Subgroups — C₃×#10599311 #14286437

Order 1920.

384 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65123	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64703	C₂	1	2	yes	maximal, complemented
5	C₃×#64675	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599311	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×D₄₀	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	—	2	2	no	—
9	C₃×C₄×D₂₀	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×#3953	C₄	1	2	yes	—
13	C₃×#3952	—	2	2	no	—
14	C₃×#3950	C₄	1	1	yes	characteristic
15	C₃×C₄×#168	C₂²	1	1	yes	characteristic
16	C₃×C₄×#168	C₂²	1	1	yes	characteristic
17	C₃×#1580	—	5	1	no	maximal, Hall
18	#65123	C₂×C₃	1	2	yes	complemented
19	C₄×D₄₀	C₂×C₃	1	1	yes	characteristic
20	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
21	#64703	C₂×C₃	1	2	yes	complemented
22	#64675	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×D₅	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₃×#685	—	2	2	no	—
30	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
31	C₂×C₃×#168	—	2	1	no	—
32	C₂×C₃×#168	D₄	1	2	yes	—
33	C₃×D₄₀	C₈	1	4	yes	complemented
34	C₃×C₈×D₅	—	4	4	no	—
35	C₃×C₄×D₈	—	5	1	no	—
36	C₃×C₈×D₄	—	5	2	no	—
37	C₃×#334	D₅	1	1	yes	characteristic, complemented
38	C₃×D₄⋊C₈	—	5	2	no	—
39	C₃×#322	—	5	1	no	—
40	C₂×D₄₀	C₃×C₄	1	1	yes	characteristic
41	C₂×C₈×D₅	—	2	2	no	—
42	C₄×D₂₀	C₂²×C₃	1	2	yes	—
43	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	#3953	C₃×C₄	1	2	yes	—
46	#3952	—	2	2	no	—
47	#3950	C₃×C₄	1	1	yes	characteristic
48	C₄×#168	C₂²×C₃	1	1	yes	characteristic
49	C₄×#168	C₂²×C₃	1	1	yes	characteristic
50	#1580	—	5	1	no	Sylow
51	C₂²×C₃×D₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
55	C₂×C₃×Dic₅	—	2	2	no	—
56	C₃×D₂₀	—	2	2	no	—
57	C₃×D₂₀	C₂×C₈	1	4	yes	complemented
58	C₃×C₄×D₅	—	4	2	no	—
59	C₃×C₄×D₅	—	4	2	no	—
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	—	4	2	no	—
62	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
63	C₃×#168	—	2	2	no	—
64	C₃×#168	—	4	1	no	—
65	C₂×C₃×D₈	—	5	1	no	—
66	C₂²×C₃×C₈	—	10	2	no	—
67	C₃×C₄×D₄	—	5	2	no	—
68	C₃×#107	—	5	1	no	—
69	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
70	C₃×#102	—	5	2	no	—
71	C₃×C₂²⋊C₈	—	10	2	no	—
72	C₃×C₄×C₈	—	5	1	no	—
73	C₃×C₄×C₈	—	5	1	no	—
74	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
75	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₄×D₅	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
80	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
81	#685	—	2	2	no	—
82	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
83	C₂×#168	—	2	1	no	—
84	C₂×#168	C₃×D₄	1	2	yes	—
85	D₄₀	C₃×C₈	1	4	yes	complemented
86	C₈×D₅	—	4	4	no	—
87	C₄×D₈	—	5	1	no	—
88	C₈×D₄	—	5	2	no	—
89	#334	C₃×D₅	1	1	yes	characteristic, complemented
90	D₄⋊C₈	—	5	2	no	—
91	#322	—	5	1	no	—
92	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
93	C₂×C₃×D₅	—	2	4	no	—
94	C₂×C₃×D₅	—	4	2	no	—
95	C₂×C₃×D₅	—	4	2	no	—
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₃×C₄×C₅	#131	1	1	yes	characteristic
99	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
100	C₃×Dic₅	—	4	2	no	—
101	C₂×C₃×D₄	—	5	2	no	—
102	C₂²×C₃×C₄	—	10	2	no	—
103	C₃×D₈	—	5	4	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	5	2	no	—
106	C₂×C₃×C₈	—	10	1	no	—
107	C₂×C₃×C₈	—	20	4	no	—
108	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	5	2	no	—
111	C₃×C₂²⋊C₄	—	10	2	no	—
112	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
113	C₂²×D₅	—	2	2	no	—
114	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
117	C₂×Dic₅	—	2	2	no	—
118	D₂₀	—	2	2	no	—
119	D₂₀	C₂×C₃×C₈	1	4	yes	complemented
120	C₄×D₅	—	4	2	no	—
121	C₄×D₅	—	4	2	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	4	2	no	—
124	C₅×C₈	C₂×C₃×C₈	1	2	yes	complemented
125	#168	—	2	2	no	—
126	#168	—	4	1	no	—
127	C₂×D₈	—	5	1	no	—
128	C₂²×C₈	—	10	2	no	—
129	C₄×D₄	—	5	2	no	—
130	#107	—	5	1	no	—
131	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
132	#102	—	5	2	no	—
133	C₂²⋊C₈	—	10	2	no	—
134	C₄×C₈	—	5	1	no	—
135	C₄×C₈	—	5	1	no	—
136	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
137	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
138	C₂×C₃×C₅	#442	1	1	yes	characteristic
139	C₂×C₃×C₅	#444	1	1	yes	characteristic
140	C₃×D₅	—	4	4	no	—
141	C₂³×C₃	—	10	2	no	—
142	C₃×D₄	—	5	4	no	—
143	C₃×D₄	—	10	2	no	—
144	C₂×C₃×C₄	—	10	2	no	—
145	C₂×C₃×C₄	—	20	2	no	—
146	C₂×C₃×C₄	—	20	2	no	—
147	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
148	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	#713	1	1	yes	characteristic
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	4	2	no	—
152	C₃×C₈	—	10	2	no	—
153	C₃×C₈	—	20	1	no	—
154	C₃×C₈	C₈×D₅	1	2	yes	complemented
155	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
156	C₂×D₅	—	2	4	no	—
157	C₂×D₅	—	4	2	no	—
158	C₂×D₅	—	4	2	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
161	C₄×C₅	C₃×#131	1	1	yes	characteristic
162	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
163	Dic₅	—	4	2	no	—
164	C₂×D₄	—	5	2	no	—
165	C₂²×C₄	—	10	2	no	—
166	D₈	—	5	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	10	1	no	—
170	C₂×C₈	—	20	4	no	—
171	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
172	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
173	C₄⋊C₄	—	5	2	no	—
174	C₂²⋊C₄	—	10	2	no	—
175	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
176	C₃×C₅	#1580	1	1	yes	characteristic, complemented, Hall
177	C₂²×C₃	—	10	4	no	—
178	C₂²×C₃	—	20	2	no	—
179	C₂²×C₃	—	20	2	no	—
180	C₂²×C₃	#4039	1	1	yes	characteristic
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	20	2	no	—
183	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
184	C₃×C₄	#4053	1	1	yes	characteristic
185	C₃×C₄	#4060	1	2	yes	—
186	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
187	C₂×C₅	C₃×#442	1	1	yes	characteristic
188	C₂×C₅	C₃×#444	1	1	yes	characteristic
189	D₅	—	4	4	no	—
190	C₂³	—	10	2	no	—
191	D₄	—	5	4	no	—
192	D₄	—	10	2	no	—
193	C₂×C₄	—	10	2	no	—
194	C₂×C₄	—	20	2	no	—
195	C₂×C₄	—	20	2	no	—
196	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
197	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
198	C₂×C₄	C₃×#713	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	—	10	2	no	—
202	C₈	—	20	1	no	—
203	C₈	C₃×C₈×D₅	1	2	yes	complemented
204	C₂×C₃	—	20	4	no	—
205	C₂×C₃	#65123	1	1	yes	characteristic
206	C₂×C₃	#65158	1	1	yes	characteristic
207	C₂×C₃	#65183	1	1	yes	characteristic
208	C₅	C₃×#1580	1	1	yes	characteristic, complemented, Sylow
209	C₂²	—	10	4	no	—
210	C₂²	—	20	2	no	—
211	C₂²	—	20	2	no	—
212	C₂²	C₃×#4039	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	20	2	no	—
215	C₄	C₂×C₃×C₈×D₅	1	1	yes	characteristic
216	C₄	C₃×#4053	1	1	yes	characteristic
217	C₄	C₃×#4060	1	2	yes	—
218	C₃	#10599311	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	20	4	no	—
220	C₂	C₃×#65123	1	1	yes	characteristic
221	C₂	C₃×#65158	1	1	yes	characteristic
222	C₂	C₃×#65183	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#334	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×D₄₀	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#334	D₅
hypocenter	C₅	C₃×#1580
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

