Subgroups — #91275

Order 480.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6769	C₂	1	2	yes	maximal, complemented
2	D₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
5	#6658	C₂	1	2	yes	maximal, complemented
6	#4050	—	3	1	no	maximal, Hall
7	D₆₀	C₂²	1	2	yes	—
8	C₄×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₃×Dic₁₀	C₂²	1	2	yes	—
10	#1203	—	2	2	no	—
11	#1200	—	2	2	no	—
12	C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	#919	—	5	1	no	maximal, Hall
15	#709	—	3	2	no	—
16	C₂×C₅×C₈	—	3	1	no	—
17	Dic₂₀	D₃	1	1	yes	characteristic, complemented
18	D₄₀	—	3	1	no	—
19	#677	—	3	2	no	—
20	C₂×D₁₅	—	2	2	no	—
21	C₂×C₅×D₃	D₄	1	1	yes	characteristic
22	C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₃×Dic₅	—	2	2	no	—
24	C₅×Dic₃	D₄	1	1	yes	characteristic
25	#239	—	5	2	no	—
26	C₃×Q₁₆	—	5	1	no	—
27	#215	—	5	2	no	—
28	D₂₄	—	5	1	no	—
29	D₃×C₈	D₅	1	1	yes	characteristic, complemented
30	C₂×C₄×C₅	—	3	1	no	—
31	C₂²⋊D₅	—	6	2	no	—
32	D₂₀	—	3	2	no	—
33	C₄×D₅	—	6	2	no	—
34	Dic₁₀	C₂×D₃	1	2	yes	—
35	C₅×C₈	—	3	1	no	—
36	C₅×C₈	C₂×D₃	1	1	yes	characteristic
37	#135	—	15	1	no	Sylow
38	C₂×C₃×C₅	C₂×D₄	1	1	yes	characteristic
39	D₁₅	—	4	2	no	—
40	C₅×D₃	—	2	1	no	—
41	C₃×Q₈	—	5	2	no	—
42	D₁₂	—	5	2	no	—
43	D₁₂	—	10	2	no	—
44	C₄×D₃	—	10	2	no	—
45	C₄×D₃	C₂×D₅	1	1	yes	characteristic
46	C₃×C₈	C₂×D₅	1	1	yes	characteristic
47	C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
48	C₂²×C₅	—	3	1	no	—
49	C₂×D₅	—	6	2	no	—
50	C₄×C₅	—	3	1	no	—
51	C₄×C₅	C₂²×D₃	1	1	yes	characteristic
52	Dic₅	—	2	2	no	—
53	#41	—	15	2	no	—
54	Q₁₆	—	5	1	no	—
55	SD₁₆	—	15	2	no	—
56	D₈	—	15	1	no	—
57	C₂×C₈	—	3	1	no	—
58	C₃×C₅	#135	1	1	yes	characteristic, complemented, Hall
59	C₂×D₃	—	10	2	no	—
60	C₂×D₃	D₂₀	1	1	yes	characteristic
61	C₃×C₄	—	10	2	no	—
62	C₃×C₄	C₂²×D₅	1	1	yes	characteristic
63	Dic₃	D₂₀	1	1	yes	characteristic
64	C₂×C₅	—	6	1	no	—
65	C₂×C₅	D₃×D₄	1	1	yes	characteristic
66	D₅	—	12	2	no	—
67	Q₈	—	5	2	no	—
68	D₄	—	15	2	no	—
69	D₄	—	30	2	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	30	2	no	—
72	C₈	—	3	1	no	—
73	C₈	D₃×D₅	1	1	yes	characteristic
74	C₂×C₃	C₂×D₂₀	1	1	yes	characteristic
75	D₃	—	2	1	no	—
76	D₃	—	20	2	no	—
77	C₅	#919	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	3	1	no	—
79	C₂²	—	30	2	no	—
80	C₄	—	3	1	no	—
81	C₄	—	10	2	no	—
82	C₄	C₂×D₃×D₅	1	1	yes	characteristic
83	C₃	#4050	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	6	1	no	—
85	C₂	—	60	2	no	—
86	C₂	D₃×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₂₀
commutator, cosocle	C₃×C₄×C₅	C₂³
Frattini	C₄	C₂×D₃×D₅
Fitting	C₃×C₅×C₈	C₂²
socle	C₂×C₃×C₅	C₂×D₄
hypercenter	C₈	D₃×D₅
hypocenter	C₃×C₅	#135
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. #6769	C₂	semidirect
2. D₁₂₀	C₂	semidirect
3. C₅×D₃×C₈	C₂	semidirect
4. C₃×Dic₂₀	C₂	semidirect
5. #6658	C₂	semidirect
17. Dic₂₀	D₃	semidirect
29. D₃×C₈	D₅	semidirect
58. C₃×C₅	#135	semidirect
77. C₅	#919	semidirect
83. C₃	#4050	semidirect

Maximal subgroups

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

subgroup	index
#6769	2
D₁₂₀	2
C₅×D₃×C₈	2
C₃×Dic₂₀	2
#6658	2
#4050	3
#919	5