Subgroups — C₂²×C₃×#170 #91981

Order 480.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#170	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#170	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃×Dic₅	C₄	1	3	yes	—
6	C₂×C₃×Dic₅	C₂²	1	3	yes	—
7	C₃×#170	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₈	—	5	1	no	maximal, Hall
9	C₂²×Dic₅	C₂×C₃	1	1	yes	characteristic
10	C₂×#170	C₂×C₃	1	6	yes	direct product factor
11	C₂²×C₃×C₅	C₈	1	4	yes	complemented
12	C₂²×C₃×C₅	C₂×C₄	1	3	yes	—
13	C₃×Dic₅	C₂×C₄	1	3	yes	—
14	C₃×Dic₅	C₂³	1	1	yes	characteristic
15	C₂²×C₃×C₄	—	5	1	no	—
16	C₂×C₃×C₈	—	5	6	no	—
17	C₂³×C₅	C₃×C₄	1	1	yes	characteristic
18	C₂×Dic₅	C₃×C₄	1	3	yes	—
19	C₂×Dic₅	C₂²×C₃	1	3	yes	—
20	#170	C₂²×C₃	1	4	yes	direct product factor
21	C₂²×C₈	—	5	1	no	Sylow
22	C₂×C₃×C₅	C₂×C₈	1	6	yes	complemented
23	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
24	C₂³×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₂×C₃×C₄	—	5	3	no	—
26	C₂×C₃×C₄	—	5	3	no	—
27	C₃×C₈	—	5	4	no	—
28	C₂²×C₅	C₃×C₈	1	4	yes	complemented
29	C₂²×C₅	C₂×C₃×C₄	1	3	yes	—
30	Dic₅	C₂×C₃×C₄	1	3	yes	—
31	Dic₅	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₄	—	5	1	no	—
33	C₂×C₈	—	5	6	no	—
34	C₃×C₅	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₃	#170	1	4	yes	direct product factor
36	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	3	yes	—
37	C₃×C₄	—	5	3	no	—
38	C₃×C₄	—	5	1	no	—
39	C₂×C₅	C₂×C₃×C₈	1	6	yes	complemented
40	C₂×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
41	C₂³	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×C₄	—	5	3	no	—
43	C₂×C₄	—	5	3	no	—
44	C₈	—	5	4	no	—
45	C₂×C₃	C₂×#170	1	6	yes	direct product factor
46	C₂×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₅	C₂²×C₃×C₈	1	1	yes	characteristic, complemented, Sylow
48	C₂²	C₃×#170	1	4	yes	direct product factor
49	C₂²	C₂×C₃×AGL(1,𝔽₅)	1	3	yes	—
50	C₄	—	5	1	no	—
51	C₄	—	5	3	no	—
52	C₃	C₂²×#170	1	1	yes	characteristic, direct product factor, Sylow
53	C₂	C₂×C₃×#170	1	6	yes	direct product factor
54	C₂	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₃×C₈
Frattini	C₂	C₂²×C₃×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂³×C₃×C₅	C₄
cosocle	Dic₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#170	C₂	direct (N × H)
3. C₂²×#170	C₃	direct (N × H)
7. C₃×#170	C₂²	direct (N × H)
10. C₂×#170	C₂×C₃	direct (N × H)
11. C₂²×C₃×C₅	C₈	semidirect
20. #170	C₂²×C₃	direct (N × H)
22. C₂×C₃×C₅	C₂×C₈	semidirect
28. C₂²×C₅	C₃×C₈	semidirect
34. C₃×C₅	C₂²×C₈	semidirect
35. C₂²×C₃	#170	direct (N × H)
39. C₂×C₅	C₂×C₃×C₈	semidirect
45. C₂×C₃	C₂×#170	direct (N × H)
47. C₅	C₂²×C₃×C₈	semidirect
48. C₂²	C₃×#170	direct (N × H)
52. C₃	C₂²×#170	direct (N × H)
53. C₂	C₂×C₃×#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₅	2
C₂×C₃×#170	2
C₂²×#170	3
C₂²×C₃×C₈	5

Subgroups — C22×C3×#170 #91981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#170 #91981