Subgroups — C₃×#10599860 #14315633

Order 1920.

292 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×#65193	C₂	1	1	yes	maximal, characteristic
2	C₃×#64989	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64730	C₂	1	2	yes	maximal, complemented
5	C₃×#64726	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599860	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4050	C₂²	1	2	yes	—
8	C₃×#4047	—	2	2	no	—
9	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×#3951	C₂²	1	2	yes	—
13	C₃×#3937	—	2	2	no	—
14	C₃×Dic₄₀	C₄	1	1	yes	characteristic, complemented
15	C₃×#3932	C₄	1	2	yes	complemented
16	C₃×D₈₀	C₄	1	1	yes	characteristic, complemented
17	C₃×#2178	—	5	1	no	maximal, Hall
18	#65193	C₂×C₃	1	1	yes	characteristic
19	#64989	C₂×C₃	1	2	yes	complemented
20	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
21	#64730	C₂×C₃	1	2	yes	complemented
22	#64726	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×#709	—	2	2	no	—
24	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₃×#681	—	2	2	no	—
28	C₃×Dic₂₀	C₂×C₄	1	2	yes	—
29	C₃×D₄₀	C₂×C₄	1	2	yes	—
30	C₃×#677	—	2	2	no	—
31	C₃×#676	—	4	2	no	—
32	C₃×C₈×D₅	—	4	2	no	—
33	C₃×C₅×C₁₆	—	2	1	no	—
34	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
36	C₃×#508	—	5	1	no	—
37	C₃×#443	—	5	2	no	—
38	C₃×#368	—	5	1	no	—
39	C₃×#359	—	5	2	no	—
40	C₃×C₄×C₁₆	D₅	1	1	yes	characteristic, complemented
41	#4050	C₂²×C₃	1	2	yes	—
42	#4047	—	2	2	no	—
43	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	#3951	C₂²×C₃	1	2	yes	—
47	#3937	—	2	2	no	—
48	Dic₄₀	C₃×C₄	1	1	yes	characteristic, complemented
49	#3932	C₃×C₄	1	2	yes	complemented
50	D₈₀	C₃×C₄	1	1	yes	characteristic, complemented
51	#2178	—	5	1	no	Sylow
52	C₂×C₃×C₄×C₅	—	2	1	no	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₃×C₂²⋊D₅	—	4	2	no	—
55	C₃×D₂₀	—	2	2	no	—
56	C₃×C₄×D₅	—	4	2	no	—
57	C₃×Dic₁₀	—	2	2	no	—
58	C₃×C₅×C₈	D₈	1	1	yes	characteristic
59	C₃×C₅×C₈	D₈	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₃×#168	—	4	2	no	—
63	C₃×#135	—	5	2	no	—
64	C₃×#131	—	10	2	no	—
65	C₃×Q₃₂	—	5	1	no	—
66	C₃×SD₃₂	—	5	2	no	—
67	C₃×D₁₆	—	5	1	no	—
68	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
69	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
70	C₃×#108	—	5	2	no	—
71	C₃×#104	—	10	2	no	—
72	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#709	—	2	2	no	—
74	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
76	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
77	#681	—	2	2	no	—
78	Dic₂₀	C₂×C₃×C₄	1	2	yes	—
79	D₄₀	C₂×C₃×C₄	1	2	yes	—
80	#677	—	2	2	no	—
81	#676	—	4	2	no	—
82	C₈×D₅	—	4	2	no	—
83	C₅×C₁₆	—	2	1	no	—
84	C₅×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
85	C₅×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
86	#508	—	5	1	no	—
87	#443	—	5	2	no	—
88	#368	—	5	1	no	—
89	#359	—	5	2	no	—
90	C₄×C₁₆	C₃×D₅	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₅	#135	1	1	yes	characteristic
92	C₂×C₃×D₅	—	4	2	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₃×Dic₅	—	4	2	no	—
97	C₃×#41	—	10	2	no	—
98	C₃×Q₁₆	—	5	2	no	—
99	C₃×SD₁₆	—	10	2	no	—
100	C₃×D₈	—	5	2	no	—
101	C₃×M₁₆	—	10	2	no	—
102	C₃×M₁₆	—	20	2	no	—
103	C₂×C₃×C₈	—	20	2	no	—
104	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
106	C₃×C₄²	D₂₀	1	1	yes	characteristic
107	C₃×C₁₆	—	2	1	no	—
108	C₃×C₁₆	C₄×D₅	1	1	yes	characteristic
109	C₃×C₁₆	C₄×D₅	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
112	C₂²⋊D₅	—	4	2	no	—
113	D₂₀	—	2	2	no	—
114	C₄×D₅	—	4	2	no	—
115	Dic₁₀	—	2	2	no	—
116	C₅×C₈	C₃×D₈	1	1	yes	characteristic
117	C₅×C₈	C₃×D₈	1	1	yes	characteristic
