Subgroups — #11750800

Order 960.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₀	C₂	1	4	yes	maximal
2	#91758	C₂	1	2	yes	maximal
3	C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	#64988	—	3	1	no	maximal, Hall
5	C₂×Dic₃₀	C₂²	1	4	yes	—
6	C₂×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₅	—	2	4	no	—
9	Dic₆₀	—	2	8	no	—
10	#4699	—	5	1	no	maximal, Hall
11	C₂×Dic₂₀	—	3	4	no	—
12	#4015	—	3	2	no	—
13	C₄×C₅×C₈	D₃	1	1	yes	characteristic
14	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
15	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
16	C₂×Dic₁₅	—	2	4	no	—
17	Dic₃₀	—	2	8	no	—
18	C₃×C₅×C₈	D₄	1	4	yes	—
19	C₂×Dic₁₂	—	5	4	no	—
20	#869	—	5	2	no	—
21	C₃×C₄×C₈	D₅	1	1	yes	characteristic
22	C₂×Dic₁₀	—	3	4	no	—
23	C₂×C₅×C₈	C₂×D₃	1	2	yes	—
24	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
25	C₄⋊Dic₅	—	6	4	no	—
26	Dic₂₀	—	6	8	no	—
27	#494	—	15	1	no	Sylow
28	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
29	C₃×C₄×C₅	Q₁₆	1	4	yes	complemented
30	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
31	Dic₁₅	—	4	4	no	—
32	C₂×Dic₆	—	5	4	no	—
33	C₂×C₃×C₈	C₂×D₅	1	2	yes	—
34	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
35	C₄⋊Dic₃	—	10	4	no	—
36	Dic₁₂	—	10	8	no	—
37	C₂×C₄×C₅	D₁₂	1	2	yes	—
38	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
39	C₂×Dic₅	—	6	4	no	—
40	Dic₁₀	—	6	8	no	—
41	C₅×C₈	D₁₂	1	4	yes	—
42	C₂×Q₁₆	—	15	4	no	—
43	C₄⋊Q₈	—	15	2	no	—
44	C₄×C₈	D₁₅	1	1	yes	characteristic
45	C₂×C₃×C₅	#127	1	1	yes	characteristic
46	C₂×C₃×C₅	C₂×Q₁₆	1	2	yes	—
47	C₂×C₃×C₄	D₂₀	1	2	yes	—
48	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
49	C₂×Dic₃	—	10	4	no	—
50	Dic₆	—	10	8	no	—
51	C₃×C₈	D₂₀	1	4	yes	—
52	C₂²×C₅	C₂×D₁₂	1	1	yes	characteristic
53	C₄×C₅	Dic₁₂	1	4	yes	complemented
54	C₄×C₅	C₂×D₁₂	1	2	yes	—
55	Dic₅	—	12	4	no	—
56	C₂×Q₈	—	15	4	no	—
57	Q₁₆	—	30	8	no	—
58	C₂×C₈	C₂×D₁₅	1	2	yes	—
59	C₄⋊C₄	—	30	4	no	—
60	C₄²	C₂×D₁₅	1	1	yes	characteristic
61	C₃×C₅	#494	1	1	yes	characteristic, complemented, Hall
62	C₂²×C₃	C₂×D₂₀	1	1	yes	characteristic
63	C₃×C₄	Dic₂₀	1	4	yes	complemented
64	C₃×C₄	C₂×D₂₀	1	2	yes	—
65	Dic₃	—	20	4	no	—
66	C₂×C₅	#874	1	1	yes	characteristic
67	C₂×C₅	C₂×Dic₁₂	1	2	yes	—
68	Q₈	—	30	8	no	—
69	C₂×C₄	—	30	4	no	—
70	C₂×C₄	D₆₀	1	2	yes	—
71	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
72	C₈	D₆₀	1	4	yes	—
73	C₂×C₃	#4020	1	1	yes	characteristic
74	C₂×C₃	C₂×Dic₂₀	1	2	yes	—
75	C₅	#4699	1	1	yes	characteristic, complemented, Sylow
76	C₂²	C₂×D₆₀	1	1	yes	characteristic
77	C₄	—	60	4	no	—
78	C₄	Dic₆₀	1	4	yes	complemented
79	C₄	C₂×D₆₀	1	2	yes	—
80	C₃	#64988	1	1	yes	characteristic, complemented, Sylow
81	C₂	#91763	1	1	yes	characteristic
82	C₂	C₂×Dic₆₀	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₁₅
Fitting	C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₄×C₈	D₁₅
hypocenter	C₃×C₅	#494
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
29. C₃×C₄×C₅	Q₁₆	semidirect
53. C₄×C₅	Dic₁₂	semidirect
61. C₃×C₅	#494	semidirect
63. C₃×C₄	Dic₂₀	semidirect
75. C₅	#4699	semidirect
78. C₄	Dic₆₀	semidirect
80. C₃	#64988	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₀	2
#91758	2
C₃×C₄×C₅×C₈	2
#64988	3
#4699	5