Subgroups — C₃×C₅×#1706 #14274740

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#492	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#486	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1706	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×SD₁₆	—	2	4	no	—
8	C₂×C₃×C₅×D₈	—	2	2	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×#127	C₂²	1	2	yes	—
11	C₃×C₅×#107	—	2	2	no	—
12	C₃×C₅×#102	—	2	4	no	—
13	C₃×C₄×C₅×C₈	C₂²	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×#1706	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
18	C₅×#493	C₂×C₃	1	1	yes	characteristic
19	C₅×#492	C₂×C₃	1	2	yes	complemented
20	C₅×#486	C₂×C₃	1	2	yes	complemented
21	C₅×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×D₄	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₃×C₅×SD₁₆	—	4	8	no	—
27	C₃×C₅×D₈	—	4	4	no	—
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
35	C₃×#493	C₂×C₅	1	1	yes	characteristic
36	C₃×#492	C₂×C₅	1	2	yes	complemented
37	C₃×#486	C₂×C₅	1	2	yes	complemented
38	C₃×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂×C₅×SD₁₆	—	2	4	no	—
40	C₂×C₅×D₈	—	2	2	no	—
41	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
42	C₅×#127	C₂²×C₃	1	2	yes	—
43	C₅×#107	—	2	2	no	—
44	C₅×#102	—	2	4	no	—
45	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	#1706	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
49	C₂³×C₃×C₅	—	4	2	no	—
50	C₃×C₅×Q₈	—	4	4	no	—
51	C₃×C₅×D₄	—	4	4	no	—
52	C₃×C₅×D₄	—	4	4	no	—
53	C₃×C₅×D₄	—	4	4	no	—
54	C₂×C₃×C₄×C₅	—	4	2	no	—
55	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	D₈	1	4	yes	complemented
60	C₃×C₅×C₈	SD₁₆	1	4	yes	complemented
61	C₂×C₃×SD₁₆	—	2	4	no	—
62	C₂×C₃×D₈	—	2	2	no	—
63	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
64	C₃×#127	C₂²×C₅	1	2	yes	—
65	C₃×#107	—	2	2	no	—
66	C₃×#102	—	2	4	no	—
67	C₃×C₄×C₈	C₂²×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	C₂×C₅×Q₈	—	2	2	no	—
71	C₂×C₅×D₄	—	2	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₅×SD₁₆	—	4	8	no	—
75	C₅×D₈	—	4	4	no	—
76	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
77	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
78	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
82	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
83	#493	C₂×C₃×C₅	1	1	yes	characteristic
84	#492	C₂×C₃×C₅	1	2	yes	complemented
85	#486	C₂×C₃×C₅	1	2	yes	complemented
86	C₈²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	C₂²×C₃×C₅	—	8	2	no	—
88	C₂²×C₃×C₅	—	8	2	no	—
89	C₂²×C₃×C₅	—	8	2	no	—
90	C₂²×C₃×C₅	#127	1	1	yes	characteristic
91	C₃×C₄×C₅	—	8	2	no	—
92	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
93	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
94	C₃×C₄×C₅	#135	1	2	yes	—
95	C₂×C₃×Q₈	—	2	2	no	—
96	C₂×C₃×D₄	—	2	2	no	—
97	C₂×C₃×D₄	—	2	2	no	—
98	C₂×C₃×D₄	—	2	2	no	—
99	C₃×SD₁₆	—	4	8	no	—
100	C₃×D₈	—	4	4	no	—
101	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
102	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
103	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
107	C₂³×C₅	—	4	2	no	—
108	C₅×Q₈	—	4	4	no	—
109	C₅×D₄	—	4	4	no	—
110	C₅×D₄	—	4	4	no	—
111	C₅×D₄	—	4	4	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	C₃×D₈	1	4	yes	complemented
118	C₅×C₈	C₃×SD₁₆	1	4	yes	complemented
119	C₂×SD₁₆	—	2	4	no	—
120	C₂×D₈	—	2	2	no	—
121	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
122	#127	C₂²×C₃×C₅	1	2	yes	—
123	#107	—	2	2	no	—
124	#102	—	2	4	no	—
125	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
