Subgroups — C₃×#10599356 #14287256

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65127	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64703	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599356	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×D₄₀	C₄	1	1	yes	characteristic
8	C₂×C₃×#676	—	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×#1589	—	5	1	no	maximal, Hall
18	#65127	C₂×C₃	1	2	yes	complemented
19	C₄×D₄₀	C₂×C₃	1	1	yes	characteristic
20	C₅×#335	C₂×C₃	1	1	yes	characteristic, complemented
21	#64703	C₂×C₃	1	2	yes	complemented
22	C₈×#168	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	1	yes	characteristic
33	C₂×C₃×#168	D₄	1	1	yes	characteristic
34	C₃×D₄₀	—	2	2	no	—
35	C₃×#676	—	4	4	no	—
36	C₃×C₄×D₈	—	5	1	no	—
37	C₃×#436	—	5	2	no	—
38	C₃×#335	D₅	1	1	yes	characteristic, complemented
39	C₃×D₄⋊C₈	—	5	2	no	—
40	C₃×C₈²	—	5	1	no	—
41	C₂×D₄₀	C₃×C₄	1	1	yes	characteristic
42	C₂×#676	—	2	2	no	—
43	C₄×D₂₀	C₂²×C₃	1	2	yes	—
44	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
45	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	#3953	C₃×C₄	1	2	yes	—
47	#3952	—	2	2	no	—
48	#3950	C₃×C₄	1	1	yes	characteristic
49	C₄×#168	C₂²×C₃	1	1	yes	characteristic
50	C₄×#168	C₂²×C₃	1	1	yes	characteristic
51	#1589	—	5	1	no	Sylow
52	C₂²×C₃×D₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
56	C₂×C₃×Dic₅	—	2	2	no	—
57	C₃×D₂₀	—	2	2	no	—
58	C₃×D₂₀	—	2	2	no	—
59	C₃×C₄×D₅	—	4	2	no	—
60	C₃×C₄×D₅	—	4	2	no	—
61	C₃×C₅×C₈	—	2	1	no	—
62	C₃×C₅×C₈	—	4	2	no	—
63	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
64	C₃×#168	—	2	2	no	—
65	C₃×#168	—	2	1	no	—
66	C₃×#168	D₈	1	2	yes	complemented
67	C₂×C₃×D₈	—	5	1	no	—
68	C₂×C₃×M₁₆	—	10	2	no	—
69	C₃×C₄×D₄	—	5	2	no	—
70	C₃×#107	—	5	1	no	—
71	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
72	C₃×#102	—	5	2	no	—
73	C₃×C₂²⋊C₈	—	10	2	no	—
74	C₃×C₄×C₈	—	5	1	no	—
75	C₃×C₄×C₈	—	5	1	no	—
76	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
77	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₄×D₅	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
83	#685	—	2	2	no	—
84	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
85	C₂×#168	—	2	1	no	—
86	C₂×#168	C₃×D₄	1	1	yes	characteristic
87	C₂×#168	C₃×D₄	1	1	yes	characteristic
88	D₄₀	—	2	2	no	—
89	#676	—	4	4	no	—
90	C₄×D₈	—	5	1	no	—
91	#436	—	5	2	no	—
92	#335	C₃×D₅	1	1	yes	characteristic, complemented
93	D₄⋊C₈	—	5	2	no	—
94	C₈²	—	5	1	no	—
95	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
96	C₂×C₃×D₅	—	4	2	no	—
97	C₂×C₃×D₅	—	4	2	no	—
98	C₂×C₃×D₅	—	4	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₃×C₄×C₅	#131	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×C₄×C₅	#135	1	1	yes	characteristic
104	C₃×Dic₅	—	4	2	no	—
105	C₂×C₃×D₄	—	5	2	no	—
106	C₂²×C₃×C₄	—	10	2	no	—
107	C₃×D₈	—	10	2	no	—
108	C₃×M₁₆	—	20	4	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	—	5	1	no	—
112	C₂×C₃×C₈	—	10	1	no	—
113	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
114	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
115	C₃×C₄⋊C₄	—	5	2	no	—
116	C₃×C₂²⋊C₄	—	10	2	no	—
117	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
118	C₂²×D₅	—	2	2	no	—
119	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
122	C₂×Dic₅	—	2	2	no	—
123	D₂₀	—	2	2	no	—
124	D₂₀	—	2	2	no	—
125	C₄×D₅	—	4	2	no	—
126	C₄×D₅	—	4	2	no	—
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	—	4	2	no	—
129	C₅×C₈	C₃×M₁₆	1	2	yes	complemented
130	#168	—	2	2	no	—
131	#168	—	2	1	no	—
132	#168	C₃×D₈	1	2	yes	complemented
133	C₂×D₈	—	5	1	no	—
134	C₂×M₁₆	—	10	2	no	—
135	C₄×D₄	—	5	2	no	—
136	#107	—	5	1	no	—
137	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
138	#102	—	5	2	no	—
139	C₂²⋊C₈	—	10	2	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
143	C₂×C₃×C₅	#436	1	1	yes	characteristic
144	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
145	C₂×C₃×C₅	#443	1	1	yes	characteristic
146	C₃×D₅	—	8	2	no	—
147	C₂³×C₃	—	10	2	no	—
148	C₃×D₄	—	10	2	no	—
149	C₃×D₄	—	10	2	no	—
150	C₂×C₃×C₄	—	10	2	no	—
151	C₂×C₃×C₄	—	20	2	no	—
152	C₂×C₃×C₄	—	20	2	no	—
153	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
154	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
155	C₂×C₃×C₄	#713	1	1	yes	characteristic
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	4	2	no	—
158	C₃×C₈	—	5	2	no	—
159	C₃×C₈	—	10	2	no	—
160	C₃×C₈	—	10	1	no	—
161	C₃×C₈	#676	1	2	yes	complemented
162	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
163	C₂×D₅	—	4	2	no	—
164	C₂×D₅	—	4	2	no	—
165	C₂×D₅	—	4	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
168	C₄×C₅	C₃×#131	1	1	yes	characteristic
169	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
170	C₄×C₅	C₃×#135	1	1	yes	characteristic
171	Dic₅	—	4	2	no	—
172	C₂×D₄	—	5	2	no	—
173	C₂²×C₄	—	10	2	no	—
174	D₈	—	10	2	no	—
175	M₁₆	—	20	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	5	1	no	—
178	C₂×C₈	—	5	1	no	—
179	C₂×C₈	—	10	1	no	—
180	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
181	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
182	C₄⋊C₄	—	5	2	no	—
183	C₂²⋊C₄	—	10	2	no	—
184	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
185	C₃×C₅	#1589	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	20	2	no	—
187	C₂²×C₃	—	20	2	no	—
188	C₂²×C₃	—	20	2	no	—
189	C₂²×C₃	#4039	1	1	yes	characteristic
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	20	2	no	—
192	C₃×C₄	C₂×#676	1	1	yes	characteristic
193	C₃×C₄	#4053	1	1	yes	characteristic
194	C₃×C₄	D₅×D₈	1	1	yes	characteristic
195	C₃×C₄	#4065	1	1	yes	characteristic
196	C₂×C₅	C₃×#436	1	1	yes	characteristic
197	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
198	C₂×C₅	C₃×#443	1	1	yes	characteristic
199	D₅	—	8	2	no	—
200	C₂³	—	10	2	no	—
201	D₄	—	10	2	no	—
202	D₄	—	10	2	no	—
203	C₂×C₄	—	10	2	no	—
204	C₂×C₄	—	20	2	no	—
205	C₂×C₄	—	20	2	no	—
206	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
207	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
208	C₂×C₄	C₃×#713	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	—	5	2	no	—
212	C₈	—	10	2	no	—
213	C₈	—	10	1	no	—
214	C₈	C₃×#676	1	2	yes	complemented
215	C₂×C₃	—	40	2	no	—
216	C₂×C₃	#65127	1	1	yes	characteristic
217	C₂×C₃	#65161	1	1	yes	characteristic
218	C₂×C₃	#65184	1	1	yes	characteristic
219	C₅	C₃×#1589	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	20	2	no	—
221	C₂²	—	20	2	no	—
222	C₂²	—	20	2	no	—
223	C₂²	C₃×#4039	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	20	2	no	—
226	C₄	C₂×C₃×#676	1	1	yes	characteristic
227	C₄	C₃×#4053	1	1	yes	characteristic
228	C₄	C₃×D₅×D₈	1	1	yes	characteristic
229	C₄	C₃×#4065	1	1	yes	characteristic
230	C₃	#10599356	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	40	2	no	—
232	C₂	C₃×#65127	1	1	yes	characteristic
233	C₂	C₃×#65161	1	1	yes	characteristic
234	C₂	C₃×#65184	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×D₄₀	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#335	D₅
hypocenter	C₅	C₃×#1589
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#335

