Subgroups — C₃×C₅×#559 #11756648

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 144 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
2	C₃×C₅×#124	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₅×Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×D₄	C₂	1	4	yes	maximal, complemented
6	C₅×#559	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
9	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₅	C₂²	1	4	yes	—
11	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
14	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₄²×C₅	C₂²	1	4	yes	—
16	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×#559	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#127	C₂×C₃	1	1	yes	characteristic
19	C₅×#124	C₂×C₃	1	4	yes	complemented
20	C₅×#123	C₂×C₃	1	4	yes	complemented
21	C₄×C₅×Q₈	C₂×C₃	1	2	yes	complemented
22	C₄×C₅×D₄	C₂×C₃	1	4	yes	complemented
23	C₂³×C₃×C₅	C₂³	1	4	yes	—
24	C₃×C₅×Q₈	—	2	4	no	—
25	C₃×C₅×D₄	—	2	8	no	—
26	C₃×C₅×D₄	—	4	2	no	—
27	C₂×C₃×C₄×C₅	—	4	4	no	—
28	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
29	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
30	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
32	C₃×#127	C₂×C₅	1	1	yes	characteristic
33	C₃×#124	C₂×C₅	1	4	yes	complemented
34	C₃×#123	C₂×C₅	1	4	yes	complemented
35	C₃×C₄×Q₈	C₂×C₅	1	2	yes	complemented
36	C₃×C₄×D₄	C₂×C₅	1	4	yes	complemented
37	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
38	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
39	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
40	C₂²×C₄×C₅	C₂²×C₃	1	4	yes	—
41	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
42	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₅×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
44	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
45	C₄²×C₅	C₂²×C₃	1	4	yes	—
46	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
47	#559	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₃×C₅	—	2	8	no	—
49	C₂²×C₃×C₅	—	2	4	no	—
50	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
51	C₃×C₄×C₅	—	2	4	no	—
52	C₃×C₄×C₅	—	2	4	no	—
53	C₃×C₄×C₅	—	2	1	no	—
54	C₃×C₄×C₅	#41	1	4	yes	—
55	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
56	C₂×C₃×D₄	C₂²×C₅	1	4	yes	—
57	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
58	C₂²×C₃×C₄	C₂²×C₅	1	4	yes	—
59	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
60	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
61	C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
62	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
63	C₃×C₄²	C₂²×C₅	1	4	yes	—
64	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
65	C₂³×C₅	C₂³×C₃	1	4	yes	—
66	C₅×Q₈	—	2	4	no	—
67	C₅×D₄	—	2	8	no	—
68	C₅×D₄	—	4	2	no	—
69	C₂×C₄×C₅	—	4	4	no	—
70	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
71	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
72	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
74	#127	C₂×C₃×C₅	1	1	yes	characteristic
75	#124	C₂×C₃×C₅	1	4	yes	complemented
76	#123	C₂×C₃×C₅	1	4	yes	complemented
77	C₄×Q₈	C₂×C₃×C₅	1	2	yes	complemented
78	C₄×D₄	C₂×C₃×C₅	1	4	yes	complemented
79	C₂×C₃×C₅	—	4	4	no	—
80	C₂×C₃×C₅	C₂×#41	1	2	yes	—
81	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂³×C₃	C₂³×C₅	1	4	yes	—
83	C₃×Q₈	—	2	4	no	—
84	C₃×D₄	—	2	8	no	—
85	C₃×D₄	—	4	2	no	—
86	C₂×C₃×C₄	—	4	4	no	—
87	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
88	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
89	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
90	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
91	C₂²×C₅	—	2	8	no	—
92	C₂²×C₅	—	2	4	no	—
93	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
94	C₄×C₅	—	2	4	no	—
95	C₄×C₅	—	2	4	no	—
96	C₄×C₅	—	2	1	no	—
97	C₄×C₅	C₃×#41	1	4	yes	—
98	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
99	C₂×D₄	C₂²×C₃×C₅	1	4	yes	—
100	C₂×D₄	C₂²×C₃×C₅	1	2	yes	—
101	C₂²×C₄	C₂²×C₃×C₅	1	4	yes	—
102	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
103	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
104	C₂²⋊C₄	C₂²×C₃×C₅	1	8	yes	—
105	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
106	C₄²	C₂²×C₃×C₅	1	4	yes	—
107	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
108	C₃×C₅	#559	1	1	yes	characteristic, direct product factor, Hall
109	C₂²×C₃	—	2	8	no	—
110	C₂²×C₃	—	2	4	no	—
111	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
112	C₃×C₄	—	2	4	no	—
113	C₃×C₄	—	2	4	no	—
114	C₃×C₄	—	2	1	no	—
115	C₃×C₄	C₅×#41	1	4	yes	—
116	C₂×C₅	—	4	4	no	—
117	C₂×C₅	C₂×C₃×#41	1	2	yes	—
118	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
119	C₂³	C₂³×C₃×C₅	1	4	yes	—
120	Q₈	—	2	4	no	—
121	D₄	—	2	8	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
125	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
126	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
127	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
128	C₂×C₃	—	4	4	no	—
129	C₂×C₃	C₂×C₅×#41	1	2	yes	—
130	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
131	C₅	C₃×#559	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²	—	2	8	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	2	4	no	—
137	C₄	—	2	1	no	—
138	C₄	C₃×C₅×#41	1	4	yes	—
139	C₃	C₅×#559	1	1	yes	characteristic, direct product factor, Sylow
140	C₂	—	4	4	no	—
141	C₂	C₂×C₃×C₅×#41	1	2	yes	—
142	C₂	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
prime radical	C₃×C₅×#127	C₂
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
2. C₃×C₅×#124	C₂	semidirect
3. C₃×C₅×#123	C₂	semidirect
4. C₃×C₄×C₅×Q₈	C₂	semidirect
5. C₃×C₄×C₅×D₄	C₂	semidirect
6. C₅×#559	C₃	direct (N × H)
17. C₃×#559	C₅	direct (N × H)
19. C₅×#124	C₂×C₃	semidirect
20. C₅×#123	C₂×C₃	semidirect
21. C₄×C₅×Q₈	C₂×C₃	semidirect
22. C₄×C₅×D₄	C₂×C₃	semidirect
33. C₃×#124	C₂×C₅	semidirect
34. C₃×#123	C₂×C₅	semidirect
35. C₃×C₄×Q₈	C₂×C₅	semidirect
36. C₃×C₄×D₄	C₂×C₅	semidirect
47. #559	C₃×C₅	direct (N × H)
75. #124	C₂×C₃×C₅	semidirect
76. #123	C₂×C₃×C₅	semidirect
77. C₄×Q₈	C₂×C₃×C₅	semidirect
78. C₄×D₄	C₂×C₃×C₅	semidirect
108. C₃×C₅	#559	direct (N × H)
131. C₅	C₃×#559	direct (N × H)
139. C₃	C₅×#559	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×#124	2
C₃×C₅×#123	2
C₃×C₄×C₅×Q₈	2
C₃×C₄×C₅×D₄	2
C₅×#559	3
C₃×#559	5

Subgroups — C3×C5×#559 #11756648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#559 #11756648