Subgroups — C₃×C₅×#1287 #14274321

Order 1920.

252 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×#432	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
4	C₅×#1287	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×C₄⋊C₈	—	2	1	no	—
7	C₃×C₅×C₄⋊C₈	—	2	1	no	—
8	C₃×C₅×C₄⋊C₈	—	2	1	no	—
9	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
10	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
11	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
12	C₃×C₅×#97	—	2	1	no	—
13	C₃×C₅×#97	C₄	1	2	yes	—
14	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
15	C₃×#1287	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×#432	C₂×C₃	1	1	yes	characteristic
17	C₅×#432	C₂×C₃	1	1	yes	characteristic
18	C₅×#431	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	Q₈	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	—	2	2	no	—
25	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₃×#432	C₂×C₅	1	1	yes	characteristic
30	C₃×#432	C₂×C₅	1	1	yes	characteristic
31	C₃×#431	C₂×C₅	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	1	no	—
34	C₅×C₄⋊C₈	—	2	1	no	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₂²⋊C₈	—	2	1	no	—
37	C₅×C₂²⋊C₈	—	2	1	no	—
38	C₅×C₂²⋊C₈	C₃×C₄	1	2	yes	—
39	C₅×#97	—	2	1	no	—
40	C₅×#97	C₃×C₄	1	2	yes	—
41	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
42	#1287	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	—	2	2	no	—
45	C₂×C₃×C₄×C₅	—	4	1	no	—
46	C₂×C₃×C₄×C₅	—	4	1	no	—
47	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₃×C₅×C₈	—	4	2	no	—
54	C₃×C₅×C₈	—	4	2	no	—
55	C₃×C₅×C₈	—	4	2	no	—
56	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
57	C₃×C₄⋊C₈	—	2	1	no	—
58	C₃×C₄⋊C₈	—	2	1	no	—
59	C₃×C₄⋊C₈	—	2	1	no	—
60	C₃×C₂²⋊C₈	—	2	1	no	—
61	C₃×C₂²⋊C₈	—	2	1	no	—
62	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
63	C₃×#97	—	2	1	no	—
64	C₃×#97	C₄×C₅	1	2	yes	—
65	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
66	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
75	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
76	#432	C₂×C₃×C₅	1	1	yes	characteristic
77	#432	C₂×C₃×C₅	1	1	yes	characteristic
78	#431	C₂×C₃×C₅	1	1	yes	characteristic
79	C₂²×C₃×C₅	—	2	1	no	—
80	C₂²×C₃×C₅	—	2	1	no	—
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	#95	1	1	yes	characteristic
83	C₃×C₄×C₅	—	2	1	no	—
84	C₃×C₄×C₅	—	2	1	no	—
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	—	2	1	no	—
87	C₃×C₄×C₅	—	2	1	no	—
88	C₃×C₄×C₅	#108	1	2	yes	—
89	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
96	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
97	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
98	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
99	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	4	1	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
109	C₅×C₈	—	4	2	no	—
110	C₅×C₈	—	4	2	no	—
111	C₅×C₈	—	4	2	no	—
112	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₂²⋊C₈	—	2	1	no	—
117	C₂²⋊C₈	—	2	1	no	—
118	C₂²⋊C₈	C₃×C₄×C₅	1	2	yes	—
119	#97	—	2	1	no	—
120	#97	C₃×C₄×C₅	1	2	yes	—
121	C₄×C₈	C₃×C₄×C₅	1	2	yes	—
122	C₂×C₃×C₅	—	4	1	no	—
123	C₂×C₃×C₅	H(3,ℤ/4)	1	1	yes	characteristic
124	C₂×C₃×C₅	#341	1	1	yes	characteristic
125	C₂×C₃×C₅	#344	1	1	yes	characteristic
126	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	4	1	no	—
129	C₂×C₃×C₄	—	4	1	no	—
130	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
131	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
136	C₃×C₈	—	4	2	no	—
137	C₃×C₈	—	4	2	no	—
138	C₃×C₈	—	4	2	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	—	2	1	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	C₃×#95	1	1	yes	characteristic
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	C₃×#108	1	2	yes	—
149	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
150	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
156	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
157	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
158	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
159	C₃×C₅	#1287	1	1	yes	characteristic, direct product factor, Hall
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	C₅×#95	1	1	yes	characteristic
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	C₅×#108	1	2	yes	—
170	C₂×C₅	—	4	1	no	—
171	C₂×C₅	C₃×H(3,ℤ/4)	1	1	yes	characteristic
172	C₂×C₅	C₃×#341	1	1	yes	characteristic
173	C₂×C₅	C₃×#344	1	1	yes	characteristic
174	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₃×C₄²×C₅	1	1	yes	characteristic
179	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₈	—	4	2	no	—
185	C₈	—	4	2	no	—
186	C₈	—	4	2	no	—
187	C₂×C₃	—	4	1	no	—
188	C₂×C₃	C₅×H(3,ℤ/4)	1	1	yes	characteristic
189	C₂×C₃	C₅×#341	1	1	yes	characteristic
190	C₂×C₃	C₅×#344	1	1	yes	characteristic
191	C₅	C₃×#1287	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	C₃×C₅×#95	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	C₃×C₅×#108	1	2	yes	—
202	C₃	C₅×#1287	1	1	yes	characteristic, direct product factor, Sylow
203	C₂	—	4	1	no	—
204	C₂	C₃×C₅×H(3,ℤ/4)	1	1	yes	characteristic
205	C₂	C₃×C₅×#341	1	1	yes	characteristic
206	C₂	C₃×C₅×#344	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#95
commutator	C₂×C₄	C₃×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂²×C₃×C₅
prime radical	C₂³×C₃×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1287	C₃	direct (N × H)
15. C₃×#1287	C₅	direct (N × H)
42. #1287	C₃×C₅	direct (N × H)
159. C₃×C₅	#1287	direct (N × H)
191. C₅	C₃×#1287	direct (N × H)
202. C₃	C₅×#1287	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×#432	2
C₃×C₅×#431	2
C₅×#1287	3
C₃×#1287	5

Subgroups — C3×C5×#1287 #14274321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1287 #14274321