Subgroups — C₃×C₅×#577 #11756666

Order 960.

516 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×C₅×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#137	C₂	1	3	yes	maximal, complemented
4	C₃×C₅×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
5	C₃×C₅×#135	C₂	1	3	yes	maximal, complemented
6	C₂×C₃×C₅×SD₁₆	C₂	1	3	yes	maximal, complemented
7	C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#577	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×#41	—	2	1	no	—
10	C₃×C₅×#41	—	2	3	no	—
11	C₃×C₅×#41	C₂²	1	3	yes	—
12	C₃×C₅×#41	C₂²	1	1	yes	characteristic
13	C₃×C₅×#41	C₂²	1	3	yes	complemented
14	C₂×C₃×C₅×Q₈	—	2	1	no	—
15	C₂×C₃×C₅×Q₈	C₂²	1	3	yes	complemented
16	C₂×C₃×C₅×D₄	—	2	3	no	—
17	C₂×C₃×C₅×D₄	C₂²	1	3	yes	—
18	C₃×C₅×Q₁₆	C₂²	1	3	yes	complemented
19	C₃×C₅×SD₁₆	C₂²	1	3	yes	complemented
20	C₃×C₅×SD₁₆	C₂²	1	6	yes	complemented
21	C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₅×D₈	C₂²	1	3	yes	complemented
23	C₃×C₅×M₁₆	C₂²	1	3	yes	complemented
24	C₂×C₃×C₅×C₈	C₂²	1	3	yes	complemented
25	C₃×#577	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₅×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#137	C₂×C₃	1	3	yes	complemented
29	C₅×AGL(1,ℤ/8)	C₂×C₃	1	3	yes	complemented
30	C₅×#135	C₂×C₃	1	3	yes	complemented
31	C₂×C₅×SD₁₆	C₂×C₃	1	3	yes	complemented
32	C₅×#131	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂³×C₃×C₅	—	2	3	no	—
34	C₃×C₅×Q₈	—	2	3	no	—
35	C₃×C₅×Q₈	D₄	1	1	yes	characteristic
36	C₃×C₅×Q₈	C₂³	1	3	yes	—
37	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
38	C₃×C₅×D₄	—	2	3	no	—
39	C₃×C₅×D₄	—	2	3	no	—
40	C₃×C₅×D₄	—	2	3	no	—
41	C₃×C₅×D₄	D₄	1	3	yes	—
42	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
43	C₃×C₅×D₄	C₂³	1	3	yes	—
44	C₂×C₃×C₄×C₅	—	2	3	no	—
45	C₂×C₃×C₄×C₅	—	2	3	no	—
46	C₂×C₃×C₄×C₅	—	2	3	no	—
47	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
48	C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₃×C₅×C₈	C₂³	1	3	yes	—
50	C₃×2₋¹⁺⁴	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic, complemented
52	C₃×#137	C₂×C₅	1	3	yes	complemented
53	C₃×AGL(1,ℤ/8)	C₂×C₅	1	3	yes	complemented
54	C₃×#135	C₂×C₅	1	3	yes	complemented
55	C₂×C₃×SD₁₆	C₂×C₅	1	3	yes	complemented
56	C₃×#131	C₂×C₅	1	1	yes	characteristic, complemented
57	C₅×#41	—	2	1	no	—
58	C₅×#41	—	2	3	no	—
59	C₅×#41	C₂²×C₃	1	3	yes	—
60	C₅×#41	C₂²×C₃	1	1	yes	characteristic
61	C₅×#41	C₂²×C₃	1	3	yes	complemented
62	C₂×C₅×Q₈	—	2	1	no	—
63	C₂×C₅×Q₈	C₂²×C₃	1	3	yes	complemented
64	C₂×C₅×D₄	—	2	3	no	—
65	C₂×C₅×D₄	C₂²×C₃	1	3	yes	—
