Subgroups — C₃×C₅×#1572 #14274606

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#431	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#334	C₂	1	2	yes	maximal, complemented
6	C₅×#1572	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
9	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
12	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
14	C₃×C₅×#97	—	2	1	no	—
15	C₃×C₄×C₅×C₈	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1572	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#432	C₂×C₃	1	1	yes	characteristic
20	C₅×#431	C₂×C₃	1	1	yes	characteristic
21	C₅×#431	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	C₂×C₃	1	2	yes	complemented
23	C₅×#334	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×#432	C₂×C₅	1	1	yes	characteristic
36	C₃×#431	C₂×C₅	1	1	yes	characteristic
37	C₃×#431	C₂×C₅	1	1	yes	characteristic
38	C₃×#335	C₂×C₅	1	2	yes	complemented
39	C₃×#334	C₂×C₅	1	2	yes	complemented
40	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₅×C₄⋊C₈	C₃×C₄	1	2	yes	—
42	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
43	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
44	C₅×C₂²⋊C₈	—	2	1	no	—
45	C₅×C₂²⋊C₈	—	2	1	no	—
46	C₅×C₂²⋊C₈	C₃×C₄	1	2	yes	—
47	C₅×#97	—	2	1	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	#1572	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	4	1	no	—
56	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×C₈	—	2	2	no	—
63	C₃×C₅×C₈	—	2	2	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₃×C₅×C₈	—	4	2	no	—
66	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
67	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
68	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
69	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
70	C₃×C₂²⋊C₈	—	2	1	no	—
71	C₃×C₂²⋊C₈	—	2	1	no	—
72	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
73	C₃×#97	—	2	1	no	—
74	C₃×C₄×C₈	—	2	1	no	—
75	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
84	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
86	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
87	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
88	#432	C₂×C₃×C₅	1	1	yes	characteristic
89	#431	C₂×C₃×C₅	1	1	yes	characteristic
90	#431	C₂×C₃×C₅	1	1	yes	characteristic
91	#335	C₂×C₃×C₅	1	2	yes	complemented
92	#334	C₂×C₃×C₅	1	2	yes	complemented
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	#131	1	2	yes	—
102	C₃×C₄×C₅	#135	1	2	yes	—
103	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
109	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
110	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
112	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
113	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
114	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	—	4	2	no	—
127	C₅×C₈	—	4	2	no	—
128	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₄⋊C₈	C₃×C₄×C₅	1	2	yes	—
130	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
131	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
132	C₂²⋊C₈	—	2	1	no	—
133	C₂²⋊C₈	—	2	1	no	—
134	C₂²⋊C₈	C₃×C₄×C₅	1	2	yes	—
135	#97	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
138	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₂×C₃×C₅	—	4	1	no	—
140	C₂×C₃×C₅	#424	1	1	yes	characteristic
141	C₂×C₃×C₅	#427	1	1	yes	characteristic
142	C₂×C₃×C₅	#430	1	1	yes	characteristic
143	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	4	2	no	—
156	C₃×C₈	—	4	2	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	C₃×#131	1	2	yes	—
166	C₄×C₅	C₃×#135	1	2	yes	—
167	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
169	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
173	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
174	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
175	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
176	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
177	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
178	C₃×C₅	#1572	1	1	yes	characteristic, direct product factor, Hall
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	C₅×#131	1	2	yes	—
188	C₃×C₄	C₅×#135	1	2	yes	—
189	C₂×C₅	—	4	1	no	—
190	C₂×C₅	C₃×#424	1	1	yes	characteristic
191	C₂×C₅	C₃×#427	1	1	yes	characteristic
192	C₂×C₅	C₃×#430	1	1	yes	characteristic
193	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	4	2	no	—
207	C₂×C₃	—	4	1	no	—
208	C₂×C₃	C₅×#424	1	1	yes	characteristic
209	C₂×C₃	C₅×#427	1	1	yes	characteristic
210	C₂×C₃	C₅×#430	1	1	yes	characteristic
211	C₅	C₃×#1572	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	C₃×C₅×#131	1	2	yes	—
221	C₄	C₃×C₅×#135	1	2	yes	—
222	C₃	C₅×#1572	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	4	1	no	—
224	C₂	C₃×C₅×#424	1	1	yes	characteristic
225	C₂	C₃×C₅×#427	1	1	yes	characteristic
226	C₂	C₃×C₅×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
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
4. C₃×C₅×#335	C₂	semidirect
5. C₃×C₅×#334	C₂	semidirect
6. C₅×#1572	C₃	direct (N × H)
18. C₃×#1572	C₅	direct (N × H)
22. C₅×#335	C₂×C₃	semidirect
23. C₅×#334	C₂×C₃	semidirect
38. C₃×#335	C₂×C₅	semidirect
39. C₃×#334	C₂×C₅	semidirect
51. #1572	C₃×C₅	direct (N × H)
91. #335	C₂×C₃×C₅	semidirect
92. #334	C₂×C₃×C₅	semidirect
178. C₃×C₅	#1572	direct (N × H)
211. C₅	C₃×#1572	direct (N × H)
222. C₃	C₅×#1572	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
C₃×C₅×#335	2
C₃×C₅×#334	2
C₅×#1572	3
C₃×#1572	5

Subgroups — C3×C5×#1572 #14274606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1572 #14274606