Subgroups — C₃×#10604861 #14315197

Order 1920.

292 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×#65791	C₂	1	2	yes	maximal, complemented
2	C₃×#65485	C₂	1	1	yes	maximal, characteristic
3	C₃×#64932	C₂	1	2	yes	maximal, complemented
4	C₃×#64889	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64885	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604861	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#135	C₄	1	1	yes	characteristic
8	C₃×#4094	C₂²	1	2	yes	—
9	C₃×#3998	—	2	2	no	—
10	C₃×#3998	C₂²	1	2	yes	—
11	C₂×C₃×#674	—	2	2	no	—
12	C₂×C₃×#674	C₂²	1	1	yes	characteristic
13	C₂×C₃×#674	C₂²	1	1	yes	characteristic
14	C₃×#3969	C₄	1	2	yes	—
15	C₃×#3951	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₃×#2170	—	5	1	no	maximal, Hall
18	#65791	C₂×C₃	1	2	yes	complemented
19	#65485	C₂×C₃	1	1	yes	characteristic
20	#64932	C₂×C₃	1	2	yes	complemented
21	#64889	C₂×C₃	1	1	yes	characteristic, complemented
22	#64885	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×#41	C₂×C₄	1	2	yes	—
24	C₃×C₅×Q₁₆	C₈	1	1	yes	characteristic
25	C₃×C₅×SD₁₆	C₈	1	2	yes	—
26	C₃×C₅×D₈	C₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₃×#681	—	2	2	no	—
30	C₃×#681	C₂×C₄	1	2	yes	—
31	C₂×C₃×#168	—	2	2	no	—
32	C₂×C₃×#168	C₂³	1	1	yes	characteristic
33	C₃×#674	—	2	2	no	—
34	C₃×#674	—	2	1	no	—
35	C₃×#674	D₄	1	1	yes	characteristic
36	C₃×#674	D₄	1	1	yes	characteristic
37	C₃×#504	—	5	2	no	—
38	C₃×#443	—	5	1	no	—
39	C₃×#364	—	5	1	no	—
40	C₃×#350	—	5	2	no	—
41	C₃×C₄×C₁₆	—	5	1	no	—
42	C₅×#135	C₃×C₄	1	1	yes	characteristic
43	#4094	C₂²×C₃	1	2	yes	—
44	#3998	—	2	2	no	—
45	#3998	C₂²×C₃	1	2	yes	—
46	C₂×#674	—	2	2	no	—
47	C₂×#674	C₂²×C₃	1	1	yes	characteristic
48	C₂×#674	C₂²×C₃	1	1	yes	characteristic
49	#3969	C₃×C₄	1	2	yes	—
50	#3951	C₃×C₄	1	1	yes	characteristic
51	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
52	#2170	—	5	1	no	Sylow
53	C₃×C₅×Q₈	C₂×C₈	1	2	yes	—
54	C₃×C₅×D₄	—	2	2	no	—
55	C₃×C₅×D₄	C₂×C₈	1	2	yes	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×Dic₅	—	2	1	no	—
59	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
61	C₃×#168	—	2	2	no	—
62	C₃×#168	C₂×D₄	1	1	yes	characteristic
63	C₃×#168	#41	1	1	yes	characteristic
64	C₃×#135	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₃×#131	—	5	2	no	—
66	C₃×M₃₂	—	5	2	no	—
67	C₃×M₃₂	—	10	2	no	—
68	C₂×C₃×C₁₆	—	5	1	no	—
69	C₂×C₃×C₁₆	—	5	1	no	—
70	C₂×C₃×C₁₆	—	10	2	no	—
71	C₃×#108	—	5	1	no	—
72	C₃×#104	—	5	2	no	—
73	C₃×C₄×C₈	—	5	1	no	—
74	C₅×#41	C₂×C₃×C₄	1	2	yes	—
75	C₅×Q₁₆	C₃×C₈	1	1	yes	characteristic
76	C₅×SD₁₆	C₃×C₈	1	2	yes	—
77	C₅×D₈	C₃×C₈	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
79	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
80	#681	—	2	2	no	—
81	#681	C₂×C₃×C₄	1	2	yes	—
82	C₂×#168	—	2	2	no	—
83	C₂×#168	C₂³×C₃	1	1	yes	characteristic
84	#674	—	2	2	no	—
85	#674	—	2	1	no	—
86	#674	C₃×D₄	1	1	yes	characteristic
87	#674	C₃×D₄	1	1	yes	characteristic
88	#504	—	5	2	no	—
89	#443	—	5	1	no	—
90	#364	—	5	1	no	—
91	#350	—	5	2	no	—
92	C₄×C₁₆	—	5	1	no	—
93	C₂²×C₃×C₅	—	2	2	no	—
94	C₂²×C₃×C₅	#131	1	1	yes	characteristic
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₃×Dic₅	—	2	2	no	—
99	C₃×#41	C₂×AGL(1,𝔽₅)	1	2	yes	—
100	C₃×Q₁₆	#170	1	1	yes	characteristic
101	C₃×SD₁₆	#170	1	2	yes	—
102	C₃×D₈	#170	1	1	yes	characteristic
103	C₃×M₁₆	—	5	2	no	—
104	C₃×M₁₆	—	10	2	no	—
105	C₂×C₃×C₈	—	5	1	no	—
106	C₂×C₃×C₈	—	10	2	no	—
107	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₃×C₄²	—	5	1	no	—
109	C₃×C₁₆	—	5	1	no	—
110	C₃×C₁₆	—	5	1	no	—
111	C₃×C₁₆	—	10	1	no	—
112	C₃×C₁₆	—	10	2	no	—
113	C₅×Q₈	C₂×C₃×C₈	1	2	yes	—
114	C₅×D₄	—	2	2	no	—