Complements

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

N	H ≅ G/N	type
1. C₃×#65127	C₂	semidirect
3. C₃×C₅×#335	C₂	semidirect
4. C₃×#64703	C₂	semidirect
5. C₃×C₈×#168	C₂	semidirect
6. #10599356	C₃	direct (N × H)
18. #65127	C₂×C₃	semidirect
20. C₅×#335	C₂×C₃	semidirect
21. #64703	C₂×C₃	semidirect
22. C₈×#168	C₂×C₃	semidirect
38. C₃×#335	D₅	semidirect
63. C₃×C₅×C₈	M₁₆	semidirect
66. C₃×#168	D₈	semidirect
92. #335	C₃×D₅	semidirect
129. C₅×C₈	C₃×M₁₆	semidirect
132. #168	C₃×D₈	semidirect
161. C₃×C₈	#676	semidirect
185. C₃×C₅	#1589	semidirect
214. C₈	C₃×#676	semidirect
219. C₅	C₃×#1589	semidirect
230. C₃	#10599356	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×#65127	2
C₃×C₄×D₄₀	2
C₃×C₅×#335	2
C₃×#64703	2
C₃×C₈×#168	2
#10599356	3
C₃×#1589	5

Subgroups — C3×#10599356 #14287256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599356 #14287256