Subgroups — C₃×C₅×#3160 #14276194

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	2	yes	maximal
2	C₃×C₅×#500	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#499	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#498	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#427	C₂	1	4	yes	maximal
6	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
7	C₅×#3160	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
10	C₃×C₅×#125	C₂²	1	4	yes	—
11	C₃×C₅×C₂²⋊Q₈	—	2	4	no	—
12	C₃×C₄×C₅×Q₈	—	2	4	no	—
13	C₃×C₅×#117	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×#107	C₂²	1	8	yes	—
16	C₃×C₅×#106	C₂²	1	8	yes	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×#3160	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#533	C₂×C₃	1	2	yes	—
22	C₅×#500	C₂×C₃	1	2	yes	complemented
23	C₅×#499	C₂×C₃	1	4	yes	complemented
24	C₅×#498	C₂×C₃	1	2	yes	complemented
25	C₅×#427	C₂×C₃	1	4	yes	—
26	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₅×Q₈	—	2	4	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
31	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
33	C₃×C₅×C₄⋊C₄	—	2	8	no	—
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
36	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
37	C₃×C₄²×C₅	—	2	4	no	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₃×#533	C₂×C₅	1	2	yes	—
42	C₃×#500	C₂×C₅	1	2	yes	complemented
43	C₃×#499	C₂×C₅	1	4	yes	complemented
44	C₃×#498	C₂×C₅	1	2	yes	complemented
45	C₃×#427	C₂×C₅	1	4	yes	—
46	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
48	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
49	C₅×#125	C₂²×C₃	1	4	yes	—
50	C₅×C₂²⋊Q₈	—	2	4	no	—
51	C₄×C₅×Q₈	—	2	4	no	—
52	C₅×#117	C₂²×C₃	1	4	yes	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×#107	C₂²×C₃	1	8	yes	—
55	C₅×#106	C₂²×C₃	1	8	yes	—
56	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
58	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
59	#3160	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×Q₈	—	4	8	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	8	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
70	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
71	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
72	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
73	C₃×#125	C₂²×C₅	1	4	yes	—
74	C₃×C₂²⋊Q₈	—	2	4	no	—
75	C₃×C₄×Q₈	—	2	4	no	—
76	C₃×#117	C₂²×C₅	1	4	yes	—
77	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
78	C₃×#107	C₂²×C₅	1	8	yes	—
79	C₃×#106	C₂²×C₅	1	8	yes	—
80	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
82	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₅×Q₈	—	2	4	no	—
84	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₅×C₈	C₃×Q₈	1	8	yes	complemented
87	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
88	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₅	—	2	4	no	—
94	C₄²×C₅	C₃×D₄	1	2	yes	—
95	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
96	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
97	#533	C₂×C₃×C₅	1	2	yes	—
98	#500	C₂×C₃×C₅	1	2	yes	complemented
99	#499	C₂×C₃×C₅	1	4	yes	complemented
100	#498	C₂×C₃×C₅	1	2	yes	complemented
101	#427	C₂×C₃×C₅	1	4	yes	—
102	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
105	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	4	8	no	—
108	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
109	C₃×C₄×C₅	#135	1	4	yes	—
110	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
114	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
115	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
116	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
117	C₃×C₄⋊C₄	—	2	8	no	—
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
120	C₃×C₂²⋊C₄	—	2	4	no	—
121	C₃×C₄²	—	2	4	no	—
122	C₃×C₄²	C₅×D₄	1	2	yes	—
123	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
124	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
125	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₅×Q₈	—	4	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	8	no	—
129	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
133	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
134	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
135	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
136	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
138	#125	C₂²×C₃×C₅	1	4	yes	—
139	C₂²⋊Q₈	—	2	4	no	—
140	C₄×Q₈	—	2	4	no	—
141	#117	C₂²×C₃×C₅	1	4	yes	—
142	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
143	#107	C₂²×C₃×C₅	1	8	yes	—
144	#106	C₂²×C₃×C₅	1	8	yes	—
145	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
147	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₂×C₃×C₅	—	2	2	no	—
149	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×#135	1	2	yes	—
151	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
152	C₃×Q₈	—	4	8	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
158	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
159	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
160	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
161	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
164	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	4	8	no	—
167	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
168	C₄×C₅	C₃×#135	1	4	yes	—
169	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
170	C₂×Q₈	—	2	4	no	—
171	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
172	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
173	C₂×C₈	C₃×C₅×Q₈	1	8	yes	complemented
174	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
175	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
176	C₄⋊C₄	—	2	8	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	C₂³×C₃×C₅	1	8	yes	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	4	no	—
181	C₄²	C₃×C₅×D₄	1	2	yes	—
182	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
183	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
184	C₃×C₅	#3160	1	1	yes	characteristic, direct product factor, Hall
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
187	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	4	8	no	—
190	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
191	C₃×C₄	C₅×#135	1	4	yes	—
192	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×C₅	C₂×C₃×#135	1	2	yes	—
196	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
197	Q₈	—	4	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
201	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
203	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
204	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
205	C₈	C₂×C₃×C₅×Q₈	1	4	yes	—
206	C₈	C₂×C₃×C₅×Q₈	1	4	yes	—
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
209	C₂×C₃	C₂×C₅×#135	1	2	yes	—
210	C₅	C₃×#3160	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
213	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	4	8	no	—
216	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
217	C₄	C₃×C₅×#135	1	4	yes	—
218	C₄	C₂²×C₃×C₅×Q₈	1	2	yes	—
219	C₃	C₅×#3160	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	2	2	no	—
221	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
222	C₂	C₂×C₃×C₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
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
2. C₃×C₅×#500	C₂	semidirect
3. C₃×C₅×#499	C₂	semidirect
4. C₃×C₅×#498	C₂	semidirect
7. C₅×#3160	C₃	direct (N × H)
20. C₃×#3160	C₅	direct (N × H)
22. C₅×#500	C₂×C₃	semidirect
23. C₅×#499	C₂×C₃	semidirect
24. C₅×#498	C₂×C₃	semidirect
30. C₂×C₃×C₅×C₈	Q₈	semidirect
42. C₃×#500	C₂×C₅	semidirect
43. C₃×#499	C₂×C₅	semidirect
44. C₃×#498	C₂×C₅	semidirect
59. #3160	C₃×C₅	direct (N × H)
86. C₂×C₅×C₈	C₃×Q₈	semidirect
98. #500	C₂×C₃×C₅	semidirect
99. #499	C₂×C₃×C₅	semidirect
100. #498	C₂×C₃×C₅	semidirect
114. C₂×C₃×C₈	C₅×Q₈	semidirect
173. C₂×C₈	C₃×C₅×Q₈	semidirect
184. C₃×C₅	#3160	direct (N × H)
210. C₅	C₃×#3160	direct (N × H)
219. C₃	C₅×#3160	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×#533	2
C₃×C₅×#500	2
C₃×C₅×#499	2
C₃×C₅×#498	2
C₃×C₅×#427	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#3160	3
C₃×#3160	5

Subgroups — C3×C5×#3160 #14276194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3160 #14276194