Subgroups — C₃×C₅×#3531 #14276565

Order 1920.

1860 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#564	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#563	C₂	1	10	yes	maximal, complemented
3	C₃×C₅×#562	C₂	1	5	yes	maximal, complemented
4	C₃×C₅×#533	C₂	1	5	yes	maximal, complemented
5	C₃×C₅×#529	C₂	1	10	yes	maximal, complemented
6	C₅×#3531	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×C₄⋊Q₈	C₂²	1	10	yes	—
8	C₃×C₅×#126	C₂²	1	20	yes	—
9	C₃×C₅×#125	C₂²	1	10	yes	—
10	C₃×C₅×#125	C₂²	1	5	yes	—
11	C₃×C₅×#125	C₂²	1	5	yes	—
12	C₃×C₅×#124	C₂²	1	10	yes	—
13	C₃×C₅×#123	C₂²	1	20	yes	—
14	C₃×C₅×C₂²⋊Q₈	C₂²	1	10	yes	—
15	C₃×C₅×C₂²⋊Q₈	C₂²	1	20	yes	—
16	C₃×C₅×#121	C₂²	1	10	yes	—
17	C₃×C₄×C₅×Q₈	C₂²	1	10	yes	—
18	C₃×C₄×C₅×D₄	C₂²	1	10	yes	—
19	C₃×C₅×#117	C₂²	1	10	yes	—
20	C₂×C₃×C₄²×C₅	C₂²	1	5	yes	—
21	C₃×#3531	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#564	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#563	C₂×C₃	1	10	yes	complemented
24	C₅×#562	C₂×C₃	1	5	yes	complemented
25	C₅×#533	C₂×C₃	1	5	yes	complemented
26	C₅×#529	C₂×C₃	1	10	yes	complemented
27	C₂×C₃×C₅×Q₈	C₂³	1	10	yes	—
28	C₂×C₃×C₅×D₄	C₂³	1	10	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	5	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	10	yes	—
31	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
32	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
33	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
34	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
35	C₃×C₅×C₄⋊C₄	C₂³	1	20	yes	—
36	C₃×C₅×C₂²⋊C₄	C₂³	1	20	yes	—
37	C₃×C₅×C₂²⋊C₄	C₂³	1	20	yes	—
38	C₃×C₄²×C₅	C₂³	1	5	yes	—
39	C₃×C₄²×C₅	C₂³	1	5	yes	—
40	C₃×C₄²×C₅	C₂³	1	10	yes	—
41	C₃×#564	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#563	C₂×C₅	1	10	yes	complemented
43	C₃×#562	C₂×C₅	1	5	yes	complemented
44	C₃×#533	C₂×C₅	1	5	yes	complemented
45	C₃×#529	C₂×C₅	1	10	yes	complemented
46	C₅×C₄⋊Q₈	C₂²×C₃	1	10	yes	—
47	C₅×#126	C₂²×C₃	1	20	yes	—
48	C₅×#125	C₂²×C₃	1	10	yes	—
49	C₅×#125	C₂²×C₃	1	5	yes	—
50	C₅×#125	C₂²×C₃	1	5	yes	—
51	C₅×#124	C₂²×C₃	1	10	yes	—
52	C₅×#123	C₂²×C₃	1	20	yes	—
53	C₅×C₂²⋊Q₈	C₂²×C₃	1	10	yes	—
54	C₅×C₂²⋊Q₈	C₂²×C₃	1	20	yes	—
55	C₅×#121	C₂²×C₃	1	10	yes	—
56	C₄×C₅×Q₈	C₂²×C₃	1	10	yes	—
57	C₄×C₅×D₄	C₂²×C₃	1	10	yes	—
58	C₅×#117	C₂²×C₃	1	10	yes	—
59	C₂×C₄²×C₅	C₂²×C₃	1	5	yes	—
60	#3531	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
61	C₂³×C₃×C₅	C₂⁴	1	5	yes	—
62	C₃×C₅×Q₈	—	4	10	no	—
63	C₃×C₅×D₄	—	4	10	no	—
64	C₂×C₃×C₄×C₅	—	2	10	no	—
65	C₂×C₃×C₄×C₅	—	4	10	no	—
66	C₂×C₃×C₄×C₅	C₂⁴	1	5	yes	—
67	C₂×C₃×C₄×C₅	C₂⁴	1	10	yes	—
68	C₂×C₃×C₄×C₅	C₂⁴	1	10	yes	—
69	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
70	C₃×C₄⋊Q₈	C₂²×C₅	1	10	yes	—
71	C₃×#126	C₂²×C₅	1	20	yes	—
72	C₃×#125	C₂²×C₅	1	5	yes	—
73	C₃×#125	C₂²×C₅	1	5	yes	—
74	C₃×#125	C₂²×C₅	1	10	yes	—
75	C₃×#124	C₂²×C₅	1	10	yes	—
76	C₃×#123	C₂²×C₅	1	20	yes	—
