Subgroups — C₃×C₅×#464 #11751439

Order 960.

224 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	C₂×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#102	C₂	1	1	yes	maximal, characteristic
8	C₅×#464	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₃×C₅×Q₁₆	—	2	2	no	—
13	C₃×C₅×SD₁₆	—	2	2	no	—
14	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	—	2	1	no	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
19	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
20	C₃×C₄²×C₅	—	2	1	no	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×#464	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₅×SD₁₆	C₂×C₃	1	1	yes	characteristic
25	C₅×#124	C₂×C₃	1	1	yes	characteristic
26	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#103	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#102	C₂×C₃	1	1	yes	characteristic
30	C₂³×C₃×C₅	—	2	1	no	—
31	C₃×C₅×Q₈	—	2	1	no	—
32	C₃×C₅×Q₈	—	2	1	no	—
33	C₃×C₅×Q₈	—	2	1	no	—
34	C₃×C₅×Q₈	D₄	1	2	yes	—
35	C₃×C₅×D₄	—	2	1	no	—
36	C₃×C₅×D₄	—	2	1	no	—
37	C₂×C₃×C₄×C₅	—	2	1	no	—
38	C₂×C₃×C₄×C₅	—	2	1	no	—
39	C₂×C₃×C₄×C₅	—	2	1	no	—
40	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₅×C₈	—	2	1	no	—
44	C₃×C₅×C₈	—	2	1	no	—
45	C₂×C₃×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×C₃×SD₁₆	C₂×C₅	1	1	yes	characteristic
47	C₃×#124	C₂×C₅	1	1	yes	characteristic
48	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×#103	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#102	C₂×C₅	1	1	yes	characteristic
52	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
55	C₅×Q₁₆	—	2	2	no	—
56	C₅×SD₁₆	—	2	2	no	—
57	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	1	no	—
60	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₅×C₂²⋊C₄	—	2	1	no	—
62	C₅×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₅	—	2	1	no	—
64	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
65	#464	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₃×C₅	—	4	1	no	—
67	C₂²×C₃×C₅	—	4	1	no	—
68	C₂²×C₃×C₅	—	4	1	no	—
69	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×C₅	—	2	2	no	—
71	C₃×C₄×C₅	—	2	1	no	—
72	C₃×C₄×C₅	—	2	1	no	—
73	C₃×C₄×C₅	—	4	1	no	—
74	C₃×C₄×C₅	—	4	1	no	—
75	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
80	C₃×Q₁₆	—	2	2	no	—
81	C₃×SD₁₆	—	2	2	no	—
82	C₂×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₄²	—	2	1	no	—
89	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	C₂³×C₅	—	2	1	no	—
91	C₅×Q₈	—	2	1	no	—
92	C₅×Q₈	—	2	1	no	—
93	C₅×Q₈	—	2	1	no	—
94	C₅×Q₈	C₃×D₄	1	2	yes	—
95	C₅×D₄	—	2	1	no	—
96	C₅×D₄	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	C₃×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₅×C₈	—	2	1	no	—
104	C₅×C₈	—	2	1	no	—
105	C₂×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	C₂×SD₁₆	C₂×C₃×C₅	1	1	yes	characteristic
107	#124	C₂×C₃×C₅	1	1	yes	characteristic
108	C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	#103	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	#102	C₂×C₃×C₅	1	1	yes	characteristic
112	C₂×C₃×C₅	—	8	1	no	—
113	C₂×C₃×C₅	#121	1	1	yes	characteristic
114	C₂×C₃×C₅	#135	1	1	yes	characteristic
115	C₂×C₃×C₅	#137	1	1	yes	characteristic
116	C₂³×C₃	—	2	1	no	—
117	C₃×Q₈	—	2	1	no	—
118	C₃×Q₈	—	2	1	no	—
119	C₃×Q₈	—	2	1	no	—
120	C₃×Q₈	C₅×D₄	1	2	yes	—
121	C₃×D₄	—	2	1	no	—
