Subgroups — C₃×C₅×#2171 #14275205

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

	subgroup	G/H
center	C₃×C₅×C₈	C₂×D₄
commutator	C₄	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₃×C₅×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₅×C₈ ◁ C₂×C₃×C₅×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#504	C₂	semidirect
3. C₃×C₅×#364	C₂	semidirect
4. C₃×C₅×#350	C₂	semidirect
5. C₃×C₅×#346	C₂	semidirect
6. C₅×#2171	C₃	direct (N × H)
17. C₃×#2171	C₅	direct (N × H)
18. C₅×#504	C₂×C₃	semidirect
20. C₅×#364	C₂×C₃	semidirect
21. C₅×#350	C₂×C₃	semidirect
22. C₅×#346	C₂×C₃	semidirect
36. C₃×#504	C₂×C₅	semidirect
38. C₃×#364	C₂×C₅	semidirect
39. C₃×#350	C₂×C₅	semidirect
40. C₃×#346	C₂×C₅	semidirect
51. #2171	C₃×C₅	direct (N × H)
86. #504	C₂×C₃×C₅	semidirect
88. #364	C₂×C₃×C₅	semidirect
89. #350	C₂×C₃×C₅	semidirect
90. #346	C₂×C₃×C₅	semidirect
164. C₃×C₅	#2171	direct (N × H)
188. C₅	C₃×#2171	direct (N × H)
195. C₃	C₅×#2171	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×#504	2
C₃×C₅×#443	2
C₃×C₅×#364	2
C₃×C₅×#350	2
C₃×C₅×#346	2
C₅×#2171	3
C₃×#2171	5

Subgroups — C3×C5×#2171 #14275205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2171 #14275205