Subgroups — C₂×D₅×SL(2,𝔽₃) #91963

Order 480.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×SL(2,𝔽₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×Q₈×D₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₃×D₅	—	4	1	no	maximal
5	C₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	C₂²×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
7	C₂×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
8	Q₈×D₅	—	3	2	no	—
9	Q₈×D₅	C₂×C₃	1	2	yes	complemented
10	C₂×C₄×D₅	—	3	1	no	—
11	C₂×Dic₁₀	—	3	1	no	—
12	C₂²×C₃×C₅	—	4	1	no	—
13	C₂×C₃×D₅	—	4	4	no	—
14	C₂×C₃×D₅	—	4	2	no	—
15	C₂×SL(2,𝔽₃)	—	5	2	no	—
16	C₂×SL(2,𝔽₃)	D₅	1	1	yes	characteristic, direct product factor
17	C₂²×D₅	A₄	1	1	yes	characteristic
18	C₅×Q₈	—	3	1	no	—
19	C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
20	C₂×C₄×C₅	—	3	1	no	—
21	C₂×Dic₅	—	3	1	no	—
22	C₄×D₅	—	3	2	no	—
23	C₄×D₅	—	3	2	no	—
24	Dic₁₀	—	3	2	no	—
25	Dic₁₀	—	3	2	no	—
26	C₂²×Q₈	—	5	1	no	Sylow
27	C₂×C₃×C₅	—	4	2	no	—
28	C₂×C₃×C₅	—	4	1	no	—
29	C₃×D₅	—	4	4	no	—
30	C₂³×C₃	—	20	1	no	—
31	SL(2,𝔽₃)	C₂×D₅	1	1	yes	characteristic, direct product factor
32	C₂²×C₅	C₂×A₄	1	1	yes	characteristic
33	C₂×D₅	SL(2,𝔽₃)	1	4	yes	direct product factor
34	C₂×D₅	C₂×A₄	1	2	yes	—
35	C₄×C₅	—	3	1	no	—
36	C₄×C₅	—	3	1	no	—
37	Dic₅	—	3	2	no	—
38	C₂×Q₈	—	5	2	no	—
39	C₂×Q₈	—	15	1	no	—
40	C₂×Q₈	—	15	2	no	—
41	C₂×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
42	C₂²×C₄	—	15	1	no	—
43	C₃×C₅	—	4	1	no	Hall
44	C₂²×C₃	—	4	1	no	—
45	C₂²×C₃	—	20	4	no	—
46	C₂²×C₃	—	20	2	no	—
47	C₂×C₅	C₂×SL(2,𝔽₃)	1	2	yes	complemented
48	C₂×C₅	C₂²×A₄	1	1	yes	characteristic
49	D₅	C₂×SL(2,𝔽₃)	1	4	yes	direct product factor
50	C₂³	—	5	1	no	—
51	Q₈	—	3	1	no	—
52	Q₈	—	15	2	no	—
53	Q₈	—	15	2	no	—
54	Q₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	15	2	no	—
57	C₂×C₄	—	15	2	no	—
58	C₂×C₄	—	15	1	no	—
59	C₂×C₃	—	4	2	no	—
60	C₂×C₃	—	4	1	no	—
61	C₂×C₃	—	20	4	no	—
62	C₅	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	5	4	no	—
64	C₂²	—	5	2	no	—
65	C₂²	D₅×A₄	1	1	yes	characteristic
66	C₄	—	3	1	no	—
67	C₄	—	3	1	no	—
68	C₄	—	15	2	no	—
69	C₃	—	4	1	no	Sylow
70	C₂	—	5	4	no	—
71	C₂	D₅×SL(2,𝔽₃)	1	2	yes	direct product factor
72	C₂	C₂×D₅×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	D₅×A₄
commutator, cosocle, hypocenter	C₅×Q₈	C₂²×C₃
Frattini	C₂	C₂×D₅×A₄
Fitting	C₂×C₅×Q₈	C₂×C₃
socle	C₂²×C₅	C₂×A₄
LCS	G ▷ C₅×Q₈
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×SL(2,𝔽₃)	C₂	semidirect
2. D₅×SL(2,𝔽₃)	C₂	direct (N × H)
3. C₂×Q₈×D₅	C₃	semidirect
5. C₅×SL(2,𝔽₃)	C₂²	semidirect
7. C₂×C₅×Q₈	C₂×C₃	semidirect
9. Q₈×D₅	C₂×C₃	semidirect
16. C₂×SL(2,𝔽₃)	D₅	direct (N × H)
19. C₅×Q₈	C₂²×C₃	semidirect
31. SL(2,𝔽₃)	C₂×D₅	direct (N × H)
33. C₂×D₅	SL(2,𝔽₃)	direct (N × H)
41. C₂×Q₈	C₃×D₅	semidirect
47. C₂×C₅	C₂×SL(2,𝔽₃)	semidirect
49. D₅	C₂×SL(2,𝔽₃)	direct (N × H)
54. Q₈	C₂×C₃×D₅	semidirect
62. C₅	C₂²×SL(2,𝔽₃)	semidirect
71. C₂	D₅×SL(2,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×SL(2,𝔽₃)	2
D₅×SL(2,𝔽₃)	2
C₂×Q₈×D₅	3
C₂²×C₃×D₅	4
C₂²×SL(2,𝔽₃)	5

Subgroups — C2×D5×SL(2,𝔽3) #91963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×SL(2,𝔽₃) #91963