Subgroups — C₃×#10599869 #14315252

Order 1920.

288 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×#65190	C₂	1	2	yes	maximal, complemented
2	C₃×#64989	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64728	C₂	1	2	yes	maximal, complemented
5	C₃×#64683	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599869	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4050	C₄	1	1	yes	characteristic
8	C₃×#4047	C₂²	1	2	yes	—
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	C₄	1	1	yes	characteristic
13	C₃×#3944	C₂²	1	2	yes	—
14	C₃×#3937	C₄	1	2	yes	—
15	C₃×#3930	—	2	2	no	—
16	C₃×D₅×C₁₆	—	2	2	no	—
17	C₃×#2171	—	5	1	no	maximal, Hall
18	#65190	C₂×C₃	1	2	yes	complemented
19	#64989	C₂×C₃	1	1	yes	characteristic
20	C₅×#346	C₂×C₃	1	1	yes	characteristic, complemented
21	#64728	C₂×C₃	1	2	yes	complemented
22	#64683	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×#709	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×#681	C₂×C₄	1	2	yes	—
28	C₃×Dic₂₀	C₈	1	1	yes	characteristic
29	C₃×D₄₀	C₈	1	1	yes	characteristic
30	C₃×#677	C₈	1	2	yes	—
31	C₃×#676	—	2	2	no	—
32	C₃×C₈×D₅	—	2	2	no	—
33	C₃×C₅×C₁₆	—	2	1	no	—
34	C₃×C₅×C₁₆	D₄	1	2	yes	—
35	C₃×#672	—	2	2	no	—
36	C₃×#504	—	5	2	no	—
37	C₃×#443	—	5	1	no	—
38	C₃×#364	—	5	1	no	—
39	C₃×#350	—	5	2	no	—
40	C₃×#346	D₅	1	1	yes	characteristic, complemented
41	#4050	C₃×C₄	1	1	yes	characteristic
42	#4047	C₂²×C₃	1	2	yes	—
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	C₃×C₄	1	1	yes	characteristic
47	#3944	C₂²×C₃	1	2	yes	—
48	#3937	C₃×C₄	1	2	yes	—
49	#3930	—	2	2	no	—
50	D₅×C₁₆	—	2	2	no	—
51	#2171	—	5	1	no	Sylow
52	C₂×C₃×C₄×C₅	—	2	1	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₂²⋊D₅	—	2	2	no	—
55	C₃×D₂₀	C₂×C₈	1	2	yes	—
56	C₃×C₄×D₅	—	2	2	no	—
57	C₃×Dic₁₀	C₂×C₈	1	2	yes	—
58	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₃×#168	—	2	2	no	—
63	C₃×#135	—	5	1	no	—
64	C₃×#131	—	5	2	no	—
65	C₃×M₃₂	—	5	2	no	—
66	C₃×M₃₂	—	10	2	no	—
67	C₂×C₃×C₁₆	—	10	2	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	1	no	—
71	C₃×#104	—	5	2	no	—
72	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#709	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
76	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
77	#681	C₂×C₃×C₄	1	2	yes	—
78	Dic₂₀	C₃×C₈	1	1	yes	characteristic
79	D₄₀	C₃×C₈	1	1	yes	characteristic
80	#677	C₃×C₈	1	2	yes	—
81	#676	—	2	2	no	—
82	C₈×D₅	—	2	2	no	—
83	C₅×C₁₆	—	2	1	no	—
84	C₅×C₁₆	C₃×D₄	1	2	yes	—
85	#672	—	2	2	no	—
86	#504	—	5	2	no	—
87	#443	—	5	1	no	—
88	#364	—	5	1	no	—
89	#350	—	5	2	no	—
90	#346	C₃×D₅	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₅	#131	1	1	yes	characteristic
92	C₂×C₃×D₅	—	2	2	no	—
93	C₃×C₄×C₅	—	4	1	no	—
94	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₃×Dic₅	—	2	2	no	—
97	C₃×#41	—	5	2	no	—
98	C₃×Q₁₆	—	5	1	no	—
99	C₃×SD₁₆	—	5	2	no	—
100	C₃×D₈	—	5	1	no	—
101	C₃×M₁₆	—	5	2	no	—
102	C₃×M₁₆	—	10	2	no	—
103	C₂×C₃×C₈	—	10	2	no	—
104	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
106	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
107	C₃×C₁₆	—	2	1	no	—
108	C₃×C₁₆	—	10	2	no	—
109	C₃×C₁₆	D₂₀	1	2	yes	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
112	C₂²⋊D₅	—	2	2	no	—
113	D₂₀	C₂×C₃×C₈	1	2	yes	—
114	C₄×D₅	—	2	2	no	—
115	Dic₁₀	C₂×C₃×C₈	1	2	yes	—
116	C₅×C₈	C₂×C₃×C₈	1	1	yes	characteristic
117	C₅×C₈	C₂×C₃×C₈	1	1	yes	characteristic
118	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
119	C₅×C₈	C₃×#41	1	1	yes	characteristic
