Subgroups — #90990

Order 480.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	#3950	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂×Dic₁₅	—	2	1	no	—
7	C₂×C₃×Dic₅	—	2	1	no	—
8	C₅×C₃⋊C₈	C₄	1	2	yes	complemented
9	#807	—	5	1	no	maximal, Hall
10	C₂×C₅×C₈	—	3	1	no	—
11	C₄⋊Dic₅	—	3	1	no	—
12	C₄⋊Dic₅	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
14	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄×C₅	Q₈	1	1	yes	characteristic
16	Dic₁₅	—	4	1	no	—
17	C₃×Dic₅	—	4	1	no	—
18	C₃×C₄⋊C₄	—	5	1	no	—
19	C₄⋊Dic₃	—	5	1	no	—
20	C₂×C₃⋊C₈	D₅	1	1	yes	characteristic
21	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
22	C₂×Dic₅	—	2	1	no	—
23	C₂×Dic₅	—	6	1	no	—
24	C₅×C₈	—	3	2	no	—
25	#107	—	15	1	no	Sylow
26	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₃×C₅	D₈	1	1	yes	characteristic
28	C₂×C₃×C₅	Q₁₆	1	1	yes	characteristic
29	C₂×C₃×C₄	—	10	1	no	—
30	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
31	C₂×Dic₃	—	10	1	no	—
32	C₃⋊C₈	Dic₅	1	2	yes	complemented
33	C₂²×C₅	#67	1	1	yes	characteristic
34	C₄×C₅	Dic₆	1	1	yes	characteristic
35	C₄×C₅	C₄×D₃	1	1	yes	characteristic
36	Dic₅	—	4	1	no	—
37	Dic₅	—	12	1	no	—
38	C₂×C₈	—	3	1	no	—
39	C₄⋊C₄	—	5	1	no	—
40	C₄⋊C₄	—	15	1	no	—
41	C₃×C₅	#107	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₃	D₂₀	1	1	yes	characteristic
43	C₃×C₄	—	20	1	no	—
44	C₃×C₄	Dic₁₀	1	1	yes	characteristic
45	C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
46	Dic₃	—	20	1	no	—
47	C₂×C₅	Dic₃⋊C₄	1	1	yes	characteristic
48	C₂×C₅	#213	1	1	yes	characteristic
49	C₂×C₅	#216	1	1	yes	characteristic
50	C₂×C₄	—	10	1	no	—
51	C₂×C₄	—	30	1	no	—
52	C₂×C₄	D₃×D₅	1	1	yes	characteristic
53	C₈	—	3	2	no	—
54	C₂×C₃	D₄₀	1	1	yes	characteristic
55	C₂×C₃	Dic₂₀	1	1	yes	characteristic
56	C₂×C₃	C₄⋊Dic₅	1	1	yes	characteristic
57	C₅	#807	1	1	yes	characteristic, complemented, Sylow
58	C₂²	#1202	1	1	yes	characteristic
59	C₄	—	20	1	no	—
60	C₄	—	60	1	no	—
61	C₄	D₃×Dic₅	1	1	yes	characteristic
62	C₄	#1204	1	1	yes	characteristic
63	C₃	#3950	1	1	yes	characteristic, complemented, Sylow
64	C₂	#6653	1	1	yes	characteristic
65	C₂	#6662	1	1	yes	characteristic
66	C₂	Dic₃⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#1202
commutator	C₃×C₄×C₅	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₅
Fitting, cosocle	C₂×C₃×C₄×C₅	C₂²
socle, prime radical	C₂²×C₃×C₅	D₄
hypocenter	C₃×C₅	#107
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. C₅×C₃⋊C₈	C₄	semidirect
32. C₃⋊C₈	Dic₅	semidirect
41. C₃×C₅	#107	semidirect
57. C₅	#807	semidirect
63. C₃	#3950	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₁₅	2
C₂×C₅×C₃⋊C₈	2
C₃×C₄⋊Dic₅	2
#3950	3
#807	5