77	C₃×C₂²⋊Q₈	C₂²×C₅	1	20	yes	—
78	C₃×C₂²⋊Q₈	C₂²×C₅	1	10	yes	—
79	C₃×#121	C₂²×C₅	1	10	yes	—
80	C₃×C₄×Q₈	C₂²×C₅	1	10	yes	—
81	C₃×C₄×D₄	C₂²×C₅	1	10	yes	—
82	C₃×#117	C₂²×C₅	1	10	yes	—
83	C₂×C₃×C₄²	C₂²×C₅	1	5	yes	—
84	C₂×C₅×Q₈	C₂³×C₃	1	10	yes	—
85	C₂×C₅×D₄	C₂³×C₃	1	10	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	5	yes	—
87	C₂²×C₄×C₅	C₂³×C₃	1	10	yes	—
88	C₅×C₄⋊C₄	C₂³×C₃	1	10	yes	—
89	C₅×C₄⋊C₄	C₂³×C₃	1	10	yes	—
90	C₅×C₄⋊C₄	C₂³×C₃	1	10	yes	—
91	C₅×C₄⋊C₄	C₂³×C₃	1	10	yes	—
92	C₅×C₄⋊C₄	C₂³×C₃	1	20	yes	—
93	C₅×C₂²⋊C₄	C₂³×C₃	1	20	yes	—
94	C₅×C₂²⋊C₄	C₂³×C₃	1	20	yes	—
95	C₄²×C₅	C₂³×C₃	1	10	yes	—
96	C₄²×C₅	C₂³×C₃	1	5	yes	—
97	C₄²×C₅	C₂³×C₃	1	5	yes	—
98	#564	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	#563	C₂×C₃×C₅	1	10	yes	complemented
100	#562	C₂×C₃×C₅	1	5	yes	complemented
101	#533	C₂×C₃×C₅	1	5	yes	complemented
102	#529	C₂×C₃×C₅	1	10	yes	complemented
103	C₂²×C₃×C₅	—	2	5	no	—
104	C₂²×C₃×C₅	—	2	10	no	—
105	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	5	no	—
107	C₃×C₄×C₅	—	2	10	no	—
108	C₃×C₄×C₅	—	2	10	no	—
109	C₃×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
110	C₂×C₃×Q₈	C₂³×C₅	1	10	yes	—
111	C₂×C₃×D₄	C₂³×C₅	1	10	yes	—
112	C₂²×C₃×C₄	C₂³×C₅	1	5	yes	—
113	C₂²×C₃×C₄	C₂³×C₅	1	10	yes	—
114	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
115	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
116	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
117	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
118	C₃×C₄⋊C₄	C₂³×C₅	1	20	yes	—
119	C₃×C₂²⋊C₄	C₂³×C₅	1	20	yes	—
120	C₃×C₂²⋊C₄	C₂³×C₅	1	20	yes	—
121	C₃×C₄²	C₂³×C₅	1	10	yes	—
122	C₃×C₄²	C₂³×C₅	1	5	yes	—
123	C₃×C₄²	C₂³×C₅	1	5	yes	—
124	C₂³×C₅	C₂⁴×C₃	1	5	yes	—
125	C₅×Q₈	—	4	10	no	—
126	C₅×D₄	—	4	10	no	—
127	C₂×C₄×C₅	—	2	10	no	—
128	C₂×C₄×C₅	—	4	10	no	—
129	C₂×C₄×C₅	C₂⁴×C₃	1	5	yes	—
130	C₂×C₄×C₅	C₂⁴×C₃	1	10	yes	—
131	C₂×C₄×C₅	C₂⁴×C₃	1	10	yes	—
132	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
133	C₄⋊Q₈	C₂²×C₃×C₅	1	10	yes	—
134	#126	C₂²×C₃×C₅	1	20	yes	—
135	#125	C₂²×C₃×C₅	1	10	yes	—
136	#125	C₂²×C₃×C₅	1	5	yes	—
137	#125	C₂²×C₃×C₅	1	5	yes	—
138	#124	C₂²×C₃×C₅	1	10	yes	—
139	#123	C₂²×C₃×C₅	1	20	yes	—
140	C₂²⋊Q₈	C₂²×C₃×C₅	1	10	yes	—
141	C₂²⋊Q₈	C₂²×C₃×C₅	1	20	yes	—
142	#121	C₂²×C₃×C₅	1	10	yes	—
143	C₄×Q₈	C₂²×C₃×C₅	1	10	yes	—
144	C₄×D₄	C₂²×C₃×C₅	1	10	yes	—
145	#117	C₂²×C₃×C₅	1	10	yes	—
146	C₂×C₄²	C₂²×C₃×C₅	1	5	yes	—
147	C₂×C₃×C₅	—	4	5	no	—
148	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
149	C₂×C₃×C₅	#585	1	2	yes	—
150	C₂³×C₃	C₂⁴×C₅	1	5	yes	—
151	C₃×Q₈	—	4	10	no	—
152	C₃×D₄	—	4	10	no	—
153	C₂×C₃×C₄	—	2	10	no	—
154	C₂×C₃×C₄	—	4	10	no	—
155	C₂×C₃×C₄	C₂⁴×C₅	1	5	yes	—
156	C₂×C₃×C₄	C₂⁴×C₅	1	10	yes	—
157	C₂×C₃×C₄	C₂⁴×C₅	1	10	yes	—
158	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
159	C₂²×C₅	—	2	5	no	—
160	C₂²×C₅	—	2	10	no	—
161	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
162	C₄×C₅	—	2	5	no	—
