Subgroups — C₃×C₅×#2112 #14275146

Order 1920.

520 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	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₅×#2112	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	—	2	4	no	—
7	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×M₃₂	—	2	4	no	—
12	C₂×C₃×C₅×C₁₆	C₂²	1	4	yes	complemented
13	C₃×#2112	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₅×C₈	C₂×C₃	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂×C₃	1	2	yes	complemented
16	C₅×C₂²⋊C₁₆	C₂×C₃	1	4	yes	complemented
17	C₂⁴×C₃×C₅	C₈	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	—	2	2	no	—
19	C₂²×C₃×C₄×C₅	—	2	2	no	—
20	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	4	no	—
26	C₂×C₃×C₅×C₈	—	2	4	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
30	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₁₆	—	2	4	no	—
32	C₂³×C₃×C₈	C₂×C₅	1	1	yes	characteristic
33	C₂×C₃×M₃₂	C₂×C₅	1	2	yes	complemented
34	C₃×C₂²⋊C₁₆	C₂×C₅	1	4	yes	complemented
35	C₂³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	2	4	no	—
37	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
39	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
40	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₅×M₃₂	—	2	4	no	—
42	C₂×C₅×C₁₆	C₂²×C₃	1	4	yes	complemented
43	#2112	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	—	2	2	no	—
45	C₂³×C₃×C₅	—	2	4	no	—
46	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
48	C₂×C₃×C₄×C₅	—	2	2	no	—
49	C₂×C₃×C₄×C₅	—	2	4	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₅×C₈	—	2	2	no	—
58	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
59	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
60	C₂³×C₃×C₄	C₄×C₅	1	1	yes	characteristic
61	C₂²×C₃×C₈	—	2	4	no	—
62	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
64	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
65	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₃×M₃₂	—	2	4	no	—
67	C₂×C₃×C₁₆	C₂²×C₅	1	4	yes	complemented
68	C₂⁴×C₅	C₃×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	C₃×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₃×C₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
81	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
82	C₅×C₁₆	—	2	4	no	—
83	C₂³×C₈	C₂×C₃×C₅	1	1	yes	characteristic
84	C₂×M₃₂	C₂×C₃×C₅	1	2	yes	complemented
85	C₂²⋊C₁₆	C₂×C₃×C₅	1	4	yes	complemented
86	C₂²×C₃×C₅	—	2	8	no	—
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	—	2	2	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
91	C₂²×C₃×C₅	M₃₂	1	4	yes	complemented
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
95	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂⁴×C₃	C₅×C₈	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	C₅×C₈	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₅×C₈	1	2	yes	—
102	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
103	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	—	2	4	no	—
107	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
108	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
110	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
111	C₃×C₁₆	—	2	4	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
115	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
122	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
124	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
127	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
128	C₂³×C₄	C₃×C₄×C₅	1	1	yes	characteristic
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₈	C₃×C₄×C₅	1	2	yes	—
132	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
133	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
134	M₃₂	—	2	4	no	—
135	C₂×C₁₆	C₂²×C₃×C₅	1	4	yes	complemented
136	C₂×C₃×C₅	—	2	2	no	—
137	C₂×C₃×C₅	—	2	4	no	—
138	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
140	C₂³×C₃	—	2	2	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
143	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
152	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
155	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
156	C₂²×C₅	—	2	8	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	C₃×C₂²⋊C₈	1	2	yes	—
161	C₂²×C₅	C₃×M₃₂	1	4	yes	complemented
162	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	C₃×C₂²⋊C₈	1	2	yes	—
165	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
167	C₂⁴	C₃×C₅×C₈	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	C₃×C₅×C₈	1	1	yes	characteristic
171	C₂²×C₄	C₃×C₅×C₈	1	2	yes	—
172	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
173	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
178	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
179	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
180	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
181	C₁₆	—	2	4	no	—
182	C₃×C₅	#2112	1	1	yes	characteristic, direct product factor, Hall
183	C₂²×C₃	—	2	8	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
188	C₂²×C₃	C₅×M₃₂	1	4	yes	complemented
189	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
192	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
194	C₂×C₅	—	2	2	no	—
195	C₂×C₅	—	2	4	no	—
196	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₅	C₂×C₃×M₃₂	1	2	yes	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	2	no	—
200	C₂³	C₂×C₃×C₅×C₈	1	2	yes	—
201	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
208	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
209	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
210	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
213	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
217	C₂×C₃	C₂×C₅×M₃₂	1	2	yes	—
218	C₅	C₃×#2112	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	8	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	C₃×C₅×C₂²⋊C₈	1	2	yes	—
224	C₂²	C₃×C₅×M₃₂	1	4	yes	complemented
225	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	C₃×C₅×C₂²⋊C₈	1	2	yes	—
228	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
230	C₃	C₅×#2112	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	2	4	no	—
232	C₂	—	2	2	no	—
233	C₂	C₂×C₃×C₅×C₂²⋊C₈	1	1	yes	characteristic
234	C₂	C₂×C₃×C₅×M₃₂	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	C₂⁴×C₃×C₅	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. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
4. C₅×#2112	C₃	direct (N × H)
12. C₂×C₃×C₅×C₁₆	C₂²	semidirect
13. C₃×#2112	C₅	direct (N × H)
15. C₂×C₅×M₃₂	C₂×C₃	semidirect
16. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
33. C₂×C₃×M₃₂	C₂×C₅	semidirect
34. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
42. C₂×C₅×C₁₆	C₂²×C₃	semidirect
43. #2112	C₃×C₅	direct (N × H)
67. C₂×C₃×C₁₆	C₂²×C₅	semidirect
84. C₂×M₃₂	C₂×C₃×C₅	semidirect
85. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
91. C₂²×C₃×C₅	M₃₂	semidirect
135. C₂×C₁₆	C₂²×C₃×C₅	semidirect
161. C₂²×C₅	C₃×M₃₂	semidirect
182. C₃×C₅	#2112	direct (N × H)
188. C₂²×C₃	C₅×M₃₂	semidirect
218. C₅	C₃×#2112	direct (N × H)
224. C₂²	C₃×C₅×M₃₂	semidirect
230. C₃	C₅×#2112	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₈	2
C₂×C₃×C₅×M₃₂	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2112	3
C₃×#2112	5

Subgroups — C3×C5×#2112 #14275146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2112 #14275146