66	C₅×Q₁₆	C₂²×C₃	1	3	yes	complemented
67	C₅×SD₁₆	C₂²×C₃	1	3	yes	complemented
68	C₅×SD₁₆	C₂²×C₃	1	6	yes	complemented
69	C₅×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₅×D₈	C₂²×C₃	1	3	yes	complemented
71	C₅×M₁₆	C₂²×C₃	1	3	yes	complemented
72	C₂×C₅×C₈	C₂²×C₃	1	3	yes	complemented
73	#577	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₃×C₅	—	2	1	no	—
75	C₂²×C₃×C₅	—	2	3	no	—
76	C₂²×C₃×C₅	—	8	3	no	—
77	C₂²×C₃×C₅	C₂×D₄	1	3	yes	—
78	C₃×C₄×C₅	—	2	1	no	—
79	C₃×C₄×C₅	—	2	3	no	—
80	C₃×C₄×C₅	C₂×D₄	1	3	yes	—
81	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₃×#41	—	2	1	no	—
83	C₃×#41	—	2	3	no	—
84	C₃×#41	C₂²×C₅	1	3	yes	—
85	C₃×#41	C₂²×C₅	1	1	yes	characteristic
86	C₃×#41	C₂²×C₅	1	3	yes	complemented
87	C₂×C₃×Q₈	—	2	1	no	—
88	C₂×C₃×Q₈	C₂²×C₅	1	3	yes	complemented
89	C₂×C₃×D₄	—	2	3	no	—
90	C₂×C₃×D₄	C₂²×C₅	1	3	yes	—
91	C₃×Q₁₆	C₂²×C₅	1	3	yes	complemented
92	C₃×SD₁₆	C₂²×C₅	1	3	yes	complemented
93	C₃×SD₁₆	C₂²×C₅	1	6	yes	complemented
94	C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
95	C₃×D₈	C₂²×C₅	1	3	yes	complemented
96	C₃×M₁₆	C₂²×C₅	1	3	yes	complemented
97	C₂×C₃×C₈	C₂²×C₅	1	3	yes	complemented
98	C₂³×C₅	—	2	3	no	—
99	C₅×Q₈	—	2	3	no	—
100	C₅×Q₈	C₃×D₄	1	1	yes	characteristic
101	C₅×Q₈	C₂³×C₃	1	3	yes	—
102	C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
103	C₅×D₄	—	2	3	no	—
104	C₅×D₄	—	2	3	no	—
105	C₅×D₄	—	2	3	no	—
106	C₅×D₄	C₃×D₄	1	3	yes	—
107	C₅×D₄	C₂³×C₃	1	1	yes	characteristic
108	C₅×D₄	C₂³×C₃	1	3	yes	—
109	C₂×C₄×C₅	—	2	3	no	—
110	C₂×C₄×C₅	—	2	3	no	—
111	C₂×C₄×C₅	—	2	3	no	—
112	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
113	C₅×C₈	C₂³×C₃	1	1	yes	characteristic
114	C₅×C₈	C₂³×C₃	1	3	yes	—
115	2₋¹⁺⁴	C₂×C₃×C₅	1	1	yes	characteristic, complemented
116	H(5,𝔽₂)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	#137	C₂×C₃×C₅	1	3	yes	complemented
118	AGL(1,ℤ/8)	C₂×C₃×C₅	1	3	yes	complemented
119	#135	C₂×C₃×C₅	1	3	yes	complemented
120	C₂×SD₁₆	C₂×C₃×C₅	1	3	yes	complemented
121	#131	C₂×C₃×C₅	1	1	yes	characteristic, complemented
122	C₂×C₃×C₅	—	2	3	no	—
123	C₂×C₃×C₅	—	4	1	no	—
124	C₂×C₃×C₅	—	4	3	no	—
125	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂³×C₃	—	2	3	no	—
127	C₃×Q₈	—	2	3	no	—
128	C₃×Q₈	C₅×D₄	1	1	yes	characteristic
129	C₃×Q₈	C₂³×C₅	1	3	yes	—
130	C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
131	C₃×D₄	—	2	3	no	—
132	C₃×D₄	—	2	3	no	—
133	C₃×D₄	—	2	3	no	—
134	C₃×D₄	C₅×D₄	1	3	yes	—
135	C₃×D₄	C₂³×C₅	1	1	yes	characteristic
136	C₃×D₄	C₂³×C₅	1	3	yes	—
137	C₂×C₃×C₄	—	2	3	no	—
138	C₂×C₃×C₄	—	2	3	no	—
139	C₂×C₃×C₄	—	2	3	no	—
