Subgroups — C₃×C₅×#535 #11756624

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 232 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	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#127	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#124	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#121	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#120	C₂	1	4	yes	maximal, complemented
7	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#535	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂⁴×C₃×C₅	C₂²	1	2	yes	—
10	C₃×C₅×#41	—	2	4	no	—
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅×D₄	—	2	4	no	—
13	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₅×D₄	C₂²	1	8	yes	complemented
18	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	complemented
20	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
23	C₃×C₄²×C₅	C₂²	1	2	yes	complemented
24	C₃×#535	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×#127	C₂×C₃	1	2	yes	complemented
28	C₅×#124	C₂×C₃	1	2	yes	complemented
29	C₅×#121	C₂×C₃	1	4	yes	complemented
30	C₅×#120	C₂×C₃	1	4	yes	complemented
31	C₅×#117	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×C₃×C₅	—	2	4	no	—
33	C₂³×C₃×C₅	—	4	4	no	—
34	C₂³×C₃×C₅	C₂³	1	4	yes	—
35	C₂³×C₃×C₅	C₂³	1	2	yes	—
36	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×Q₈	—	2	2	no	—
38	C₃×C₅×D₄	—	2	4	no	—
39	C₃×C₅×D₄	—	2	4	no	—
40	C₃×C₅×D₄	—	2	4	no	—
41	C₃×C₅×D₄	—	2	2	no	—
42	C₃×C₅×D₄	—	4	8	no	—
43	C₂×C₃×C₄×C₅	—	2	4	no	—
44	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
45	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×#41	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×C₃×D₄	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#127	C₂×C₅	1	2	yes	complemented
52	C₃×#124	C₂×C₅	1	2	yes	complemented
53	C₃×#121	C₂×C₅	1	4	yes	complemented
54	C₃×#120	C₂×C₅	1	4	yes	complemented
55	C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂⁴×C₅	C₂²×C₃	1	2	yes	—
57	C₅×#41	—	2	4	no	—
58	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
59	C₂×C₅×D₄	—	2	4	no	—
60	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
61	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
62	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
63	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
64	C₂×C₅×D₄	C₂²×C₃	1	8	yes	complemented
65	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	complemented
67	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
68	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₅×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
70	C₄²×C₅	C₂²×C₃	1	2	yes	complemented
71	#535	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	—	2	4	no	—
75	C₂²×C₃×C₅	—	2	4	no	—
76	C₂²×C₃×C₅	—	2	8	no	—
77	C₂²×C₃×C₅	—	4	8	no	—
78	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
80	C₃×C₄×C₅	—	2	2	no	—
81	C₃×C₄×C₅	—	2	4	no	—
82	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂⁴×C₃	C₂²×C₅	1	2	yes	—
85	C₃×#41	—	2	4	no	—
86	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
87	C₂×C₃×D₄	—	2	4	no	—
88	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
89	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
90	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
92	C₂×C₃×D₄	C₂²×C₅	1	8	yes	complemented
93	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	complemented
95	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
96	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
97	C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
98	C₃×C₄²	C₂²×C₅	1	2	yes	complemented
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	—	4	4	no	—
101	C₂³×C₅	C₂³×C₃	1	4	yes	—
102	C₂³×C₅	C₂³×C₃	1	2	yes	—
103	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₅×Q₈	—	2	2	no	—
105	C₅×D₄	—	2	4	no	—
106	C₅×D₄	—	2	4	no	—
107	C₅×D₄	—	2	4	no	—
108	C₅×D₄	—	2	2	no	—
109	C₅×D₄	—	4	8	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
112	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
113	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
114	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂×#41	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	C₂²×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
118	#127	C₂×C₃×C₅	1	2	yes	complemented
119	#124	C₂×C₃×C₅	1	2	yes	complemented
120	#121	C₂×C₃×C₅	1	4	yes	complemented
121	#120	C₂×C₃×C₅	1	4	yes	complemented
122	#117	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	C₂×C₃×C₅	—	2	2	no	—
124	C₂×C₃×C₅	—	4	2	no	—
125	C₂×C₃×C₅	—	4	4	no	—
126	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
128	C₂³×C₃	—	2	4	no	—
129	C₂³×C₃	—	4	4	no	—
130	C₂³×C₃	C₂³×C₅	1	4	yes	—
131	C₂³×C₃	C₂³×C₅	1	2	yes	—
132	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
133	C₃×Q₈	—	2	2	no	—
134	C₃×D₄	—	2	4	no	—
135	C₃×D₄	—	2	4	no	—
136	C₃×D₄	—	2	4	no	—
137	C₃×D₄	—	2	2	no	—
138	C₃×D₄	—	4	8	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	C₅×D₄	1	4	yes	complemented
141	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