120	#168	—	2	2	no	—
121	#135	—	5	1	no	—
122	#131	—	5	2	no	—
123	M₃₂	—	5	2	no	—
124	M₃₂	—	10	2	no	—
125	C₂×C₁₆	—	10	2	no	—
126	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
127	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
128	#108	—	5	1	no	—
129	#104	—	5	2	no	—
130	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
133	C₃×D₅	—	4	2	no	—
134	C₃×Q₈	—	5	2	no	—
135	C₃×D₄	—	5	2	no	—
136	C₃×D₄	—	10	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	10	2	no	—
139	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
140	C₃×C₈	—	10	2	no	—
141	C₃×C₈	C₈×D₅	1	1	yes	characteristic
142	C₃×C₈	C₈×D₅	1	1	yes	characteristic
143	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
144	C₃×C₈	#709	1	1	yes	characteristic
145	C₂²×C₅	C₃×#131	1	1	yes	characteristic
146	C₂×D₅	—	2	2	no	—
147	C₄×C₅	—	4	1	no	—
148	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
149	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
150	Dic₅	—	2	2	no	—
151	#41	—	5	2	no	—
152	Q₁₆	—	5	1	no	—
153	SD₁₆	—	5	2	no	—
154	D₈	—	5	1	no	—
155	M₁₆	—	5	2	no	—
156	M₁₆	—	10	2	no	—
157	C₂×C₈	—	10	2	no	—
158	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
160	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
161	C₁₆	—	2	1	no	—
162	C₁₆	—	10	2	no	—
163	C₁₆	C₃×D₂₀	1	2	yes	—
164	C₃×C₅	#2171	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	—	10	2	no	—
166	C₂²×C₃	#4047	1	1	yes	characteristic
167	C₃×C₄	—	4	1	no	—
168	C₃×C₄	—	10	2	no	—
169	C₃×C₄	C₄×D₂₀	1	1	yes	characteristic
170	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
173	D₅	—	4	2	no	—
174	Q₈	—	5	2	no	—
175	D₄	—	5	2	no	—
176	D₄	—	10	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	10	2	no	—
179	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
180	C₈	—	10	2	no	—
181	C₈	C₃×C₈×D₅	1	1	yes	characteristic
182	C₈	C₃×C₈×D₅	1	1	yes	characteristic
183	C₈	C₂×C₃×D₂₀	1	1	yes	characteristic
184	C₈	C₃×#709	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	20	2	no	—
187	C₂×C₃	C₈×D₂₀	1	1	yes	characteristic
188	C₅	C₃×#2171	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	10	2	no	—
190	C₂²	C₃×#4047	1	1	yes	characteristic
191	C₄	—	4	1	no	—
192	C₄	—	10	2	no	—
193	C₄	C₃×C₄×D₂₀	1	1	yes	characteristic
194	C₄	C₂×C₃×C₈×D₅	1	1	yes	characteristic
195	C₃	#10599869	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	20	2	no	—
198	C₂	C₃×C₈×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂×D₂₀
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×#346	C₂
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₃×#4050	C₄
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#346	D₅
hypocenter	C₅	C₃×#2171
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₁₆ ◁ C₃×#346

Complements

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

N	H ≅ G/N	type
1. C₃×#65190	C₂	semidirect
3. C₃×C₅×#346	C₂	semidirect
4. C₃×#64728	C₂	semidirect
5. C₃×#64683	C₂	semidirect
6. #10599869	C₃	direct (N × H)
18. #65190	C₂×C₃	semidirect
20. C₅×#346	C₂×C₃	semidirect
21. #64728	C₂×C₃	semidirect
22. #64683	C₂×C₃	semidirect
40. C₃×#346	D₅	semidirect
90. #346	C₃×D₅	semidirect
164. C₃×C₅	#2171	semidirect
188. C₅	C₃×#2171	semidirect
195. C₃	#10599869	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×#65190	2
C₃×#64989	2
C₃×C₅×#346	2
C₃×#64728	2
C₃×#64683	2
#10599869	3
C₃×#2171	5

Subgroups — C3×#10599869 #14315252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599869 #14315252