Subgroups — C₂×C₅×#13049955 #27189198

Order 13020.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#13049955	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅×#10620253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₇×D₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅×#10620253	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×#13049955	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×C₇×D₃₁	C₂×C₃	1	2	yes	complemented
7	C₂×C₅×C₇×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₅×D₃₁⋊C₃	—	7	1	no	maximal, Hall
9	C₂×#10620253	C₂×C₅	1	1	yes	characteristic, complemented
10	#13049955	C₂×C₅	1	2	yes	direct product factor
11	C₅×C₇×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₅×C₃₁⋊C₃	—	7	1	no	—
13	C₅×D₃₁⋊C₃	—	7	2	no	—
14	C₂×C₇×D₃₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	#10620253	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₂×C₅×D₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
17	C₅×C₃₁⋊C₃	—	7	1	no	Hall
18	C₂×C₇×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
19	C₇×D₃₁	C₂×C₃×C₅	1	2	yes	complemented
20	C₂²×C₅×ΓL(1,𝔽₈)	—	31	1	no	maximal, Hall
21	C₂×D₃₁⋊C₃	—	7	1	no	Hall
22	C₂×C₅×C₃₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₅×D₃₁	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₇×C₃₁	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Hall
25	C₂×C₅×ΓL(1,𝔽₈)	—	31	2	no	—
26	C₂×C₅×ΓL(1,𝔽₈)	—	31	1	no	—
27	C₂×C₃₁⋊C₃	—	7	1	no	—
28	D₃₁⋊C₃	—	7	2	no	—
29	C₅×C₃₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₅×C₇	—	31	1	no	Hall
31	C₂×D₃₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
32	C₅×ΓL(1,𝔽₈)	—	31	1	no	Hall
33	C₃₁⋊C₃	—	7	1	no	Hall
34	C₂²×ΓL(1,𝔽₈)	—	31	1	no	Hall
35	C₂×C₅×C₇	—	31	2	no	—
36	C₂×C₅×C₇	D₃₁⋊C₃	1	1	yes	characteristic, complemented
37	C₂×C₃₁	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	D₃₁	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
39	C₂²×C₃×C₅	—	217	1	no	Hall
40	C₂×ΓL(1,𝔽₈)	—	31	2	no	—
41	C₂×ΓL(1,𝔽₈)	—	31	1	no	—
42	C₅×C₇	C₂×D₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
43	C₃₁	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₅	—	217	2	no	—
45	C₂×C₃×C₅	—	217	1	no	—
46	C₂²×C₇	—	31	1	no	Hall
47	ΓL(1,𝔽₈)	—	31	1	no	Hall
48	C₂²×C₅	—	31	1	no	Hall
49	C₃×C₅	—	217	1	no	Hall
50	C₂×C₇	—	31	2	no	—
51	C₂×C₇	C₅×D₃₁⋊C₃	1	1	yes	characteristic, complemented
52	C₂²×C₃	—	217	1	no	Hall
53	C₂×C₅	—	31	2	no	—
54	C₂×C₅	#13049955	1	1	yes	characteristic, direct product factor
55	C₇	C₂×C₅×D₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃	—	217	2	no	—
57	C₂×C₃	—	217	1	no	—
58	C₅	C₂×#13049955	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	31	1	no	Sylow
60	C₃	—	217	1	no	Sylow
61	C₂	—	31	2	no	—
62	C₂	C₅×#13049955	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	#13049955
commutator, cosocle, hypocenter	C₇×C₃₁	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇×C₃₁	C₂×C₃
LCS	G ▷ C₇×C₃₁
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×#13049955	C₂	direct (N × H)
2. C₂×C₅×#10620253	C₂	semidirect
3. C₂×C₅×C₇×D₃₁	C₃	semidirect
4. C₅×#10620253	C₂²	semidirect
5. C₂×#13049955	C₅	direct (N × H)
6. C₅×C₇×D₃₁	C₂×C₃	semidirect
7. C₂×C₅×C₇×C₃₁	C₂×C₃	semidirect
9. C₂×#10620253	C₂×C₅	semidirect
10. #13049955	C₂×C₅	direct (N × H)
11. C₅×C₇×C₃₁	C₂²×C₃	semidirect
14. C₂×C₇×D₃₁	C₃×C₅	semidirect
15. #10620253	C₂²×C₅	semidirect
16. C₂×C₅×D₃₁	ΓL(1,𝔽₈)	semidirect
18. C₂×C₇×C₃₁	C₂×C₃×C₅	semidirect
19. C₇×D₃₁	C₂×C₃×C₅	semidirect
22. C₂×C₅×C₃₁	C₂×ΓL(1,𝔽₈)	semidirect
23. C₅×D₃₁	C₂×ΓL(1,𝔽₈)	semidirect
24. C₇×C₃₁	C₂²×C₃×C₅	semidirect
29. C₅×C₃₁	C₂²×ΓL(1,𝔽₈)	semidirect
31. C₂×D₃₁	C₅×ΓL(1,𝔽₈)	semidirect
36. C₂×C₅×C₇	D₃₁⋊C₃	semidirect
37. C₂×C₃₁	C₂×C₅×ΓL(1,𝔽₈)	semidirect
38. D₃₁	C₂×C₅×ΓL(1,𝔽₈)	semidirect
42. C₅×C₇	C₂×D₃₁⋊C₃	semidirect
43. C₃₁	C₂²×C₅×ΓL(1,𝔽₈)	semidirect
51. C₂×C₇	C₅×D₃₁⋊C₃	semidirect
54. C₂×C₅	#13049955	direct (N × H)
55. C₇	C₂×C₅×D₃₁⋊C₃	semidirect
58. C₅	C₂×#13049955	direct (N × H)
62. C₂	C₅×#13049955	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×#13049955	2
C₂×C₅×#10620253	2
C₂×C₅×C₇×D₃₁	3
C₂×#13049955	5
C₂×C₅×D₃₁⋊C₃	7
C₂²×C₅×ΓL(1,𝔽₈)	31

Subgroups — C2×C5×#13049955 #27189198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#13049955 #27189198