Subgroups — C₃×C₅×#537 #11756626

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 180 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×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#121	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#537	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#41	—	2	8	no	—
8	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
10	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄×C₅	C₂²	1	4	yes	—
13	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₃×#537	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#41	C₂×C₃	1	2	yes	complemented
20	C₅×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
21	C₅×#121	C₂×C₃	1	4	yes	complemented
22	C₅×#121	C₂×C₃	1	4	yes	complemented
23	C₂×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂³×C₃×C₅	C₂³	1	4	yes	—
25	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₅×Q₈	—	2	4	no	—
27	C₃×C₅×D₄	—	2	8	no	—
28	C₃×C₅×D₄	—	2	4	no	—
29	C₃×C₅×D₄	—	4	4	no	—
30	C₂×C₃×C₄×C₅	—	2	8	no	—
31	C₂×C₃×C₄×C₅	—	4	2	no	—
32	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
33	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
35	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂×C₃×#41	C₂×C₅	1	2	yes	complemented
37	C₃×C₂²⋊Q₈	C₂×C₅	1	4	yes	complemented
38	C₃×#121	C₂×C₅	1	4	yes	complemented
39	C₃×#121	C₂×C₅	1	4	yes	complemented
40	C₂×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
41	C₅×#41	—	2	8	no	—
42	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
43	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
44	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
45	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
46	C₂²×C₄×C₅	C₂²×C₃	1	4	yes	—
47	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
48	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	C₅×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
52	#537	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₃×C₅	—	2	1	no	—
54	C₂²×C₃×C₅	—	2	1	no	—
55	C₂²×C₃×C₅	—	2	4	no	—
56	C₂²×C₃×C₅	—	2	4	no	—
57	C₂²×C₃×C₅	—	2	4	no	—
58	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
60	C₃×C₄×C₅	—	2	4	no	—
61	C₃×C₄×C₅	—	2	4	no	—
62	C₃×C₄×C₅	C₂×D₄	1	4	yes	—
63	C₃×#41	—	2	8	no	—
64	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
65	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
66	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
67	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
68	C₂²×C₃×C₄	C₂²×C₅	1	4	yes	—
69	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
71	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
72	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
73	C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
74	C₂³×C₅	C₂³×C₃	1	4	yes	—
75	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
76	C₅×Q₈	—	2	4	no	—
77	C₅×D₄	—	2	8	no	—
78	C₅×D₄	—	2	4	no	—
79	C₅×D₄	—	4	4	no	—
80	C₂×C₄×C₅	—	2	8	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
83	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
84	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
85	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
86	C₂×#41	C₂×C₃×C₅	1	2	yes	complemented
87	C₂²⋊Q₈	C₂×C₃×C₅	1	4	yes	complemented
88	#121	C₂×C₃×C₅	1	4	yes	complemented
89	#121	C₂×C₃×C₅	1	4	yes	complemented
90	C₂×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
91	C₂×C₃×C₅	—	2	2	no	—
92	C₂×C₃×C₅	—	4	4	no	—
93	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂³×C₃	C₂³×C₅	1	4	yes	—
97	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
98	C₃×Q₈	—	2	4	no	—
99	C₃×D₄	—	2	8	no	—
100	C₃×D₄	—	2	4	no	—
101	C₃×D₄	—	4	4	no	—
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	4	2	no	—
104	C₂×C₃×C₄	C₅×D₄	1	4	yes	complemented
105	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
106	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
107	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	—	2	1	no	—
110	C₂²×C₅	—	2	4	no	—
111	C₂²×C₅	—	2	4	no	—
112	C₂²×C₅	—	2	4	no	—
113	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
114	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
115	C₄×C₅	—	2	4	no	—
116	C₄×C₅	—	2	4	no	—
117	C₄×C₅	C₂×C₃×D₄	1	4	yes	—
118	#41	—	2	8	no	—
119	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
120	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
121	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
122	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
123	C₂²×C₄	C₂²×C₃×C₅	1	4	yes	—
124	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	—
125	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
126	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
127	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
128	C₂²⋊C₄	C₂²×C₃×C₅	1	8	yes	—
129	C₃×C₅	#537	1	1	yes	characteristic, direct product factor, Hall
130	C₂²×C₃	—	2	1	no	—
131	C₂²×C₃	—	2	1	no	—
132	C₂²×C₃	—	2	4	no	—
133	C₂²×C₃	—	2	4	no	—
134	C₂²×C₃	—	2	4	no	—
135	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
136	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
137	C₃×C₄	—	2	4	no	—
138	C₃×C₄	—	2	4	no	—
139	C₃×C₄	C₂×C₅×D₄	1	4	yes	—
140	C₂×C₅	—	2	2	no	—
141	C₂×C₅	—	4	4	no	—
142	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
143	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
145	C₂³	C₂³×C₃×C₅	1	4	yes	—
146	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
147	Q₈	—	2	4	no	—
148	D₄	—	2	8	no	—
149	D₄	—	2	4	no	—
150	D₄	—	4	4	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₃×C₅×D₄	1	4	yes	complemented
154	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
155	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
156	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	—	4	4	no	—
159	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
160	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
162	C₅	C₃×#537	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	2	4	no	—
168	C₂²	C₂×C₃×C₅×D₄	1	2	yes	—
169	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	2	4	no	—
172	C₄	C₂×C₃×C₅×D₄	1	4	yes	—
173	C₃	C₅×#537	1	1	yes	characteristic, direct product factor, Sylow
174	C₂	—	2	2	no	—
175	C₂	—	4	4	no	—
176	C₂	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
177	C₂	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
178	C₂	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic

	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. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
3. C₃×C₅×#121	C₂	semidirect
4. C₃×C₅×#121	C₂	semidirect
5. C₂×C₃×C₅×C₄⋊C₄	C₂	semidirect
6. C₅×#537	C₃	direct (N × H)
18. C₃×#537	C₅	direct (N × H)
19. C₂×C₅×#41	C₂×C₃	semidirect
20. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
21. C₅×#121	C₂×C₃	semidirect
22. C₅×#121	C₂×C₃	semidirect
23. C₂×C₅×C₄⋊C₄	C₂×C₃	semidirect
32. C₂×C₃×C₄×C₅	D₄	semidirect
36. C₂×C₃×#41	C₂×C₅	semidirect
37. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
38. C₃×#121	C₂×C₅	semidirect
39. C₃×#121	C₂×C₅	semidirect
40. C₂×C₃×C₄⋊C₄	C₂×C₅	semidirect
52. #537	C₃×C₅	direct (N × H)
82. C₂×C₄×C₅	C₃×D₄	semidirect
86. C₂×#41	C₂×C₃×C₅	semidirect
87. C₂²⋊Q₈	C₂×C₃×C₅	semidirect
88. #121	C₂×C₃×C₅	semidirect
89. #121	C₂×C₃×C₅	semidirect
90. C₂×C₄⋊C₄	C₂×C₃×C₅	semidirect
104. C₂×C₃×C₄	C₅×D₄	semidirect
129. C₃×C₅	#537	direct (N × H)
153. C₂×C₄	C₃×C₅×D₄	semidirect
162. C₅	C₃×#537	direct (N × H)
173. C₃	C₅×#537	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×C₂²⋊Q₈	2
C₃×C₅×#121	2
C₂×C₃×C₅×C₄⋊C₄	2
C₅×#537	3
C₃×#537	5

Subgroups — C3×C5×#537 #11756626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#537 #11756626