163	C₄×C₅	—	2	10	no	—
164	C₄×C₅	—	2	10	no	—
165	C₄×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
166	C₂×Q₈	C₂³×C₃×C₅	1	10	yes	—
167	C₂×D₄	C₂³×C₃×C₅	1	10	yes	—
168	C₂²×C₄	C₂³×C₃×C₅	1	5	yes	—
169	C₂²×C₄	C₂³×C₃×C₅	1	10	yes	—
170	C₄⋊C₄	C₂³×C₃×C₅	1	10	yes	—
171	C₄⋊C₄	C₂³×C₃×C₅	1	10	yes	—
172	C₄⋊C₄	C₂³×C₃×C₅	1	10	yes	—
173	C₄⋊C₄	C₂³×C₃×C₅	1	10	yes	—
174	C₄⋊C₄	C₂³×C₃×C₅	1	20	yes	—
175	C₂²⋊C₄	C₂³×C₃×C₅	1	20	yes	—
176	C₂²⋊C₄	C₂³×C₃×C₅	1	20	yes	—
177	C₄²	C₂³×C₃×C₅	1	10	yes	—
178	C₄²	C₂³×C₃×C₅	1	5	yes	—
179	C₄²	C₂³×C₃×C₅	1	5	yes	—
180	C₃×C₅	#3531	1	1	yes	characteristic, direct product factor, Hall
181	C₂²×C₃	—	2	5	no	—
182	C₂²×C₃	—	2	10	no	—
183	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
184	C₃×C₄	—	2	5	no	—
185	C₃×C₄	—	2	10	no	—
186	C₃×C₄	—	2	10	no	—
187	C₃×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
188	C₂×C₅	—	4	5	no	—
189	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×C₅	C₃×#585	1	2	yes	—
191	C₂³	C₂⁴×C₃×C₅	1	5	yes	—
192	Q₈	—	4	10	no	—
193	D₄	—	4	10	no	—
194	C₂×C₄	—	2	10	no	—
195	C₂×C₄	—	4	10	no	—
196	C₂×C₄	C₂⁴×C₃×C₅	1	5	yes	—
197	C₂×C₄	C₂⁴×C₃×C₅	1	10	yes	—
198	C₂×C₄	C₂⁴×C₃×C₅	1	10	yes	—
199	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
200	C₂×C₃	—	4	5	no	—
201	C₂×C₃	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
202	C₂×C₃	C₅×#585	1	2	yes	—
203	C₅	C₃×#3531	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	2	5	no	—
205	C₂²	—	2	10	no	—
206	C₂²	C₂⁵×C₃×C₅	1	1	yes	characteristic
207	C₄	—	2	5	no	—
208	C₄	—	2	10	no	—
209	C₄	—	2	10	no	—
210	C₄	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
211	C₃	C₅×#3531	1	1	yes	characteristic, direct product factor, Sylow
212	C₂	—	4	5	no	—
213	C₂	C₂×C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂	C₃×C₅×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃×C₅
socle	C₂²×C₃×C₅	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#564	C₂	semidirect
2. C₃×C₅×#563	C₂	semidirect
3. C₃×C₅×#562	C₂	semidirect
4. C₃×C₅×#533	C₂	semidirect
5. C₃×C₅×#529	C₂	semidirect
6. C₅×#3531	C₃	direct (N × H)
21. C₃×#3531	C₅	direct (N × H)
22. C₅×#564	C₂×C₃	semidirect
23. C₅×#563	C₂×C₃	semidirect
24. C₅×#562	C₂×C₃	semidirect
25. C₅×#533	C₂×C₃	semidirect
26. C₅×#529	C₂×C₃	semidirect
41. C₃×#564	C₂×C₅	semidirect
42. C₃×#563	C₂×C₅	semidirect
43. C₃×#562	C₂×C₅	semidirect
44. C₃×#533	C₂×C₅	semidirect
45. C₃×#529	C₂×C₅	semidirect
60. #3531	C₃×C₅	direct (N × H)
98. #564	C₂×C₃×C₅	semidirect
99. #563	C₂×C₃×C₅	semidirect
100. #562	C₂×C₃×C₅	semidirect
101. #533	C₂×C₃×C₅	semidirect
102. #529	C₂×C₃×C₅	semidirect
180. C₃×C₅	#3531	direct (N × H)
203. C₅	C₃×#3531	direct (N × H)
211. C₃	C₅×#3531	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×#564	2
C₃×C₅×#563	2
C₃×C₅×#562	2
C₃×C₅×#533	2
C₃×C₅×#529	2
C₅×#3531	3
C₃×#3531	5

Subgroups — C3×C5×#3531 #14276565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3531 #14276565