Subgroups — #90963

Order 480.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₃₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#208	C₂	1	1	yes	maximal, characteristic
3	C₃×#681	C₂	1	1	yes	maximal, characteristic
4	#3956	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂×Dic₁₅	C₄	1	2	yes	—
7	Dic₃₀	—	2	1	no	—
8	Dic₃₀	—	2	1	no	—
9	C₃×#168	—	2	1	no	—
10	C₅×C₃⋊C₈	—	2	1	no	—
11	#824	—	5	1	no	maximal, Hall
12	C₂×Dic₁₀	—	3	1	no	—
13	C₅×M₁₆	—	3	1	no	—
14	#681	D₃	1	1	yes	characteristic
15	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₃×C₄×C₅	D₄	1	1	yes	characteristic
17	C₃×C₄×C₅	D₄	1	1	yes	characteristic
18	Dic₁₅	—	2	2	no	—
19	C₂×Dic₆	—	5	1	no	—
20	C₃×M₁₆	—	5	1	no	—
21	#208	D₅	1	1	yes	characteristic
22	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×Dic₅	—	3	2	no	—
24	Dic₁₀	—	6	1	no	—
25	Dic₁₀	—	6	1	no	—
26	C₅×C₈	—	6	1	no	—
27	#168	—	2	1	no	—
28	#101	—	15	1	no	Sylow
29	C₂×C₃×C₅	—	2	1	no	—
30	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
32	C₂×Dic₃	—	5	2	no	—
33	Dic₆	—	10	1	no	—
34	Dic₆	—	10	1	no	—
35	C₃×C₈	—	10	1	no	—
36	C₃⋊C₈	—	2	1	no	—
37	C₂²×C₅	C₄×D₃	1	1	yes	characteristic
38	C₄×C₅	D₁₂	1	1	yes	characteristic
39	C₄×C₅	#67	1	1	yes	characteristic
40	Dic₅	—	6	2	no	—
41	C₂×Q₈	—	15	1	no	—
42	M₁₆	—	3	1	no	—
43	M₁₆	—	5	1	no	—
44	C₃×C₅	#101	1	1	yes	characteristic, complemented, Hall
45	C₂²×C₃	C₄×D₅	1	1	yes	characteristic
46	C₃×C₄	D₂₀	1	1	yes	characteristic
47	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
48	Dic₃	—	10	2	no	—
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
51	Q₈	—	30	1	no	—
52	Q₈	—	30	1	no	—
53	C₂×C₄	—	15	2	no	—
54	C₂×C₄	D₃×D₅	1	1	yes	characteristic
55	C₈	—	6	1	no	—
56	C₈	—	10	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	#685	1	1	yes	characteristic
59	C₅	#824	1	1	yes	characteristic, complemented, Sylow
60	C₂²	#1200	1	1	yes	characteristic
61	C₄	—	30	2	no	—
62	C₄	#1202	1	1	yes	characteristic
63	C₄	#1203	1	1	yes	characteristic
64	C₃	#3956	1	1	yes	characteristic, complemented, Sylow
65	C₂	—	2	1	no	—
66	C₂	#6667	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6667
commutator, prime radical	C₂²×C₃×C₅	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₅
Fitting, cosocle	C₂×C₃×C₄×C₅	C₂²
socle	C₂×C₃×C₅	C₂²⋊C₄
hypocenter	C₃×C₅	#101
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
44. C₃×C₅	#101	semidirect
59. C₅	#824	semidirect
64. C₃	#3956	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₃₀	2
C₅×#208	2
C₃×#681	2
#3956	3
#824	5