122	C₃×D₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	C₅×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₅×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	—	2	1	no	—
131	C₂²×C₅	—	4	1	no	—
132	C₂²×C₅	—	4	1	no	—
133	C₂²×C₅	—	4	1	no	—
134	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
135	C₄×C₅	—	2	2	no	—
136	C₄×C₅	—	2	1	no	—
137	C₄×C₅	—	2	1	no	—
138	C₄×C₅	—	4	1	no	—
139	C₄×C₅	—	4	1	no	—
140	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
141	C₄×C₅	C₃×#41	1	1	yes	characteristic
142	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
143	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₂×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
145	Q₁₆	—	2	2	no	—
146	SD₁₆	—	2	2	no	—
147	C₂×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂²⋊C₄	—	2	1	no	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
155	C₃×C₅	#464	1	1	yes	characteristic, direct product factor, Hall
156	C₂²×C₃	—	4	1	no	—
157	C₂²×C₃	—	4	1	no	—
158	C₂²×C₃	—	4	1	no	—
159	C₂²×C₃	C₂×C₅×D₄	1	1	yes	characteristic
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	4	1	no	—
164	C₃×C₄	—	4	1	no	—
165	C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₃×C₄	C₅×#41	1	1	yes	characteristic
167	C₂×C₅	—	8	1	no	—
168	C₂×C₅	C₃×#121	1	1	yes	characteristic
169	C₂×C₅	C₃×#135	1	1	yes	characteristic
170	C₂×C₅	C₃×#137	1	1	yes	characteristic
171	C₂³	—	2	1	no	—
172	Q₈	—	2	1	no	—
173	Q₈	—	2	1	no	—
174	Q₈	—	2	1	no	—
175	Q₈	C₃×C₅×D₄	1	2	yes	—
176	D₄	—	2	1	no	—
177	D₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	C₃×C₅×D₄	1	1	yes	characteristic
182	C₂×C₄	C₃×C₅×D₄	1	1	yes	characteristic
183	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₂×C₃	—	8	1	no	—
187	C₂×C₃	C₅×#121	1	1	yes	characteristic
188	C₂×C₃	C₅×#135	1	1	yes	characteristic
189	C₂×C₃	C₅×#137	1	1	yes	characteristic
190	C₅	C₃×#464	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	4	1	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	1	no	—
194	C₂²	C₂×C₃×C₅×D₄	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	1	no	—
200	C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	C₄	C₃×C₅×#41	1	1	yes	characteristic
202	C₃	C₅×#464	1	1	yes	characteristic, direct product factor, Sylow
203	C₂	—	8	1	no	—
204	C₂	C₃×C₅×#121	1	1	yes	characteristic
205	C₂	C₃×C₅×#135	1	1	yes	characteristic
206	C₂	C₃×C₅×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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. C₂×C₃×C₅×Q₁₆	C₂	semidirect
4. C₃×C₄×C₅×Q₈	C₂	semidirect
5. C₃×C₅×C₄⋊C₈	C₂	semidirect
6. C₃×C₅×#103	C₂	semidirect
8. C₅×#464	C₃	direct (N × H)
22. C₃×#464	C₅	direct (N × H)
23. C₂×C₅×Q₁₆	C₂×C₃	semidirect
26. C₄×C₅×Q₈	C₂×C₃	semidirect
27. C₅×C₄⋊C₈	C₂×C₃	semidirect
28. C₅×#103	C₂×C₃	semidirect
45. C₂×C₃×Q₁₆	C₂×C₅	semidirect
48. C₃×C₄×Q₈	C₂×C₅	semidirect
49. C₃×C₄⋊C₈	C₂×C₅	semidirect
50. C₃×#103	C₂×C₅	semidirect
65. #464	C₃×C₅	direct (N × H)
105. C₂×Q₁₆	C₂×C₃×C₅	semidirect
108. C₄×Q₈	C₂×C₃×C₅	semidirect
109. C₄⋊C₈	C₂×C₃×C₅	semidirect
110. #103	C₂×C₃×C₅	semidirect
155. C₃×C₅	#464	direct (N × H)
190. C₅	C₃×#464	direct (N × H)
202. C₃	C₅×#464	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×Q₁₆	2
C₂×C₃×C₅×SD₁₆	2
C₃×C₅×#124	2
C₃×C₄×C₅×Q₈	2
C₃×C₅×C₄⋊C₈	2
C₃×C₅×#103	2
C₃×C₅×#102	2
C₅×#464	3
C₃×#464	5

Subgroups — C3×C5×#464 #11751439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#464 #11751439