118	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₅×C₈	C₃×#41	1	1	yes	characteristic
120	#168	—	4	2	no	—
121	#135	—	5	2	no	—
122	#131	—	10	2	no	—
123	Q₃₂	—	5	1	no	—
124	SD₃₂	—	5	2	no	—
125	D₁₆	—	5	1	no	—
126	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
127	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
128	#108	—	5	2	no	—
129	#104	—	10	2	no	—
130	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
133	C₃×D₅	—	8	2	no	—
134	C₃×Q₈	—	10	2	no	—
135	C₃×D₄	—	10	2	no	—
136	C₃×D₄	—	20	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	20	2	no	—
139	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
140	C₃×C₈	—	20	2	no	—
141	C₃×C₈	D₄₀	1	1	yes	characteristic
142	C₃×C₈	D₄₀	1	1	yes	characteristic
143	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
144	C₃×C₈	#709	1	1	yes	characteristic
145	C₂²×C₅	C₃×#135	1	1	yes	characteristic
146	C₂×D₅	—	4	2	no	—
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
149	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
150	Dic₅	—	4	2	no	—
151	#41	—	10	2	no	—
152	Q₁₆	—	5	2	no	—
153	SD₁₆	—	10	2	no	—
154	D₈	—	5	2	no	—
155	M₁₆	—	10	2	no	—
156	M₁₆	—	20	2	no	—
157	C₂×C₈	—	20	2	no	—
158	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
160	C₄²	C₃×D₂₀	1	1	yes	characteristic
161	C₁₆	—	2	1	no	—
162	C₁₆	C₃×C₄×D₅	1	1	yes	characteristic
163	C₁₆	C₃×C₄×D₅	1	1	yes	characteristic
164	C₃×C₅	#2178	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	—	20	2	no	—
166	C₂²×C₃	#4050	1	1	yes	characteristic
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	20	2	no	—
169	C₃×C₄	C₄×D₂₀	1	1	yes	characteristic
170	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
173	D₅	—	8	2	no	—
174	Q₈	—	10	2	no	—
175	D₄	—	10	2	no	—
176	D₄	—	20	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	20	2	no	—
179	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
180	C₈	—	20	2	no	—
181	C₈	C₃×D₄₀	1	1	yes	characteristic
182	C₈	C₃×D₄₀	1	1	yes	characteristic
183	C₈	C₂×C₃×C₄×D₅	1	1	yes	characteristic
184	C₈	C₃×#709	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	40	2	no	—
187	C₂×C₃	C₄×D₄₀	1	1	yes	characteristic
188	C₅	C₃×#2178	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	20	2	no	—
190	C₂²	C₃×#4050	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	20	2	no	—
193	C₄	C₃×C₄×D₂₀	1	1	yes	characteristic
194	C₄	C₂×C₃×D₄₀	1	1	yes	characteristic
195	C₃	#10599860	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	40	2	no	—
198	C₂	C₃×C₄×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	D₄₀
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×C₁₆	C₂
socle	C₂×C₃×C₅	C₄×D₈
prime radical	C₃×#65193	C₂
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×C₄×C₁₆	D₅
hypocenter	C₅	C₃×#2178
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈ ◁ C₃×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×#64989	C₂	semidirect
3. C₃×C₄×C₅×C₁₆	C₂	semidirect
4. C₃×#64730	C₂	semidirect
5. C₃×#64726	C₂	semidirect
6. #10599860	C₃	direct (N × H)
14. C₃×Dic₄₀	C₄	semidirect
15. C₃×#3932	C₄	semidirect
16. C₃×D₈₀	C₄	semidirect
19. #64989	C₂×C₃	semidirect
20. C₄×C₅×C₁₆	C₂×C₃	semidirect
21. #64730	C₂×C₃	semidirect
22. #64726	C₂×C₃	semidirect
40. C₃×C₄×C₁₆	D₅	semidirect
48. Dic₄₀	C₃×C₄	semidirect
49. #3932	C₃×C₄	semidirect
50. D₈₀	C₃×C₄	semidirect
90. C₄×C₁₆	C₃×D₅	semidirect
164. C₃×C₅	#2178	semidirect
188. C₅	C₃×#2178	semidirect
195. C₃	#10599860	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×#65193	2
C₃×#64989	2
C₃×C₄×C₅×C₁₆	2
C₃×#64730	2
C₃×#64726	2
#10599860	3
C₃×#2178	5

Subgroups — C3×#10599860 #14315633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599860 #14315633