Subgroups — C₃×C₅×#2164 #14275198

Order 1920.

276 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2164	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#117	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
12	C₃×C₅×#97	C₄	1	1	yes	characteristic
13	C₃×C₅×#97	C₄	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#2164	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#405	C₂×C₃	1	1	yes	characteristic
18	C₅×C₄⋊C₁₆	C₂×C₃	1	2	yes	complemented
19	C₅×C₂²⋊C₁₆	C₂×C₃	1	2	yes	—
20	C₅×#346	C₂×C₃	1	1	yes	characteristic, complemented
21	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	C₈	1	2	yes	—
30	C₃×C₅×C₂²⋊C₄	C₈	1	2	yes	—
31	C₃×C₄²×C₅	C₂×C₄	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	2	no	—
35	C₃×#405	C₂×C₅	1	1	yes	characteristic
36	C₃×C₄⋊C₁₆	C₂×C₅	1	2	yes	complemented
37	C₃×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
38	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×#117	C₃×C₄	1	1	yes	characteristic
43	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
44	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
45	C₅×#97	C₃×C₄	1	1	yes	characteristic
46	C₅×#97	C₃×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	#2164	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	1	no	—
52	C₂×C₃×C₄×C₅	—	2	1	no	—
53	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₅×C₈	—	2	1	no	—
58	C₃×C₅×C₈	—	2	1	no	—
59	C₃×C₅×C₈	M₁₆	1	2	yes	—
60	C₃×C₅×C₈	#41	1	2	yes	—
61	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₃×#117	C₄×C₅	1	1	yes	characteristic
64	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
65	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
66	C₃×#97	C₄×C₅	1	1	yes	characteristic
67	C₃×#97	C₄×C₅	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
71	C₅×M₁₆	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	C₃×C₈	1	2	yes	—
78	C₅×C₂²⋊C₄	C₃×C₈	1	2	yes	—
79	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₅×C₁₆	—	2	2	no	—
82	C₅×C₁₆	—	2	2	no	—
83	#405	C₂×C₃×C₅	1	1	yes	characteristic
84	C₄⋊C₁₆	C₂×C₃×C₅	1	2	yes	complemented
85	C₂²⋊C₁₆	C₂×C₃×C₅	1	2	yes	—
86	#346	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₃×C₄×C₅	—	2	1	no	—
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	#117	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
97	C₃×M₁₆	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
103	C₃×C₄⋊C₄	C₅×C₈	1	2	yes	—
104	C₃×C₂²⋊C₄	C₅×C₈	1	2	yes	—
105	C₃×C₄²	C₂×C₄×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	2	no	—
109	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	C₃×M₁₆	1	2	yes	—
119	C₅×C₈	C₃×#41	1	2	yes	—
120	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
121	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
122	#117	C₃×C₄×C₅	1	1	yes	characteristic
123	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
124	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
125	#97	C₃×C₄×C₅	1	1	yes	characteristic
126	#97	C₃×C₄×C₅	1	1	yes	characteristic
127	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₂×C₃×C₅	—	4	1	no	—
130	C₂×C₃×C₅	#431	1	1	yes	characteristic
131	C₂×C₃×C₅	#504	1	2	yes	—
132	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
136	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₃×C₈	—	2	1	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	C₅×M₁₆	1	2	yes	—
142	C₃×C₈	C₅×#41	1	2	yes	—
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	C₃×#117	1	1	yes	characteristic
150	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
156	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
157	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
158	C₄⋊C₄	C₃×C₅×C₈	1	2	yes	—
159	C₂²⋊C₄	C₃×C₅×C₈	1	2	yes	—
160	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
161	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	2	no	—
164	C₃×C₅	#2164	1	1	yes	characteristic, direct product factor, Hall
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	C₅×#117	1	1	yes	characteristic
172	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
173	C₂×C₅	—	4	1	no	—
174	C₂×C₅	C₃×#431	1	1	yes	characteristic
175	C₂×C₅	C₃×#504	1	2	yes	—
176	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
180	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
181	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	C₃×C₅×M₁₆	1	2	yes	—
186	C₈	C₃×C₅×#41	1	2	yes	—
187	C₂×C₃	—	4	1	no	—
188	C₂×C₃	C₅×#431	1	1	yes	characteristic
189	C₂×C₃	C₅×#504	1	2	yes	—
190	C₅	C₃×#2164	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	C₃×C₅×#117	1	1	yes	characteristic
198	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
199	C₃	C₅×#2164	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	4	1	no	—
201	C₂	C₃×C₅×#431	1	1	yes	characteristic
202	C₂	C₃×C₅×#504	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₃×C₅×#346	C₂	semidirect
5. C₃×C₄×C₅×C₁₆	C₂	semidirect
6. C₅×#2164	C₃	direct (N × H)
16. C₃×#2164	C₅	direct (N × H)
18. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
20. C₅×#346	C₂×C₃	semidirect
21. C₄×C₅×C₁₆	C₂×C₃	semidirect
36. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
38. C₃×#346	C₂×C₅	semidirect
39. C₃×C₄×C₁₆	C₂×C₅	semidirect
49. #2164	C₃×C₅	direct (N × H)
84. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
86. #346	C₂×C₃×C₅	semidirect
87. C₄×C₁₆	C₂×C₃×C₅	semidirect
164. C₃×C₅	#2164	direct (N × H)
190. C₅	C₃×#2164	direct (N × H)
199. C₃	C₅×#2164	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×#405	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×C₂²⋊C₁₆	2
C₃×C₅×#346	2
C₃×C₄×C₅×C₁₆	2
C₅×#2164	3
C₃×#2164	5

Subgroups — C3×C5×#2164 #14275198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2164 #14275198