Subgroups — C₃×C₅×#2162 #14275196

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	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×C₄×C₅×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₅×#2162	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	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	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#2162	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	C₂×C₃	1	2	yes	complemented
17	C₅×C₂²⋊C₁₆	C₂×C₃	1	2	yes	—
18	C₄×C₅×C₁₆	C₂×C₃	1	2	yes	complemented
19	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	—	2	2	no	—
23	C₂×C₃×C₅×C₈	—	2	2	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₈	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₈	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₅×C₁₆	—	2	4	no	—
33	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
34	C₃×C₄⋊C₁₆	C₂×C₅	1	2	yes	complemented
35	C₃×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
36	C₃×C₄×C₁₆	C₂×C₅	1	2	yes	complemented
37	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	C₂²×C₃	1	4	yes	—
41	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	#2162	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	—	2	2	no	—
48	C₂×C₃×C₄×C₅	—	2	1	no	—
49	C₂×C₃×C₄×C₅	—	2	1	no	—
50	C₂×C₃×C₄×C₅	C₁₆	1	4	yes	complemented
51	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
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	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₄×C₅	1	1	yes	characteristic
64	C₂×C₃×C₁₆	C₂²×C₅	1	4	yes	—
65	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
66	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₃×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₃×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
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	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
78	C₄²×C₅	C₃×C₈	1	1	yes	characteristic
79	C₄²×C₅	C₃×C₈	1	1	yes	characteristic
80	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	C₅×C₁₆	—	2	4	no	—
83	C₂×C₄×C₈	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	C₄×C₁₆	C₂×C₃×C₅	1	2	yes	complemented
87	C₂²×C₃×C₅	—	2	1	no	—
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	C₂×C₁₆	1	2	yes	—
90	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₃×C₄×C₅	—	2	1	no	—
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	C₂×C₁₆	1	2	yes	—
94	C₃×C₄×C₅	C₂×C₁₆	1	2	yes	—
95	C₃×C₄×C₅	M₃₂	1	2	yes	—
96	C₃×C₄×C₅	#117	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₅×C₈	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₅×C₈	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
107	C₃×C₄²	C₅×C₈	1	1	yes	characteristic
108	C₃×C₄²	C₅×C₈	1	1	yes	characteristic
109	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₃×C₁₆	—	2	4	no	—
112	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	C₃×C₁₆	1	4	yes	complemented
117	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	C₃×M₁₆	1	2	yes	—
126	C₅×C₈	C₃×#41	1	2	yes	—
127	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₂×C₄²	C₃×C₄×C₅	1	1	yes	characteristic
130	C₂×C₁₆	C₂²×C₃×C₅	1	4	yes	—
131	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
132	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
133	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
134	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₂×C₃×C₅	—	2	2	no	—
136	C₂×C₃×C₅	#431	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
138	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
139	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	C₅×C₁₆	1	4	yes	complemented
144	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	C₅×M₁₆	1	2	yes	—
153	C₃×C₈	C₅×#41	1	2	yes	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	C₂×C₃×C₁₆	1	2	yes	—
157	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	C₂×C₃×C₁₆	1	2	yes	—
161	C₄×C₅	C₂×C₃×C₁₆	1	2	yes	—
162	C₄×C₅	C₃×M₃₂	1	2	yes	—
163	C₄×C₅	C₃×#117	1	1	yes	characteristic
164	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₃×C₅×C₈	1	1	yes	characteristic
166	C₂²×C₄	C₃×C₅×C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
171	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
172	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
174	C₄²	C₃×C₅×C₈	1	1	yes	characteristic
175	C₄²	C₃×C₅×C₈	1	1	yes	characteristic
176	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
177	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₃×C₅	#2162	1	1	yes	characteristic, direct product factor, Hall
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	C₂×C₅×C₁₆	1	2	yes	—
183	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	C₂×C₅×C₁₆	1	2	yes	—
187	C₃×C₄	C₂×C₅×C₁₆	1	2	yes	—
188	C₃×C₄	C₅×M₃₂	1	2	yes	—
189	C₃×C₄	C₅×#117	1	1	yes	characteristic
190	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	C₃×#431	1	1	yes	characteristic
193	C₂×C₅	C₂²×C₃×C₁₆	1	1	yes	characteristic
194	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
195	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	C₃×C₅×C₁₆	1	4	yes	complemented
200	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	C₃×C₅×M₁₆	1	2	yes	—
209	C₈	C₃×C₅×#41	1	2	yes	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	C₅×#431	1	1	yes	characteristic
212	C₂×C₃	C₂²×C₅×C₁₆	1	1	yes	characteristic
213	C₂×C₃	C₂×C₅×M₃₂	1	1	yes	characteristic
214	C₅	C₃×#2162	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₂×C₃×C₅×C₁₆	1	2	yes	—
218	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×C₃×C₅×C₁₆	1	2	yes	—
222	C₄	C₂×C₃×C₅×C₁₆	1	2	yes	—
223	C₄	C₃×C₅×M₃₂	1	2	yes	—
224	C₄	C₃×C₅×#117	1	1	yes	characteristic
225	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
226	C₃	C₅×#2162	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	2	2	no	—
228	C₂	C₃×C₅×#431	1	1	yes	characteristic
229	C₂	C₂²×C₃×C₅×C₁₆	1	1	yes	characteristic
230	C₂	C₂×C₃×C₅×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅×C₈	C₂²
commutator	C₂	C₂²×C₃×C₅×C₁₆
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×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×C₄×C₅×C₁₆	C₂	semidirect
5. C₅×#2162	C₃	direct (N × H)
14. C₃×#2162	C₅	direct (N × H)
16. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
18. C₄×C₅×C₁₆	C₂×C₃	semidirect
34. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
36. C₃×C₄×C₁₆	C₂×C₅	semidirect
45. #2162	C₃×C₅	direct (N × H)
50. C₂×C₃×C₄×C₅	C₁₆	semidirect
84. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
86. C₄×C₁₆	C₂×C₃×C₅	semidirect
116. C₂×C₄×C₅	C₃×C₁₆	semidirect
143. C₂×C₃×C₄	C₅×C₁₆	semidirect
179. C₃×C₅	#2162	direct (N × H)
199. C₂×C₄	C₃×C₅×C₁₆	semidirect
214. C₅	C₃×#2162	direct (N × H)
226. C₃	C₅×#2162	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×C₂²⋊C₁₆	2
C₃×C₄×C₅×C₁₆	2
C₅×#2162	3
C₃×#2162	5

Subgroups — C3×C5×#2162 #14275196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2162 #14275196