142	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
143	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	—	2	4	no	—
149	C₂²×C₅	—	2	8	no	—
150	C₂²×C₅	—	4	8	no	—
151	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
152	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
156	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
157	C₂⁴	C₂²×C₃×C₅	1	2	yes	—
158	#41	—	2	4	no	—
159	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
162	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
163	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
164	C₂×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
165	C₂×D₄	C₂²×C₃×C₅	1	8	yes	complemented
166	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	complemented
168	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	complemented
169	C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
170	C₂²⋊C₄	C₂²×C₃×C₅	1	8	yes	complemented
171	C₄²	C₂²×C₃×C₅	1	2	yes	complemented
172	C₃×C₅	#535	1	1	yes	characteristic, direct product factor, Hall
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	4	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	8	no	—
178	C₂²×C₃	—	4	8	no	—
179	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
180	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
184	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
185	C₂×C₅	—	2	2	no	—
186	C₂×C₅	—	4	4	no	—
187	C₂×C₅	—	4	2	no	—
188	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₂×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
190	C₂³	—	2	4	no	—
191	C₂³	—	4	4	no	—
192	C₂³	C₂³×C₃×C₅	1	4	yes	—
193	C₂³	C₂³×C₃×C₅	1	2	yes	—
194	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
195	Q₈	—	2	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	4	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	C₃×C₅×D₄	1	4	yes	complemented
203	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
204	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
205	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
206	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	—	4	4	no	—
209	C₂×C₃	—	4	2	no	—
210	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
211	C₂×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
212	C₅	C₃×#535	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	8	no	—
218	C₂²	—	4	8	no	—
219	C₂²	C₂×C₃×C₅×D₄	1	2	yes	—
220	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	C₂×C₃×C₅×D₄	1	2	yes	—
224	C₄	C₂×C₃×C₅×D₄	1	2	yes	—
225	C₃	C₅×#535	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	—	4	2	no	—
229	C₂	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
230	C₂	C₃×C₅×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×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. C₂²×C₃×C₅×D₄	C₂	semidirect
3. C₃×C₅×#127	C₂	semidirect
4. C₃×C₅×#124	C₂	semidirect
5. C₃×C₅×#121	C₂	semidirect
6. C₃×C₅×#120	C₂	semidirect
7. C₃×C₅×#117	C₂	semidirect
8. C₅×#535	C₃	direct (N × H)
11. C₂×C₃×C₅×Q₈	C₂²	semidirect
16. C₂×C₃×C₅×D₄	C₂²	semidirect
17. C₂×C₃×C₅×D₄	C₂²	semidirect
19. C₂²×C₃×C₄×C₅	C₂²	semidirect
20. C₃×C₅×C₄⋊C₄	C₂²	semidirect
22. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
23. C₃×C₄²×C₅	C₂²	semidirect
24. C₃×#535	C₅	direct (N × H)
25. C₂×C₅×#41	C₂×C₃	semidirect
26. C₂²×C₅×D₄	C₂×C₃	semidirect
27. C₅×#127	C₂×C₃	semidirect
28. C₅×#124	C₂×C₃	semidirect
29. C₅×#121	C₂×C₃	semidirect
30. C₅×#120	C₂×C₃	semidirect
31. C₅×#117	C₂×C₃	semidirect
44. C₂×C₃×C₄×C₅	D₄	semidirect
49. C₂×C₃×#41	C₂×C₅	semidirect
50. C₂²×C₃×D₄	C₂×C₅	semidirect
51. C₃×#127	C₂×C₅	semidirect
52. C₃×#124	C₂×C₅	semidirect
53. C₃×#121	C₂×C₅	semidirect
54. C₃×#120	C₂×C₅	semidirect
55. C₃×#117	C₂×C₅	semidirect
58. C₂×C₅×Q₈	C₂²×C₃	semidirect
63. C₂×C₅×D₄	C₂²×C₃	semidirect
64. C₂×C₅×D₄	C₂²×C₃	semidirect
66. C₂²×C₄×C₅	C₂²×C₃	semidirect
67. C₅×C₄⋊C₄	C₂²×C₃	semidirect
69. C₅×C₂²⋊C₄	C₂²×C₃	semidirect
70. C₄²×C₅	C₂²×C₃	semidirect
71. #535	C₃×C₅	direct (N × H)
86. C₂×C₃×Q₈	C₂²×C₅	semidirect
91. C₂×C₃×D₄	C₂²×C₅	semidirect
92. C₂×C₃×D₄	C₂²×C₅	semidirect
94. C₂²×C₃×C₄	C₂²×C₅	semidirect
95. C₃×C₄⋊C₄	C₂²×C₅	semidirect
97. C₃×C₂²⋊C₄	C₂²×C₅	semidirect
98. C₃×C₄²	C₂²×C₅	semidirect
111. C₂×C₄×C₅	C₃×D₄	semidirect
116. C₂×#41	C₂×C₃×C₅	semidirect
117. C₂²×D₄	C₂×C₃×C₅	semidirect
118. #127	C₂×C₃×C₅	semidirect
119. #124	C₂×C₃×C₅	semidirect
120. #121	C₂×C₃×C₅	semidirect
121. #120	C₂×C₃×C₅	semidirect
122. #117	C₂×C₃×C₅	semidirect
140. C₂×C₃×C₄	C₅×D₄	semidirect
159. C₂×Q₈	C₂²×C₃×C₅	semidirect
164. C₂×D₄	C₂²×C₃×C₅	semidirect
165. C₂×D₄	C₂²×C₃×C₅	semidirect
167. C₂²×C₄	C₂²×C₃×C₅	semidirect
168. C₄⋊C₄	C₂²×C₃×C₅	semidirect
170. C₂²⋊C₄	C₂²×C₃×C₅	semidirect
171. C₄²	C₂²×C₃×C₅	semidirect
172. C₃×C₅	#535	direct (N × H)
202. C₂×C₄	C₃×C₅×D₄	semidirect
212. C₅	C₃×#535	direct (N × H)
225. C₃	C₅×#535	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
C₂²×C₃×C₅×D₄	2
C₃×C₅×#127	2
C₃×C₅×#124	2
C₃×C₅×#121	2
C₃×C₅×#120	2
C₃×C₅×#117	2
C₅×#535	3
C₃×#535	5

Subgroups — C3×C5×#535 #11756624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#535 #11756624