115	C₅×D₄	C₂×C₃×C₈	1	2	yes	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₂×Dic₅	—	2	1	no	—
119	C₅×C₈	C₂×C₃×C₈	1	1	yes	characteristic
120	C₅×C₈	C₂×C₃×C₈	1	1	yes	characteristic
121	#168	—	2	2	no	—
122	#168	C₂×C₃×D₄	1	1	yes	characteristic
123	#168	C₃×#41	1	1	yes	characteristic
124	#135	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
125	#131	—	5	2	no	—
126	M₃₂	—	5	2	no	—
127	M₃₂	—	10	2	no	—
128	C₂×C₁₆	—	5	1	no	—
129	C₂×C₁₆	—	5	1	no	—
130	C₂×C₁₆	—	10	2	no	—
131	#108	—	5	1	no	—
132	#104	—	5	2	no	—
133	C₄×C₈	—	5	1	no	—
134	C₂×C₃×C₅	—	2	1	no	—
135	C₂×C₃×C₅	—	4	2	no	—
136	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
137	C₃×Q₈	C₂×#170	1	2	yes	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	C₂×#170	1	2	yes	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	10	1	no	—
142	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₃×C₈	—	5	1	no	—
144	C₃×C₈	—	5	1	no	—
145	C₃×C₈	—	10	2	no	—
146	C₃×C₈	C₂×#170	1	1	yes	characteristic
147	C₃×C₈	C₂×#170	1	1	yes	characteristic
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	C₃×#131	1	1	yes	characteristic
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
152	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
153	Dic₅	—	2	2	no	—
154	#41	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
155	Q₁₆	C₃×#170	1	1	yes	characteristic
156	SD₁₆	C₃×#170	1	2	yes	—
157	D₈	C₃×#170	1	1	yes	characteristic
158	M₁₆	—	5	2	no	—
159	M₁₆	—	10	2	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	10	2	no	—
162	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₄²	—	5	1	no	—
164	C₁₆	—	5	1	no	—
165	C₁₆	—	5	1	no	—
166	C₁₆	—	10	1	no	—
167	C₁₆	—	10	2	no	—
168	C₃×C₅	#2170	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	#4131	1	1	yes	characteristic
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	10	2	no	—
173	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
174	C₃×C₄	C₂²×#170	1	1	yes	characteristic
175	C₂×C₅	—	2	1	no	—
176	C₂×C₅	—	4	2	no	—
177	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
178	Q₈	C₂×C₃×#170	1	2	yes	—
179	D₄	—	2	2	no	—
180	D₄	C₂×C₃×#170	1	2	yes	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	10	1	no	—
183	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₈	—	5	1	no	—
185	C₈	—	5	1	no	—
186	C₈	—	10	2	no	—
187	C₈	C₂×C₃×#170	1	1	yes	characteristic
188	C₈	C₂×C₃×#170	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	4	2	no	—
191	C₂×C₃	D₄×#170	1	1	yes	characteristic
192	C₅	C₃×#2170	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	2	no	—
194	C₂²	C₃×#4131	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	10	2	no	—
197	C₄	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₄	C₂²×C₃×#170	1	1	yes	characteristic
199	C₃	#10604861	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	C₃×D₄×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting, prime radical	C₃×C₅×#135	C₄
socle	C₂×C₃×C₅	C₈×D₄
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₃×#135	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2170
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
1. C₃×#65791	C₂	semidirect
3. C₃×#64932	C₂	semidirect
4. C₃×#64889	C₂	semidirect
5. C₃×#64885	C₂	semidirect
6. #10604861	C₃	direct (N × H)
18. #65791	C₂×C₃	semidirect
20. #64932	C₂×C₃	semidirect
21. #64889	C₂×C₃	semidirect
22. #64885	C₂×C₃	semidirect
168. C₃×C₅	#2170	semidirect
192. C₅	C₃×#2170	semidirect
199. C₃	#10604861	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×#65791	2
C₃×#65485	2
C₃×#64932	2
C₃×#64889	2
C₃×#64885	2
#10604861	3
C₃×#2170	5

Subgroups — C3×#10604861 #14315197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604861 #14315197