Subgroups — C₃×C₅×#519 #11756608

Order 960.

532 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₈	C₂	1	8	yes	maximal, complemented
3	C₃×C₅×#117	C₂	1	6	yes	maximal
4	C₅×#519	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×Q₈	C₄	1	8	yes	complemented
6	C₂×C₃×C₅×Q₈	C₂²	1	4	yes	—
7	C₂²×C₃×C₄×C₅	C₂²	1	3	yes	—
8	C₃×C₅×C₄⋊C₄	C₂²	1	12	yes	—
9	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
10	C₃×C₄²×C₅	C₂²	1	12	yes	—
11	C₃×#519	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
13	C₄×C₅×Q₈	C₂×C₃	1	8	yes	complemented
14	C₅×#117	C₂×C₃	1	6	yes	—
15	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
16	C₃×C₅×Q₈	C₂×C₄	1	16	yes	—
17	C₂×C₃×C₄×C₅	C₂×C₄	1	12	yes	—
18	C₂×C₃×C₄×C₅	C₂³	1	8	yes	—
19	C₂×C₃×C₄×C₅	C₂³	1	6	yes	—
20	C₂²×C₃×Q₈	C₂×C₅	1	1	yes	characteristic
21	C₃×C₄×Q₈	C₂×C₅	1	8	yes	complemented
22	C₃×#117	C₂×C₅	1	6	yes	—
23	C₂×C₅×Q₈	C₃×C₄	1	8	yes	complemented
24	C₂×C₅×Q₈	C₂²×C₃	1	4	yes	—
25	C₂²×C₄×C₅	C₂²×C₃	1	3	yes	—
26	C₅×C₄⋊C₄	C₂²×C₃	1	12	yes	—
27	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
28	C₄²×C₅	C₂²×C₃	1	12	yes	—
29	#519	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₃×C₅	—	2	2	no	—
31	C₂²×C₃×C₅	C₂²×C₄	1	2	yes	—
32	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
33	C₃×C₄×C₅	—	2	8	no	—
34	C₃×C₄×C₅	C₂²×C₄	1	12	yes	—
35	C₂×C₃×Q₈	C₄×C₅	1	8	yes	complemented
36	C₂×C₃×Q₈	C₂²×C₅	1	4	yes	—
37	C₂²×C₃×C₄	C₂²×C₅	1	3	yes	—
38	C₃×C₄⋊C₄	C₂²×C₅	1	12	yes	—
39	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
40	C₃×C₄²	C₂²×C₅	1	12	yes	—
41	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
42	C₅×Q₈	C₂×C₃×C₄	1	16	yes	—
43	C₂×C₄×C₅	C₂×C₃×C₄	1	12	yes	—
44	C₂×C₄×C₅	C₂³×C₃	1	8	yes	—
45	C₂×C₄×C₅	C₂³×C₃	1	6	yes	—
46	C₂²×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
47	C₄×Q₈	C₂×C₃×C₅	1	8	yes	complemented
48	#117	C₂×C₃×C₅	1	6	yes	—
49	C₂×C₃×C₅	—	2	2	no	—
50	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
51	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
52	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
53	C₃×Q₈	C₂×C₄×C₅	1	16	yes	—
54	C₂×C₃×C₄	C₂×C₄×C₅	1	12	yes	—
55	C₂×C₃×C₄	C₂³×C₅	1	8	yes	—
56	C₂×C₃×C₄	C₂³×C₅	1	6	yes	—
57	C₂²×C₅	—	2	2	no	—
58	C₂²×C₅	C₂²×C₃×C₄	1	2	yes	—
59	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
60	C₄×C₅	—	2	8	no	—
61	C₄×C₅	C₂²×C₃×C₄	1	12	yes	—
62	C₂×Q₈	C₃×C₄×C₅	1	8	yes	complemented
63	C₂×Q₈	C₂²×C₃×C₅	1	4	yes	—
64	C₂²×C₄	C₂²×C₃×C₅	1	3	yes	—
65	C₄⋊C₄	C₂²×C₃×C₅	1	12	yes	—
66	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
67	C₄²	C₂²×C₃×C₅	1	12	yes	—
68	C₃×C₅	#519	1	1	yes	characteristic, direct product factor, Hall
69	C₂²×C₃	—	2	2	no	—
70	C₂²×C₃	C₂²×C₄×C₅	1	2	yes	—
71	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
72	C₃×C₄	—	2	8	no	—
73	C₃×C₄	C₂²×C₄×C₅	1	12	yes	—
74	C₂×C₅	—	2	2	no	—
75	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
76	C₂×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
77	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
78	Q₈	C₂×C₃×C₄×C₅	1	16	yes	—
79	C₂×C₄	C₂×C₃×C₄×C₅	1	12	yes	—
80	C₂×C₄	C₂³×C₃×C₅	1	8	yes	—
81	C₂×C₄	C₂³×C₃×C₅	1	6	yes	—
82	C₂×C₃	—	2	2	no	—
83	C₂×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
84	C₂×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
85	C₅	C₃×#519	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²	—	2	2	no	—
87	C₂²	C₂²×C₃×C₄×C₅	1	2	yes	—
88	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
89	C₄	—	2	8	no	—
90	C₄	C₂²×C₃×C₄×C₅	1	12	yes	—
91	C₃	C₅×#519	1	1	yes	characteristic, direct product factor, Sylow
92	C₂	—	2	2	no	—
93	C₂	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
94	C₂	C₃×C₅×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator	C₂	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂³
LCS	G ▷ C₂ ▷ 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×C₄×C₅×Q₈	C₂	semidirect
4. C₅×#519	C₃	direct (N × H)
5. C₂×C₃×C₅×Q₈	C₄	semidirect
11. C₃×#519	C₅	direct (N × H)
13. C₄×C₅×Q₈	C₂×C₃	semidirect
21. C₃×C₄×Q₈	C₂×C₅	semidirect
23. C₂×C₅×Q₈	C₃×C₄	semidirect
29. #519	C₃×C₅	direct (N × H)
35. C₂×C₃×Q₈	C₄×C₅	semidirect
47. C₄×Q₈	C₂×C₃×C₅	semidirect
62. C₂×Q₈	C₃×C₄×C₅	semidirect
68. C₃×C₅	#519	direct (N × H)
85. C₅	C₃×#519	direct (N × H)
91. C₃	C₅×#519	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×Q₈	2
C₃×C₄×C₅×Q₈	2
C₃×C₅×#117	2
C₅×#519	3
C₃×#519	5

Subgroups — C3×C5×#519 #11756608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#519 #11756608