126	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
127	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₂×C₃×C₅	—	16	2	no	—
129	C₂×C₃×C₅	#492	1	1	yes	characteristic
130	C₂×C₃×C₅	#493	1	1	yes	characteristic
131	C₂×C₃×C₅	#495	1	1	yes	characteristic
132	C₂³×C₃	—	4	2	no	—
133	C₃×Q₈	—	4	4	no	—
134	C₃×D₄	—	4	4	no	—
135	C₃×D₄	—	4	4	no	—
136	C₃×D₄	—	4	4	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	C₅×D₈	1	4	yes	complemented
143	C₃×C₈	C₅×SD₁₆	1	4	yes	complemented
144	C₂²×C₅	—	8	2	no	—
145	C₂²×C₅	—	8	2	no	—
146	C₂²×C₅	—	8	2	no	—
147	C₂²×C₅	C₃×#127	1	1	yes	characteristic
148	C₄×C₅	—	8	2	no	—
149	C₄×C₅	C₂×C₃×D₈	1	2	yes	—
150	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
151	C₄×C₅	C₃×#135	1	2	yes	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	SD₁₆	—	4	8	no	—
157	D₈	—	4	4	no	—
158	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
159	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
160	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
164	C₃×C₅	#1706	1	1	yes	characteristic, direct product factor, Hall
165	C₂²×C₃	—	8	2	no	—
166	C₂²×C₃	—	8	2	no	—
167	C₂²×C₃	—	8	2	no	—
168	C₂²×C₃	C₅×#127	1	1	yes	characteristic
169	C₃×C₄	—	8	2	no	—
170	C₃×C₄	C₂×C₅×D₈	1	2	yes	—
171	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
172	C₃×C₄	C₅×#135	1	2	yes	—
173	C₂×C₅	—	16	2	no	—
174	C₂×C₅	C₃×#492	1	1	yes	characteristic
175	C₂×C₅	C₃×#493	1	1	yes	characteristic
176	C₂×C₅	C₃×#495	1	1	yes	characteristic
177	C₂³	—	4	2	no	—
178	Q₈	—	4	4	no	—
179	D₄	—	4	4	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	C₃×C₅×D₈	1	4	yes	complemented
188	C₈	C₃×C₅×SD₁₆	1	4	yes	complemented
189	C₂×C₃	—	16	2	no	—
190	C₂×C₃	C₅×#492	1	1	yes	characteristic
191	C₂×C₃	C₅×#493	1	1	yes	characteristic
192	C₂×C₃	C₅×#495	1	1	yes	characteristic
193	C₅	C₃×#1706	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	8	2	no	—
195	C₂²	—	8	2	no	—
196	C₂²	—	8	2	no	—
197	C₂²	C₃×C₅×#127	1	1	yes	characteristic
198	C₄	—	8	2	no	—
199	C₄	C₂×C₃×C₅×D₈	1	2	yes	—
200	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
201	C₄	C₃×C₅×#135	1	2	yes	—
202	C₃	C₅×#1706	1	1	yes	characteristic, direct product factor, Sylow
203	C₂	—	16	2	no	—
204	C₂	C₃×C₅×#492	1	1	yes	characteristic
205	C₂	C₃×C₅×#493	1	1	yes	characteristic
206	C₂	C₃×C₅×#495	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#500	C₂	semidirect
3. C₃×C₅×#492	C₂	semidirect
4. C₃×C₅×#486	C₂	semidirect
5. C₃×C₅×C₈²	C₂	semidirect
6. C₅×#1706	C₃	direct (N × H)
16. C₃×#1706	C₅	direct (N × H)
17. C₅×#500	C₂×C₃	semidirect
19. C₅×#492	C₂×C₃	semidirect
20. C₅×#486	C₂×C₃	semidirect
21. C₅×C₈²	C₂×C₃	semidirect
34. C₃×#500	C₂×C₅	semidirect
36. C₃×#492	C₂×C₅	semidirect
37. C₃×#486	C₂×C₅	semidirect
38. C₃×C₈²	C₂×C₅	semidirect
48. #1706	C₃×C₅	direct (N × H)
59. C₃×C₅×C₈	D₈	semidirect
60. C₃×C₅×C₈	SD₁₆	semidirect
82. #500	C₂×C₃×C₅	semidirect
84. #492	C₂×C₃×C₅	semidirect
85. #486	C₂×C₃×C₅	semidirect
86. C₈²	C₂×C₃×C₅	semidirect
117. C₅×C₈	C₃×D₈	semidirect
118. C₅×C₈	C₃×SD₁₆	semidirect
142. C₃×C₈	C₅×D₈	semidirect
143. C₃×C₈	C₅×SD₁₆	semidirect
164. C₃×C₅	#1706	direct (N × H)
187. C₈	C₃×C₅×D₈	semidirect
188. C₈	C₃×C₅×SD₁₆	semidirect
193. C₅	C₃×#1706	direct (N × H)
202. C₃	C₅×#1706	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×#500	2
C₃×C₅×#493	2
C₃×C₅×#492	2
C₃×C₅×#486	2
C₃×C₅×C₈²	2
C₅×#1706	3
C₃×#1706	5

Subgroups — C3×C5×#1706 #14274740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1706 #14274740