Subgroups — C₃×C₅×#540 #11756629

Order 960.

480 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#121	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#121	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×C₅×D₄	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#540	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
11	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	4	yes	—
14	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×#540	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#125	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×#123	C₂×C₃	1	4	yes	complemented
26	C₅×#121	C₂×C₃	1	4	yes	complemented
27	C₅×#121	C₂×C₃	1	2	yes	complemented
28	C₄×C₅×D₄	C₂×C₃	1	2	yes	complemented
29	C₅×#117	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
31	C₂³×C₃×C₅	C₂³	1	4	yes	—
32	C₃×C₅×D₄	—	2	4	no	—
33	C₃×C₅×D₄	—	4	4	no	—
34	C₃×C₅×D₄	—	4	4	no	—
35	C₂×C₃×C₄×C₅	—	2	2	no	—
36	C₂×C₃×C₄×C₅	—	4	4	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
44	C₃×#123	C₂×C₅	1	4	yes	complemented
45	C₃×#121	C₂×C₅	1	4	yes	complemented
46	C₃×#121	C₂×C₅	1	2	yes	complemented
47	C₃×C₄×D₄	C₂×C₅	1	2	yes	complemented
48	C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
50	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
51	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
52	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²×C₃	1	4	yes	—
54	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
58	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
59	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
60	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
62	#540	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×C₃×C₅	—	2	1	no	—
64	C₂²×C₃×C₅	—	2	4	no	—
65	C₂²×C₃×C₅	—	2	4	no	—
66	C₂²×C₃×C₅	—	2	2	no	—
67	C₂²×C₃×C₅	—	2	4	no	—
68	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
69	C₃×C₄×C₅	—	2	1	no	—
70	C₃×C₄×C₅	—	2	2	no	—
71	C₃×C₄×C₅	—	2	4	no	—
72	C₃×C₄×C₅	—	2	2	no	—
73	C₃×C₄×C₅	#41	1	2	yes	—
74	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
75	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
76	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
77	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂²×C₅	1	4	yes	—
79	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
81	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
82	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
83	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
84	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
85	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₂³×C₅	C₂³×C₃	1	4	yes	—
89	C₅×D₄	—	2	4	no	—
90	C₅×D₄	—	4	4	no	—
91	C₅×D₄	—	4	4	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	4	4	no	—
94	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
95	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
96	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
98	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	#127	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	#125	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	#123	C₂×C₃×C₅	1	4	yes	complemented
102	#121	C₂×C₃×C₅	1	4	yes	complemented
103	#121	C₂×C₃×C₅	1	2	yes	complemented
104	C₄×D₄	C₂×C₃×C₅	1	2	yes	complemented
105	#117	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	C₂×C₃×C₅	—	4	1	no	—
107	C₂×C₃×C₅	—	4	4	no	—
108	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
109	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
110	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
111	C₂³×C₃	C₂³×C₅	1	4	yes	—
112	C₃×D₄	—	2	4	no	—
113	C₃×D₄	—	4	4	no	—
114	C₃×D₄	—	4	4	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	4	4	no	—
117	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
118	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
119	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
121	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₂²×C₅	—	2	1	no	—
123	C₂²×C₅	—	2	4	no	—
124	C₂²×C₅	—	2	4	no	—
125	C₂²×C₅	—	2	2	no	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
128	C₄×C₅	—	2	1	no	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	—	2	2	no	—
132	C₄×C₅	C₃×#41	1	2	yes	—
133	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
134	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
135	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
136	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂²×C₄	C₂²×C₃×C₅	1	4	yes	—
138	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
139	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
140	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
141	C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
142	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
143	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
144	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₃×C₅	#540	1	1	yes	characteristic, direct product factor, Hall
147	C₂²×C₃	—	2	1	no	—
148	C₂²×C₃	—	2	4	no	—
149	C₂²×C₃	—	2	4	no	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	C₅×#41	1	2	yes	—
158	C₂×C₅	—	4	1	no	—
159	C₂×C₅	—	4	4	no	—
160	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
161	C₂×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
162	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
163	C₂³	C₂³×C₃×C₅	1	4	yes	—
164	D₄	—	2	4	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
170	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
171	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
172	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
173	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
174	C₂×C₃	—	4	1	no	—
175	C₂×C₃	—	4	4	no	—
176	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
177	C₂×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
178	C₅	C₃×#540	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²	—	2	1	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	—	2	4	no	—
188	C₄	—	2	2	no	—
189	C₄	C₃×C₅×#41	1	2	yes	—
190	C₃	C₅×#540	1	1	yes	characteristic, direct product factor, Sylow
191	C₂	—	4	1	no	—
192	C₂	—	4	4	no	—
193	C₂	C₂×C₃×C₅×#41	1	1	yes	characteristic
194	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. C₃×C₅×#127	C₂	semidirect
2. C₃×C₅×#125	C₂	semidirect
3. C₃×C₅×#123	C₂	semidirect
4. C₃×C₅×#121	C₂	semidirect
5. C₃×C₅×#121	C₂	semidirect
6. C₃×C₄×C₅×D₄	C₂	semidirect
7. C₃×C₅×#117	C₂	semidirect
8. C₅×#540	C₃	direct (N × H)
22. C₃×#540	C₅	direct (N × H)
23. C₅×#127	C₂×C₃	semidirect
24. C₅×#125	C₂×C₃	semidirect
25. C₅×#123	C₂×C₃	semidirect
26. C₅×#121	C₂×C₃	semidirect
27. C₅×#121	C₂×C₃	semidirect
28. C₄×C₅×D₄	C₂×C₃	semidirect
29. C₅×#117	C₂×C₃	semidirect
42. C₃×#127	C₂×C₅	semidirect
43. C₃×#125	C₂×C₅	semidirect
44. C₃×#123	C₂×C₅	semidirect
45. C₃×#121	C₂×C₅	semidirect
46. C₃×#121	C₂×C₅	semidirect
47. C₃×C₄×D₄	C₂×C₅	semidirect
48. C₃×#117	C₂×C₅	semidirect
62. #540	C₃×C₅	direct (N × H)
99. #127	C₂×C₃×C₅	semidirect
100. #125	C₂×C₃×C₅	semidirect
101. #123	C₂×C₃×C₅	semidirect
102. #121	C₂×C₃×C₅	semidirect
103. #121	C₂×C₃×C₅	semidirect
104. C₄×D₄	C₂×C₃×C₅	semidirect
105. #117	C₂×C₃×C₅	semidirect
146. C₃×C₅	#540	direct (N × H)
178. C₅	C₃×#540	direct (N × H)
190. C₃	C₅×#540	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×#127	2
C₃×C₅×#125	2
C₃×C₅×#123	2
C₃×C₅×#121	2
C₃×C₄×C₅×D₄	2
C₃×C₅×#117	2
C₅×#540	3
C₃×#540	5

Subgroups — C3×C5×#540 #11756629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#540 #11756629