Subgroups — C₅×#66662 #13132585

Order 1680.

60 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×#1161	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₃×C₅×Dic₇	—	4	1	no	maximal
4	#66662	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
6	C₅×D₂₈	—	3	1	no	—
7	C₄×C₅×D₇	—	3	1	no	—
8	C₅×#41⋊C₃	—	7	1	no	maximal, Hall
9	C₂×C₃×C₅×C₇	—	4	1	no	—
10	C₇×SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic, complemented
11	C₂×C₅×D₇	—	3	1	no	—
12	C₄×C₅×C₇	—	3	1	no	—
13	C₅×Dic₇	A₄	1	1	yes	characteristic
14	C₅×SL(2,𝔽₃)	D₇	1	1	yes	characteristic, complemented
15	#1161	C₃×C₅	1	1	yes	characteristic, complemented, Hall
16	C₃×C₅×C₇	—	4	1	no	Hall
17	C₃×Dic₇	—	4	1	no	—
18	C₅×#41	—	7	1	no	Hall
19	C₂×C₅×C₇	C₂×A₄	1	1	yes	characteristic
20	C₅×D₇	—	6	1	no	—
21	C₃×C₄×C₅	—	28	1	no	—
22	C₇×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
23	D₂₈	—	3	1	no	—
24	C₄×D₇	—	3	1	no	—
25	#41⋊C₃	—	7	1	no	Hall
26	C₂×C₃×C₇	—	4	1	no	—
27	C₅×Q₈	C₃×D₇	1	1	yes	characteristic
28	C₅×D₄	—	21	1	no	—
29	C₂×C₄×C₅	—	21	1	no	—
30	C₅×C₇	#41⋊C₃	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃×C₅	—	4	1	no	—
32	C₂×D₇	—	3	1	no	—
33	C₄×C₇	—	3	1	no	—
34	Dic₇	C₅×A₄	1	1	yes	characteristic
35	SL(2,𝔽₃)	C₅×D₇	1	1	yes	characteristic, complemented
36	C₃×C₇	—	4	1	no	Hall
37	C₂²×C₅	—	21	1	no	—
38	C₄×C₅	—	3	1	no	—
39	C₄×C₅	—	7	1	no	—
40	#41	—	7	1	no	Sylow
41	C₃×C₅	—	4	1	no	Hall
42	C₂×C₇	C₂×C₅×A₄	1	1	yes	characteristic
43	D₇	—	6	1	no	—
44	C₃×C₄	—	28	1	no	—
45	C₂×C₅	—	42	1	no	—
46	C₂×C₅	A₄×D₇	1	1	yes	characteristic
47	Q₈	C₃×C₅×D₇	1	1	yes	characteristic
48	D₄	—	21	1	no	—
49	C₂×C₄	—	21	1	no	—
50	C₇	C₅×#41⋊C₃	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃	—	4	1	no	—
52	C₅	#66662	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	21	1	no	—
54	C₄	—	3	1	no	—
55	C₄	—	7	1	no	—
56	C₃	—	4	1	no	Sylow
57	C₂	—	42	1	no	—
58	C₂	C₅×A₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	A₄×D₇
commutator, cosocle, hypocenter	C₇×Q₈	C₂×C₃×C₅
Frattini	C₂	C₅×A₄×D₇
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×#1161	C₃	semidirect
4. #66662	C₅	direct (N × H)
10. C₇×SL(2,𝔽₃)	C₂×C₅	semidirect
14. C₅×SL(2,𝔽₃)	D₇	semidirect
15. #1161	C₃×C₅	semidirect
30. C₅×C₇	#41⋊C₃	semidirect
35. SL(2,𝔽₃)	C₅×D₇	semidirect
50. C₇	C₅×#41⋊C₃	semidirect
52. C₅	#66662	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×#1161	3
C₃×C₅×Dic₇	4
#66662	5
C₅×#41⋊C₃	7

Subgroups — C5×#66662 #13132585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66662 #13132585