Subgroups — C₅×D₇⋊SL(2,𝔽₃) #13132589

Order 1680.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇⋊SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×Q₈×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₅×AGL(1,𝔽₇)	—	4	1	no	maximal
4	D₇⋊SL(2,𝔽₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
6	C₄×C₅×D₇	—	3	1	no	—
7	C₅×Dic₁₄	—	3	1	no	—
8	C₂×C₅×SL(2,𝔽₃)	—	7	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₈)	—	4	1	no	—
10	C₅×AGL(1,𝔽₇)	—	4	2	no	—
11	C₇⋊SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic, complemented
12	C₂×C₅×D₇	A₄	1	1	yes	characteristic
13	C₄×C₅×C₇	—	3	1	no	—
14	C₅×Dic₇	—	3	1	no	—
15	C₅×SL(2,𝔽₃)	—	7	1	no	—
16	Q₈×D₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
17	C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
18	C₂×AGL(1,𝔽₇)	—	4	1	no	—
19	C₂×C₅×Q₈	—	7	1	no	Hall
20	C₂×C₅×C₇	C₂×A₄	1	1	yes	characteristic
21	C₅×D₇	SL(2,𝔽₃)	1	2	yes	complemented
22	C₂²×C₃×C₅	—	28	1	no	—
23	C₇×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
24	C₄×D₇	—	3	1	no	—
25	Dic₁₄	—	3	1	no	—
26	C₂×SL(2,𝔽₃)	—	7	1	no	Hall
27	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
28	AGL(1,𝔽₇)	—	4	2	no	—
29	C₅×Q₈	—	21	1	no	—
30	C₅×Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
31	C₂×C₄×C₅	—	21	1	no	—
32	C₅×C₇	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₅	—	28	2	no	—
34	C₂×C₃×C₅	—	28	1	no	—
35	C₂×D₇	C₅×A₄	1	1	yes	characteristic
36	C₄×C₇	—	3	1	no	—
37	Dic₇	—	3	1	no	—
38	SL(2,𝔽₃)	—	7	1	no	—
39	ΓL(1,𝔽₈)	—	4	1	no	Hall
40	C₂²×C₅	—	7	1	no	—
41	C₄×C₅	—	3	1	no	—
42	C₄×C₅	—	21	1	no	—
43	C₂×Q₈	—	7	1	no	Sylow
44	C₃×C₅	—	28	1	no	Hall
45	C₂×C₇	C₂×C₅×A₄	1	1	yes	characteristic
46	D₇	C₅×SL(2,𝔽₃)	1	2	yes	complemented
47	C₂²×C₃	—	28	1	no	—
48	C₂×C₅	—	7	2	no	—
49	C₂×C₅	D₇⋊A₄	1	1	yes	characteristic
50	Q₈	—	21	1	no	—
51	Q₈	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
52	C₂×C₄	—	21	1	no	—
53	C₇	C₂×C₅×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃	—	28	2	no	—
55	C₂×C₃	—	28	1	no	—
56	C₅	D₇⋊SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²	—	7	1	no	—
58	C₄	—	3	1	no	—
59	C₄	—	21	1	no	—
60	C₃	—	28	1	no	Sylow
61	C₂	—	7	2	no	—
62	C₂	C₅×D₇⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₇⋊A₄
commutator, cosocle, hypocenter	C₇×Q₈	C₂×C₃×C₅
Frattini	C₂	C₅×D₇⋊A₄
Fitting	C₅×C₇×Q₈	C₂×C₃
socle	C₂×C₅×C₇	C₂×A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₇⋊SL(2,𝔽₃)	C₂	semidirect
2. C₅×Q₈×D₇	C₃	semidirect
4. D₇⋊SL(2,𝔽₃)	C₅	direct (N × H)
5. C₅×C₇×Q₈	C₂×C₃	semidirect
11. C₇⋊SL(2,𝔽₃)	C₂×C₅	semidirect
16. Q₈×D₇	C₃×C₅	semidirect
21. C₅×D₇	SL(2,𝔽₃)	semidirect
23. C₇×Q₈	C₂×C₃×C₅	semidirect
30. C₅×Q₈	AGL(1,𝔽₇)	semidirect
32. C₅×C₇	C₂×SL(2,𝔽₃)	semidirect
46. D₇	C₅×SL(2,𝔽₃)	semidirect
51. Q₈	C₅×AGL(1,𝔽₇)	semidirect
53. C₇	C₂×C₅×SL(2,𝔽₃)	semidirect
56. C₅	D₇⋊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
C₅×C₇⋊SL(2,𝔽₃)	2
C₅×Q₈×D₇	3
C₂×C₅×AGL(1,𝔽₇)	4
D₇⋊SL(2,𝔽₃)	5
C₂×C₅×SL(2,𝔽₃)	7

Subgroups — C5×D7⋊SL(2,𝔽3) #13132589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₇⋊SL(2,𝔽₃) #13132589