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

N	H ≅ G/N	type
1. C₃×#65123	C₂	semidirect
3. C₃×C₅×#334	C₂	semidirect
4. C₃×#64703	C₂	semidirect
5. C₃×#64675	C₂	semidirect
6. #10599311	C₃	direct (N × H)
18. #65123	C₂×C₃	semidirect
20. C₅×#334	C₂×C₃	semidirect
21. #64703	C₂×C₃	semidirect
22. #64675	C₂×C₃	semidirect
33. C₃×D₄₀	C₈	semidirect
37. C₃×#334	D₅	semidirect
57. C₃×D₂₀	C₂×C₈	semidirect
62. C₃×C₅×C₈	C₂×C₈	semidirect
85. D₄₀	C₃×C₈	semidirect
89. #334	C₃×D₅	semidirect
119. D₂₀	C₂×C₃×C₈	semidirect
124. C₅×C₈	C₂×C₃×C₈	semidirect
154. C₃×C₈	C₈×D₅	semidirect
176. C₃×C₅	#1580	semidirect
203. C₈	C₃×C₈×D₅	semidirect
208. C₅	C₃×#1580	semidirect
218. C₃	#10599311	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×#65123	2
C₃×C₄×D₄₀	2
C₃×C₅×#334	2
C₃×#64703	2
C₃×#64675	2
#10599311	3
C₃×#1580	5

Subgroups — C3×#10599311 #14286437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599311 #14286437