Subgroups — C₃×C₅×#532 #11756621

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#124	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#121	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×C₅×D₄	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#532	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×#41	—	2	8	no	—
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	4	yes	—
16	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	2	yes	—
22	C₃×#532	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×#41	C₂×C₃	1	2	yes	complemented
24	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#124	C₂×C₃	1	2	yes	complemented
27	C₅×#121	C₂×C₃	1	4	yes	complemented
28	C₄×C₅×D₄	C₂×C₃	1	4	yes	complemented
29	C₂×C₄²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₅	C₂³	1	4	yes	—
31	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₃×C₅×Q₈	—	2	4	no	—
33	C₃×C₅×D₄	—	2	8	no	—
34	C₃×C₅×D₄	—	2	4	no	—
35	C₃×C₅×D₄	—	2	8	no	—
36	C₂×C₃×C₄×C₅	—	2	8	no	—
37	C₂×C₃×C₄×C₅	—	2	4	no	—
38	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₃×#41	C₂×C₅	1	2	yes	complemented
45	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
46	C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
47	C₃×#124	C₂×C₅	1	2	yes	complemented
48	C₃×#121	C₂×C₅	1	4	yes	complemented
49	C₃×C₄×D₄	C₂×C₅	1	4	yes	complemented
50	C₂×C₃×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
51	C₅×#41	—	2	8	no	—
52	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
54	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
55	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₃	1	4	yes	—
58	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
60	C₅×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
61	C₄²×C₅	C₂²×C₃	1	2	yes	—
62	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	#532	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₃×C₅	—	2	2	no	—
66	C₂²×C₃×C₅	—	2	4	no	—
67	C₂²×C₃×C₅	—	2	8	no	—
68	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
70	C₃×C₄×C₅	—	2	4	no	—
71	C₃×C₄×C₅	—	2	2	no	—
72	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₃×C₄×C₅	#41	1	4	yes	complemented
75	C₃×#41	—	2	8	no	—
76	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
77	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
78	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
79	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
80	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
81	C₂²×C₃×C₄	C₂²×C₅	1	4	yes	—
82	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
84	C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
85	C₃×C₄²	C₂²×C₅	1	2	yes	—
86	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
88	C₂³×C₅	C₂³×C₃	1	4	yes	—
89	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₅×Q₈	—	2	4	no	—
91	C₅×D₄	—	2	8	no	—
92	C₅×D₄	—	2	4	no	—
93	C₅×D₄	—	2	8	no	—
94	C₂×C₄×C₅	—	2	8	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
97	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
98	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
99	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
102	C₂×#41	C₂×C₃×C₅	1	2	yes	complemented
103	C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#127	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	#124	C₂×C₃×C₅	1	2	yes	complemented
106	#121	C₂×C₃×C₅	1	4	yes	complemented
107	C₄×D₄	C₂×C₃×C₅	1	4	yes	complemented
108	C₂×C₄²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	C₂×C₃×C₅	—	2	2	no	—
110	C₂×C₃×C₅	—	4	4	no	—
111	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₃×C₅	C₂×#41	1	2	yes	—
113	C₂³×C₃	C₂³×C₅	1	4	yes	—
114	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
115	C₃×Q₈	—	2	4	no	—
116	C₃×D₄	—	2	8	no	—
117	C₃×D₄	—	2	4	no	—
118	C₃×D₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	C₅×D₄	1	4	yes	complemented
122	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
123	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