140	C₂×C₃×C₄	C₂³×C₅	1	3	yes	—
141	C₃×C₈	C₂³×C₅	1	1	yes	characteristic
142	C₃×C₈	C₂³×C₅	1	3	yes	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	—	2	3	no	—
145	C₂²×C₅	—	8	3	no	—
146	C₂²×C₅	C₂×C₃×D₄	1	3	yes	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	3	no	—
149	C₄×C₅	C₂×C₃×D₄	1	3	yes	—
150	C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
151	#41	—	2	1	no	—
152	#41	—	2	3	no	—
153	#41	C₂²×C₃×C₅	1	3	yes	—
154	#41	C₂²×C₃×C₅	1	1	yes	characteristic
155	#41	C₂²×C₃×C₅	1	3	yes	complemented
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	C₂²×C₃×C₅	1	3	yes	complemented
158	C₂×D₄	—	2	3	no	—
159	C₂×D₄	C₂²×C₃×C₅	1	3	yes	—
160	Q₁₆	C₂²×C₃×C₅	1	3	yes	complemented
161	SD₁₆	C₂²×C₃×C₅	1	3	yes	complemented
162	SD₁₆	C₂²×C₃×C₅	1	6	yes	complemented
163	SD₁₆	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
164	D₈	C₂²×C₃×C₅	1	3	yes	complemented
165	M₁₆	C₂²×C₃×C₅	1	3	yes	complemented
166	C₂×C₈	C₂²×C₃×C₅	1	3	yes	complemented
167	C₃×C₅	#577	1	1	yes	characteristic, direct product factor, Hall
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	3	no	—
170	C₂²×C₃	—	8	3	no	—
171	C₂²×C₃	C₂×C₅×D₄	1	3	yes	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	3	no	—
174	C₃×C₄	C₂×C₅×D₄	1	3	yes	—
175	C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
176	C₂×C₅	—	2	3	no	—
177	C₂×C₅	—	4	3	no	—
178	C₂×C₅	—	4	1	no	—
179	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
180	C₂³	—	2	3	no	—
181	Q₈	—	2	3	no	—
182	Q₈	C₃×C₅×D₄	1	1	yes	characteristic
183	Q₈	C₂³×C₃×C₅	1	3	yes	—
184	Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
185	D₄	—	2	3	no	—
186	D₄	—	2	3	no	—
187	D₄	—	2	3	no	—
188	D₄	C₃×C₅×D₄	1	3	yes	—
189	D₄	C₂³×C₃×C₅	1	1	yes	characteristic
190	D₄	C₂³×C₃×C₅	1	3	yes	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	C₂³×C₃×C₅	1	3	yes	—
195	C₈	C₂³×C₃×C₅	1	1	yes	characteristic
196	C₈	C₂³×C₃×C₅	1	3	yes	—
197	C₂×C₃	—	2	3	no	—
198	C₂×C₃	—	4	3	no	—
199	C₂×C₃	—	4	1	no	—
200	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
201	C₅	C₃×#577	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²	—	2	1	no	—
203	C₂²	—	2	3	no	—
204	C₂²	—	8	3	no	—
205	C₂²	C₂×C₃×C₅×D₄	1	3	yes	—
206	C₄	—	2	1	no	—
207	C₄	—	2	3	no	—
208	C₄	C₂×C₃×C₅×D₄	1	3	yes	—
209	C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
210	C₃	C₅×#577	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	3	no	—
212	C₂	—	4	3	no	—
213	C₂	—	4	1	no	—
214	C₂	C₂²×C₃×C₅×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₃×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×2₋¹⁺⁴	C₂	semidirect
