Subgroups — C₃×#10601716 #14315190

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×C₅×#443	C₂	1	1	yes	maximal, characteristic
2	C₃×#65429	C₂	1	2	yes	maximal, complemented
3	C₃×#64781	C₂	1	2	yes	maximal, complemented
4	C₃×#64683	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601716	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#135	C₄	1	1	yes	characteristic
8	C₃×C₅×#131	C₂²	1	2	yes	—
9	C₃×C₅×#108	C₄	1	1	yes	characteristic
10	C₃×C₅×#104	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×#3944	—	2	2	no	—
13	C₃×#3944	C₂²	1	2	yes	—
14	C₂×C₃×#672	—	2	2	no	—
15	C₂×C₃×#672	C₂²	1	1	yes	characteristic
16	C₂×C₃×#672	C₂²	1	1	yes	characteristic
17	C₃×#2170	—	5	1	no	maximal, Hall
18	C₅×#443	C₂×C₃	1	1	yes	characteristic
19	#65429	C₂×C₃	1	2	yes	complemented
20	#64781	C₂×C₃	1	2	yes	complemented
21	#64683	C₂×C₃	1	1	yes	characteristic, complemented
22	C₄×#672	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	C₃×C₅×M₁₆	—	2	2	no	—
28	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×#672	—	2	2	no	—
34	C₃×#672	—	2	1	no	—
35	C₃×#672	D₄	1	1	yes	characteristic
36	C₃×#672	D₄	1	1	yes	characteristic
37	C₃×#504	—	5	2	no	—
38	C₃×#443	D₅	1	1	yes	characteristic
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	C₅×#131	C₂²×C₃	1	2	yes	—
44	C₅×#108	C₃×C₄	1	1	yes	characteristic
45	C₅×#104	C₃×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	#3944	—	2	2	no	—
48	#3944	C₂²×C₃	1	2	yes	—
49	C₂×#672	—	2	2	no	—
50	C₂×#672	C₂²×C₃	1	1	yes	characteristic
51	C₂×#672	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	1	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	2	2	no	—
60	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
61	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
62	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₃×C₅×C₈	#41	1	1	yes	characteristic
64	C₃×#135	Dic₅	1	1	yes	characteristic
65	C₃×#131	C₂×D₅	1	2	yes	—
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	Dic₅	1	1	yes	characteristic
72	C₃×#104	Dic₅	1	2	yes	—
73	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
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	C₅×M₁₆	—	2	2	no	—
79	C₅×M₁₆	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
83	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	#672	—	2	2	no	—
85	#672	—	2	1	no	—
86	#672	C₃×D₄	1	1	yes	characteristic
87	#672	C₃×D₄	1	1	yes	characteristic
88	#504	—	5	2	no	—
89	#443	C₃×D₅	1	1	yes	characteristic
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	—	2	2	no	—
97	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₃×#41	C₂×Dic₅	1	2	yes	—
100	C₃×Q₁₆	#168	1	1	yes	characteristic
101	C₃×SD₁₆	#168	1	2	yes	—
102	C₃×D₈	#168	1	1	yes	characteristic
103	C₃×M₁₆	—	2	2	no	—
104	C₃×M₁₆	C₂×Dic₅	1	2	yes	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
108	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
109	C₃×C₁₆	—	5	1	no	—
110	C₃×C₁₆	—	5	1	no	—
111	C₃×C₁₆	—	10	2	no	—
112	C₃×C₁₆	—	10	1	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	1	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₅×C₈	—	2	2	no	—
120	C₅×C₈	C₂×C₃×C₈	1	1	yes	characteristic
121	C₅×C₈	C₂×C₃×C₈	1	1	yes	characteristic
122	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
123	C₅×C₈	C₃×#41	1	1	yes	characteristic
124	#135	C₃×Dic₅	1	1	yes	characteristic
125	#131	C₂×C₃×D₅	1	2	yes	—
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	C₃×Dic₅	1	1	yes	characteristic
132	#104	C₃×Dic₅	1	2	yes	—
133	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
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×#168	1	2	yes	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	C₂×#168	1	2	yes	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	C₂×#168	1	1	yes	characteristic
145	C₃×C₈	C₂×#168	1	1	yes	characteristic
146	C₃×C₈	D₄×D₅	1	1	yes	characteristic
147	C₃×C₈	#711	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	—	2	2	no	—
152	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
154	#41	C₂×C₃×Dic₅	1	2	yes	—
155	Q₁₆	C₃×#168	1	1	yes	characteristic
156	SD₁₆	C₃×#168	1	2	yes	—
157	D₈	C₃×#168	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	C₂×C₃×Dic₅	1	2	yes	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
163	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
164	C₁₆	—	5	1	no	—
165	C₁₆	—	5	1	no	—
166	C₁₆	—	10	2	no	—
167	C₁₆	—	10	1	no	—
168	C₃×C₅	#2170	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	#4094	1	1	yes	characteristic
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	C₂²×#168	1	1	yes	characteristic
174	C₃×C₄	D₄×Dic₅	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×#168	1	2	yes	—
179	D₄	—	2	2	no	—
180	D₄	C₂×C₃×#168	1	2	yes	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	C₂×C₃×#168	1	1	yes	characteristic
186	C₈	C₂×C₃×#168	1	1	yes	characteristic
187	C₈	C₃×D₄×D₅	1	1	yes	characteristic
188	C₈	C₃×#711	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	4	2	no	—
191	C₂×C₃	D₄×#168	1	1	yes	characteristic
192	C₅	C₃×#2170	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	2	no	—
194	C₂²	C₃×#4094	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	C₂²×C₃×#168	1	1	yes	characteristic
198	C₄	C₃×D₄×Dic₅	1	1	yes	characteristic
199	C₃	#10601716	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	C₃×D₄×#168	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	D₄×D₅
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×#443	C₂
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₃×C₅×#135	C₄
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#443	D₅
hypocenter	C₅	C₃×#2170
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×#443

Complements

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

N	H ≅ G/N	type
2. C₃×#65429	C₂	semidirect
3. C₃×#64781	C₂	semidirect
4. C₃×#64683	C₂	semidirect
5. C₃×C₄×#672	C₂	semidirect
6. #10601716	C₃	direct (N × H)
19. #65429	C₂×C₃	semidirect
20. #64781	C₂×C₃	semidirect
21. #64683	C₂×C₃	semidirect
22. C₄×#672	C₂×C₃	semidirect
168. C₃×C₅	#2170	semidirect
192. C₅	C₃×#2170	semidirect
199. C₃	#10601716	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×C₅×#443	2
C₃×#65429	2
C₃×#64781	2
C₃×#64683	2
C₃×C₄×#672	2
#10601716	3
C₃×#2170	5

Subgroups — C3×#10601716 #14315190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601716 #14315190