124	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	—	2	8	no	—
130	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
131	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
132	C₄×C₅	—	2	4	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
135	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
136	C₄×C₅	C₃×#41	1	4	yes	complemented
137	#41	—	2	8	no	—
138	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
139	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
140	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
141	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
142	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	—
143	C₂²×C₄	C₂²×C₃×C₅	1	4	yes	—
144	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
146	C₂²⋊C₄	C₂²×C₃×C₅	1	8	yes	—
147	C₄²	C₂²×C₃×C₅	1	2	yes	—
148	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₃×C₅	#532	1	1	yes	characteristic, direct product factor, Hall
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
155	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
159	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
160	C₃×C₄	C₅×#41	1	4	yes	complemented
161	C₂×C₅	—	2	2	no	—
162	C₂×C₅	—	4	4	no	—
163	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
164	C₂×C₅	C₂×C₃×#41	1	2	yes	—
165	C₂³	C₂³×C₃×C₅	1	4	yes	—
166	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	D₄	—	2	8	no	—
169	D₄	—	2	4	no	—
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₃×C₅×D₄	1	4	yes	complemented
174	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
175	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
176	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
177	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
178	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	—	4	4	no	—
181	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
182	C₂×C₃	C₂×C₅×#41	1	2	yes	—
183	C₅	C₃×#532	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	8	no	—
187	C₂²	C₂×C₃×C₅×D₄	1	2	yes	—
188	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	C₂×C₃×C₅×D₄	1	2	yes	—
192	C₄	C₂×C₃×C₅×D₄	1	2	yes	—
193	C₄	C₃×C₅×#41	1	4	yes	complemented
194	C₃	C₅×#532	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	4	4	no	—
197	C₂	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
198	C₂	C₂×C₃×C₅×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂⁴
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. C₂×C₃×C₅×#41	C₂	semidirect
2. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₃×C₅×#127	C₂	semidirect
4. C₃×C₅×#124	C₂	semidirect
5. C₃×C₅×#121	C₂	semidirect
6. C₃×C₄×C₅×D₄	C₂	semidirect
7. C₂×C₃×C₄²×C₅	C₂	semidirect
8. C₅×#532	C₃	direct (N × H)
22. C₃×#532	C₅	direct (N × H)
23. C₂×C₅×#41	C₂×C₃	semidirect
24. C₅×C₄⋊Q₈	C₂×C₃	semidirect
25. C₅×#127	C₂×C₃	semidirect
26. C₅×#124	C₂×C₃	semidirect
27. C₅×#121	C₂×C₃	semidirect
28. C₄×C₅×D₄	C₂×C₃	semidirect
29. C₂×C₄²×C₅	C₂×C₃	semidirect
38. C₂×C₃×C₄×C₅	D₄	semidirect
44. C₂×C₃×#41	C₂×C₅	semidirect
45. C₃×C₄⋊Q₈	C₂×C₅	semidirect
46. C₃×#127	C₂×C₅	semidirect
47. C₃×#124	C₂×C₅	semidirect
48. C₃×#121	C₂×C₅	semidirect
49. C₃×C₄×D₄	C₂×C₅	semidirect
50. C₂×C₃×C₄²	C₂×C₅	semidirect
64. #532	C₃×C₅	direct (N × H)
74. C₃×C₄×C₅	#41	semidirect
96. C₂×C₄×C₅	C₃×D₄	semidirect
102. C₂×#41	C₂×C₃×C₅	semidirect
103. C₄⋊Q₈	C₂×C₃×C₅	semidirect
104. #127	C₂×C₃×C₅	semidirect
105. #124	C₂×C₃×C₅	semidirect
106. #121	C₂×C₃×C₅	semidirect
107. C₄×D₄	C₂×C₃×C₅	semidirect
108. C₂×C₄²	C₂×C₃×C₅	semidirect
121. C₂×C₃×C₄	C₅×D₄	semidirect
136. C₄×C₅	C₃×#41	semidirect
150. C₃×C₅	#532	direct (N × H)
160. C₃×C₄	C₅×#41	semidirect
173. C₂×C₄	C₃×C₅×D₄	semidirect
183. C₅	C₃×#532	direct (N × H)
193. C₄	C₃×C₅×#41	semidirect
194. C₃	C₅×#532	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×#41	2
C₃×C₅×C₄⋊Q₈	2
C₃×C₅×#127	2
C₃×C₅×#124	2
C₃×C₅×#121	2
C₃×C₄×C₅×D₄	2
C₂×C₃×C₄²×C₅	2
C₅×#532	3
C₃×#532	5

Subgroups — C3×C5×#532 #11756621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#532 #11756621