2. C₃×C₅×H(5,𝔽₂)	C₂	semidirect
3. C₃×C₅×#137	C₂	semidirect
4. C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
5. C₃×C₅×#135	C₂	semidirect
6. C₂×C₃×C₅×SD₁₆	C₂	semidirect
7. C₃×C₅×#131	C₂	semidirect
8. C₅×#577	C₃	direct (N × H)
13. C₃×C₅×#41	C₂²	semidirect
15. C₂×C₃×C₅×Q₈	C₂²	semidirect
18. C₃×C₅×Q₁₆	C₂²	semidirect
19. C₃×C₅×SD₁₆	C₂²	semidirect
20. C₃×C₅×SD₁₆	C₂²	semidirect
21. C₃×C₅×SD₁₆	C₂²	semidirect
22. C₃×C₅×D₈	C₂²	semidirect
23. C₃×C₅×M₁₆	C₂²	semidirect
24. C₂×C₃×C₅×C₈	C₂²	semidirect
25. C₃×#577	C₅	direct (N × H)
26. C₅×2₋¹⁺⁴	C₂×C₃	semidirect
27. C₅×H(5,𝔽₂)	C₂×C₃	semidirect
28. C₅×#137	C₂×C₃	semidirect
29. C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
30. C₅×#135	C₂×C₃	semidirect
31. C₂×C₅×SD₁₆	C₂×C₃	semidirect
32. C₅×#131	C₂×C₃	semidirect
50. C₃×2₋¹⁺⁴	C₂×C₅	semidirect
51. C₃×H(5,𝔽₂)	C₂×C₅	semidirect
52. C₃×#137	C₂×C₅	semidirect
53. C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
54. C₃×#135	C₂×C₅	semidirect
55. C₂×C₃×SD₁₆	C₂×C₅	semidirect
56. C₃×#131	C₂×C₅	semidirect
61. C₅×#41	C₂²×C₃	semidirect
63. C₂×C₅×Q₈	C₂²×C₃	semidirect
66. C₅×Q₁₆	C₂²×C₃	semidirect
67. C₅×SD₁₆	C₂²×C₃	semidirect
68. C₅×SD₁₆	C₂²×C₃	semidirect
69. C₅×SD₁₆	C₂²×C₃	semidirect
70. C₅×D₈	C₂²×C₃	semidirect
71. C₅×M₁₆	C₂²×C₃	semidirect
72. C₂×C₅×C₈	C₂²×C₃	semidirect
73. #577	C₃×C₅	direct (N × H)
86. C₃×#41	C₂²×C₅	semidirect
88. C₂×C₃×Q₈	C₂²×C₅	semidirect
91. C₃×Q₁₆	C₂²×C₅	semidirect
92. C₃×SD₁₆	C₂²×C₅	semidirect
93. C₃×SD₁₆	C₂²×C₅	semidirect
94. C₃×SD₁₆	C₂²×C₅	semidirect
95. C₃×D₈	C₂²×C₅	semidirect
96. C₃×M₁₆	C₂²×C₅	semidirect
97. C₂×C₃×C₈	C₂²×C₅	semidirect
115. 2₋¹⁺⁴	C₂×C₃×C₅	semidirect
116. H(5,𝔽₂)	C₂×C₃×C₅	semidirect
117. #137	C₂×C₃×C₅	semidirect
118. AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
119. #135	C₂×C₃×C₅	semidirect
120. C₂×SD₁₆	C₂×C₃×C₅	semidirect
121. #131	C₂×C₃×C₅	semidirect
155. #41	C₂²×C₃×C₅	semidirect
157. C₂×Q₈	C₂²×C₃×C₅	semidirect
160. Q₁₆	C₂²×C₃×C₅	semidirect
161. SD₁₆	C₂²×C₃×C₅	semidirect
162. SD₁₆	C₂²×C₃×C₅	semidirect
163. SD₁₆	C₂²×C₃×C₅	semidirect
164. D₈	C₂²×C₃×C₅	semidirect
165. M₁₆	C₂²×C₃×C₅	semidirect
166. C₂×C₈	C₂²×C₃×C₅	semidirect
167. C₃×C₅	#577	direct (N × H)
201. C₅	C₃×#577	direct (N × H)
210. C₃	C₅×#577	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×C₅×2₋¹⁺⁴	2
C₃×C₅×H(5,𝔽₂)	2
C₃×C₅×#137	2
C₃×C₅×AGL(1,ℤ/8)	2
C₃×C₅×#135	2
C₂×C₃×C₅×SD₁₆	2
C₃×C₅×#131	2
C₅×#577	3
C₃×#577	5

Subgroups — C3×C5×#577 #11756666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#577 #11756666