Subgroups — C₃×C₅×#1566 #14274600

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#423	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#334	C₂	1	4	yes	maximal, complemented
4	C₅×#1566	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×M₁₆	—	2	2	no	—
6	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊C₈	—	2	2	no	—
9	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
13	C₃×#1566	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₅×#423	C₂×C₃	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
16	C₅×#334	C₂×C₃	1	4	yes	complemented
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
20	C₃×C₅×M₁₆	—	4	4	no	—
21	C₂×C₃×C₅×C₈	—	2	4	no	—
22	C₂×C₃×C₅×C₈	—	2	4	no	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₃×#423	C₂×C₅	1	2	yes	—
30	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
31	C₃×#334	C₂×C₅	1	4	yes	complemented
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
37	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
38	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
39	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
40	#1566	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₃×C₅×C₈	—	2	2	no	—
52	C₃×C₅×C₈	—	4	4	no	—
53	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
54	C₂×C₃×M₁₆	—	2	2	no	—
55	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
56	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
57	C₃×C₄⋊C₈	—	2	2	no	—
58	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
59	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
60	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
62	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
65	C₅×M₁₆	—	4	4	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
70	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
71	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
72	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
73	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
74	#423	C₂×C₃×C₅	1	2	yes	—
75	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
76	#334	C₂×C₃×C₅	1	4	yes	complemented
77	C₂²×C₃×C₅	—	2	1	no	—
78	C₂²×C₃×C₅	—	2	1	no	—
79	C₂²×C₃×C₅	#106	1	2	yes	—
80	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₃×C₄×C₅	—	2	1	no	—
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	#106	1	2	yes	—
84	C₃×C₄×C₅	#108	1	2	yes	—
85	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
86	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
87	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
90	C₃×M₁₆	—	4	4	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
94	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
95	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
96	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
97	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
98	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
99	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
106	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	—	4	4	no	—
111	C₅×C₈	C₃×M₁₆	1	4	yes	complemented
112	C₂×M₁₆	—	2	2	no	—
113	C₂²×C₈	C₃×C₄×C₅	1	2	yes	—
114	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
117	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
118	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
119	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
120	C₂×C₃×C₅	—	2	2	no	—
121	C₂×C₃×C₅	#423	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
124	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
131	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	—	4	4	no	—
136	C₃×C₈	C₅×M₁₆	1	4	yes	complemented
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	C₃×#106	1	2	yes	—
140	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	C₃×#106	1	2	yes	—
144	C₄×C₅	C₃×#108	1	2	yes	—
145	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
146	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
147	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
149	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
150	M₁₆	—	4	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
154	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
155	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
156	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
157	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
158	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
159	C₃×C₅	#1566	1	1	yes	characteristic, direct product factor, Hall
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	C₅×#106	1	2	yes	—
163	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	C₅×#106	1	2	yes	—
167	C₃×C₄	C₅×#108	1	2	yes	—
168	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
169	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
170	C₂×C₅	—	2	2	no	—
171	C₂×C₅	C₃×#423	1	1	yes	characteristic
172	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
173	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
174	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
181	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
183	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	4	4	no	—
186	C₈	C₃×C₅×M₁₆	1	4	yes	complemented
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	C₅×#423	1	1	yes	characteristic
189	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
190	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
191	C₅	C₃×#1566	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	C₃×C₅×#106	1	2	yes	—
195	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	C₃×C₅×#106	1	2	yes	—
199	C₄	C₃×C₅×#108	1	2	yes	—
200	C₄	C₂×C₃×C₅×M₁₆	1	2	yes	—
201	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
202	C₃	C₅×#1566	1	1	yes	characteristic, direct product factor, Sylow
203	C₂	—	2	2	no	—
204	C₂	C₃×C₅×#423	1	1	yes	characteristic
205	C₂	C₂×C₃×C₅×#106	1	1	yes	characteristic
206	C₂	C₂×C₃×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#334	C₂	semidirect
4. C₅×#1566	C₃	direct (N × H)
13. C₃×#1566	C₅	direct (N × H)
16. C₅×#334	C₂×C₃	semidirect
31. C₃×#334	C₂×C₅	semidirect
40. #1566	C₃×C₅	direct (N × H)
53. C₃×C₅×C₈	M₁₆	semidirect
76. #334	C₂×C₃×C₅	semidirect
111. C₅×C₈	C₃×M₁₆	semidirect
136. C₃×C₈	C₅×M₁₆	semidirect
159. C₃×C₅	#1566	direct (N × H)
186. C₈	C₃×C₅×M₁₆	semidirect
191. C₅	C₃×#1566	direct (N × H)
202. C₃	C₅×#1566	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×#423	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#334	2
C₅×#1566	3
C₃×#1566	5

Subgroups — C3×C5×#1566 #14274600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1566 #14274600