Subgroups — #90959

Order 480.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
2	#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	#3965	—	3	1	no	maximal, Hall
5	C₂²×C₅×D₃	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₅×D₁₂	—	2	1	no	—
8	C₅×D₁₂	—	2	1	no	—
9	C₃⋊#168	—	2	1	no	—
10	C₃×#168	—	2	1	no	—
11	#823	—	5	1	no	maximal, Hall
12	C₂×C₅×D₄	—	3	1	no	—
13	#681	—	3	1	no	—
14	#681	D₃	1	1	yes	characteristic, complemented
15	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₅×D₃	—	2	2	no	—
17	C₂×C₅×D₃	—	4	2	no	—
18	C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	C₃×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×D₁₂	D₅	1	1	yes	characteristic
21	C₃×M₁₆	—	5	1	no	—
22	#208	—	5	1	no	—
23	C₂³×C₅	—	3	2	no	—
24	C₅×D₄	—	6	1	no	—
25	C₅×D₄	—	6	1	no	—
26	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
27	#168	—	2	1	no	—
28	#168	—	6	1	no	—
29	#100	—	15	1	no	Sylow
30	C₂×C₃×C₅	—	2	1	no	—
31	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
32	C₅×D₃	—	4	2	no	—
33	C₂²×D₃	Dic₅	1	2	yes	—
34	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
35	D₁₂	—	2	1	no	—
36	D₁₂	—	2	1	no	—
37	C₃×C₈	—	10	1	no	—
38	C₃⋊C₈	—	10	1	no	—
39	C₂²×C₅	—	6	2	no	—
40	C₂²×C₅	—	12	2	no	—
41	C₂²×C₅	C₄×D₃	1	1	yes	characteristic
42	C₄×C₅	D₁₂	1	1	yes	characteristic
43	C₄×C₅	#67	1	1	yes	characteristic
44	C₂×D₄	—	3	1	no	—
45	M₁₆	—	5	1	no	—
46	M₁₆	—	15	1	no	—
47	C₃×C₅	#100	1	1	yes	characteristic, complemented, Hall
48	C₂²×C₃	C₂×Dic₅	1	1	yes	characteristic
49	C₂×D₃	—	2	2	no	—
50	C₂×D₃	—	4	2	no	—
51	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
52	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
53	C₂×C₅	—	2	1	no	—
54	C₂×C₅	—	12	2	no	—
55	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
56	C₂³	—	3	2	no	—
57	D₄	—	6	1	no	—
58	D₄	—	6	1	no	—
59	C₂×C₄	D₃×D₅	1	1	yes	characteristic
60	C₈	—	10	1	no	—
61	C₈	—	30	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
64	D₃	—	4	2	no	—
65	C₅	#823	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	6	2	no	—
67	C₂²	—	12	2	no	—
68	C₂²	D₃×Dic₅	1	1	yes	characteristic
69	C₄	#1201	1	1	yes	characteristic
70	C₄	#1203	1	1	yes	characteristic
71	C₃	#3965	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	2	1	no	—
73	C₂	—	12	2	no	—
74	C₂	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₃)⋊Dic₅
commutator	C₂²×C₃×C₅	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₅
Fitting, cosocle	C₂×C₃×C₄×C₅	C₂²
socle	C₂×C₃×C₅	C₂²⋊C₄
prime radical	C₂×C₅×D₁₂	C₂
hypocenter	C₃×C₅	#100
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
2. #6710	C₂	semidirect
3. C₃×#681	C₂	semidirect
14. #681	D₃	semidirect
47. C₃×C₅	#100	semidirect
65. C₅	#823	semidirect
71. C₃	#3965	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₁₂	2
#6710	2
C₃×#681	2
#3965	3
#823	5