Subgroups — D₃×SL(2,𝔽₅) #10623948

Order 720.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×SL(2,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SL(2,𝔽₅)	—	3	1	no	maximal
3	D₃×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
4	D₃×Dic₅	—	6	1	no	maximal
5	SL(2,𝔽₅)	D₃	1	1	yes	characteristic, direct product factor
6	C₃×SL(2,𝔽₃)	—	5	1	no	—
7	D₃×Dic₃	—	10	1	no	maximal
8	C₂×C₅×D₃	—	6	1	no	—
9	Dic₁₅	—	6	1	no	—
10	C₃×Dic₅	—	6	1	no	—
11	D₃×Q₈	—	5	1	no	—
12	C₂×SL(2,𝔽₃)	—	15	1	no	—
13	C₂×Dic₅	—	18	1	no	—
14	C₂×C₃×D₃	—	10	1	no	—
15	C₃⋊Dic₃	—	10	1	no	—
16	C₃×Dic₃	—	10	1	no	—
17	C₂×C₃×C₅	—	6	1	no	—
18	C₅×D₃	—	6	2	no	—
19	C₃×Q₈	—	5	1	no	—
20	C₂×Dic₃	—	30	1	no	—
21	C₄×D₃	—	15	1	no	—
22	Dic₆	—	15	1	no	—
23	SL(2,𝔽₃)	—	5	1	no	—
24	SL(2,𝔽₃)	—	10	1	no	—
25	C₂²×C₅	—	18	1	no	—
26	Dic₅	—	6	1	no	—
27	Dic₅	—	18	1	no	—
28	C₂×C₃²	—	10	1	no	—
29	C₃×D₃	—	10	2	no	—
30	C₂×Q₈	—	15	1	no	Sylow
31	C₃×C₅	—	6	1	no	—
32	C₂²×C₃	—	30	1	no	—
33	C₂×D₃	A₅	1	1	yes	maximal normal, characteristic
34	C₃×C₄	—	15	1	no	—
35	Dic₃	—	10	1	no	—
36	Dic₃	—	15	1	no	—
37	Dic₃	—	30	1	no	—
38	Dic₃	—	60	1	no	—
39	C₂×C₅	—	6	1	no	—
40	C₂×C₅	—	18	2	no	—
41	C₃²	—	10	1	no	Sylow
42	Q₈	—	5	1	no	—
43	Q₈	—	45	1	no	—
44	C₂×C₄	—	45	1	no	—
45	C₂×C₃	—	10	1	no	—
46	C₂×C₃	—	20	1	no	—
47	C₂×C₃	—	30	2	no	—
48	C₂×C₃	C₂×A₅	1	1	yes	characteristic
49	D₃	SL(2,𝔽₅)	1	2	yes	direct product factor
50	C₅	—	6	1	no	Sylow
51	C₂²	—	3	1	no	—
52	C₄	—	15	1	no	—
53	C₄	—	45	1	no	—
54	C₃	—	10	1	no	—
55	C₃	—	20	1	no	—
56	C₃	C₂×SL(2,𝔽₅)	1	1	yes	characteristic, complemented
57	C₂	—	3	2	no	—
58	C₂	D₃×A₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×A₅
commutator, hypocenter	C₃×SL(2,𝔽₅)	C₂
Fitting, socle, cosocle	C₂×C₃	C₂×A₅
LCS	G ▷ C₃×SL(2,𝔽₅)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₃×SL(2,𝔽₅)	C₂	semidirect
5. SL(2,𝔽₅)	D₃	direct (N × H)
49. D₃	SL(2,𝔽₅)	direct (N × H)
56. C₃	C₂×SL(2,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃×SL(2,𝔽₅)	2
C₂×SL(2,𝔽₅)	3
D₃×SL(2,𝔽₃)	5
D₃×Dic₅	6
D₃×Dic₃	10

Subgroups — D3×SL(2,𝔽5) #10623948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×SL(2,𝔽₅) #10623948