Subgroups — #91105

Order 480.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#3953	—	3	1	no	maximal, Hall
5	C₂²×D₁₅	—	2	1	no	—
6	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₅	—	2	1	no	—
8	D₆₀	—	2	1	no	—
9	D₆₀	C₄	1	2	yes	complemented
10	C₃×C₅×C₈	—	2	1	no	—
11	#821	—	5	1	no	maximal, Hall
12	C₂×D₂₀	—	3	1	no	—
13	C₂×C₅×C₈	D₃	1	1	yes	characteristic, complemented
14	C₄⋊Dic₅	—	3	1	no	—
15	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
16	C₂×D₁₅	—	2	2	no	—
17	C₂×D₁₅	—	4	1	no	—
18	C₂×D₁₅	—	4	1	no	—
19	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₃×C₄×C₅	D₄	1	1	yes	characteristic
21	Dic₁₅	—	4	1	no	—
22	C₂×D₁₂	—	5	1	no	—
23	C₂×C₃×C₈	D₅	1	1	yes	characteristic, complemented
24	C₄⋊Dic₃	—	5	1	no	—
25	C₂²×D₅	—	6	1	no	—
26	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₅	—	6	1	no	—
28	D₂₀	—	3	2	no	—
29	D₂₀	—	6	1	no	—
30	C₅×C₈	—	2	1	no	—
31	#102	—	15	1	no	Sylow
32	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₅	D₈	1	1	yes	characteristic
34	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
35	D₁₅	—	4	2	no	—
36	C₂²×D₃	—	10	1	no	—
37	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
38	C₂×Dic₃	—	10	1	no	—
39	D₁₂	—	5	2	no	—
40	D₁₂	—	10	1	no	—
41	C₃×C₈	—	2	1	no	—
42	C₂²×C₅	D₁₂	1	1	yes	characteristic
43	C₂×D₅	—	6	2	no	—
44	C₂×D₅	—	12	1	no	—
45	C₂×D₅	—	12	1	no	—
46	C₄×C₅	C₄×D₃	1	1	yes	characteristic
47	C₄×C₅	#67	1	1	yes	characteristic
48	Dic₅	—	12	1	no	—
49	C₂×D₄	—	15	1	no	—
50	C₂×C₈	D₁₅	1	1	yes	characteristic, complemented
51	C₄⋊C₄	—	15	1	no	—
52	C₃×C₅	#102	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₃	D₂₀	1	1	yes	characteristic
54	C₂×D₃	—	10	2	no	—
55	C₂×D₃	—	20	1	no	—
56	C₂×D₃	—	20	1	no	—
57	C₃×C₄	C₄×D₅	1	1	yes	characteristic
58	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
59	Dic₃	—	20	1	no	—
60	C₂×C₅	#204	1	1	yes	characteristic
61	C₂×C₅	D₂₄	1	1	yes	characteristic
62	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
63	D₅	—	12	2	no	—
64	C₂³	—	30	1	no	—
65	D₄	—	15	2	no	—
66	D₄	—	30	1	no	—
67	C₂×C₄	—	30	1	no	—
68	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₂×C₃	#677	1	1	yes	characteristic
71	C₂×C₃	D₄₀	1	1	yes	characteristic
72	C₂×C₃	#685	1	1	yes	characteristic
73	D₃	—	20	2	no	—
74	C₅	#821	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	30	2	no	—
76	C₂²	—	60	1	no	—
77	C₂²	—	60	1	no	—
78	C₂²	D₆₀	1	1	yes	characteristic
79	C₄	—	60	1	no	—
80	C₄	C₄×D₁₅	1	1	yes	characteristic
81	C₄	#1220	1	1	yes	characteristic
82	C₃	#3953	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	60	2	no	—
84	C₂	#6706	1	1	yes	characteristic
85	C₂	D₁₂₀	1	1	yes	characteristic
86	C₂	#6714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₆₀
commutator	C₃×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₅
Fitting	C₂×C₃×C₅×C₈	C₂
socle	C₂²×C₃×C₅	D₄
prime radical	C₂×D₆₀	C₂
cosocle	C₂×C₃×C₄×C₅	C₂²
hypercenter	C₂×C₈	D₁₅
hypocenter	C₃×C₅	#102
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₈	C₂	semidirect
3. C₄⋊Dic₁₅	C₂	semidirect
9. D₆₀	C₄	semidirect
13. C₂×C₅×C₈	D₃	semidirect
23. C₂×C₃×C₈	D₅	semidirect
50. C₂×C₈	D₁₅	semidirect
52. C₃×C₅	#102	semidirect
74. C₅	#821	semidirect
82. C₃	#3953	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₆₀	2
C₂×C₃×C₅×C₈	2
C₄⋊Dic₁₅	2
#3953	3
#821	5