Subgroups — C₅×#41×ΓL(1,𝔽₈) #13132823

Order 1680.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₄×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
3	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
4	C₅×C₇×#41	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
6	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
8	#41×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×C₇×D₄	C₂×C₃	1	3	yes	complemented
11	C₂×C₄×C₅×C₇	C₂×C₃	1	3	yes	complemented
12	C₃×C₅×#41	—	7	1	no	maximal, Hall
13	C₂×C₅×ΓL(1,𝔽₈)	—	2	3	no	—
14	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
15	Q₈×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented
16	D₄×ΓL(1,𝔽₈)	C₂×C₅	1	3	yes	complemented
17	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	1	3	yes	complemented
18	C₂²×C₅×C₇	C₂²×C₃	1	3	yes	—
19	C₄×C₅×C₇	C₂²×C₃	1	3	yes	—
20	C₄×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
21	C₃×C₅×Q₈	—	7	1	no	—
22	C₃×C₅×D₄	—	7	3	no	—
23	C₂×C₃×C₄×C₅	—	7	3	no	—
24	C₇×#41	C₃×C₅	1	1	yes	characteristic, complemented, Hall
25	C₅×ΓL(1,𝔽₈)	#41	1	1	yes	characteristic, direct product factor, Hall
26	C₂²×ΓL(1,𝔽₈)	C₂²×C₅	1	3	yes	—
27	C₄×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
28	C₄×ΓL(1,𝔽₈)	C₂²×C₅	1	3	yes	—
29	C₅×#41	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₅×C₇	—	2	3	no	—
31	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₃×C₅	—	7	3	no	—
33	C₃×C₄×C₅	—	7	3	no	—
34	C₃×C₄×C₅	—	7	1	no	—
35	C₇×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
36	C₇×D₄	C₂×C₃×C₅	1	3	yes	complemented
37	C₂×C₄×C₇	C₂×C₃×C₅	1	3	yes	complemented
38	C₃×#41	—	7	1	no	Hall
39	C₂×ΓL(1,𝔽₈)	—	2	3	no	—
40	C₂×ΓL(1,𝔽₈)	C₂³×C₅	1	1	yes	characteristic
41	C₅×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
42	C₅×D₄	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
43	C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
44	C₅×C₇	C₃×#41	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	—	7	1	no	—
46	C₂×C₃×C₅	—	14	3	no	—
47	C₂²×C₇	C₂²×C₃×C₅	1	3	yes	—
48	C₄×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
49	C₄×C₇	C₂²×C₃×C₅	1	3	yes	—
50	C₃×Q₈	—	7	1	no	—
51	C₃×D₄	—	7	3	no	—
52	C₂×C₃×C₄	—	7	3	no	—
53	ΓL(1,𝔽₈)	C₅×#41	1	1	yes	characteristic, direct product factor, Hall
54	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
55	C₄×C₅	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
56	C₄×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
57	#41	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
58	C₃×C₅	—	7	1	no	Hall
59	C₂×C₇	—	2	3	no	—
60	C₂×C₇	C₂³×C₃×C₅	1	1	yes	characteristic
61	C₂²×C₃	—	7	3	no	—
62	C₃×C₄	—	7	1	no	—
63	C₃×C₄	—	7	3	no	—
64	C₂×C₅	—	2	3	no	—
65	C₂×C₅	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
66	Q₈	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
67	D₄	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	complemented
68	C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	complemented
69	C₇	C₃×C₅×#41	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	1	no	—
71	C₂×C₃	—	14	3	no	—
72	C₅	#41×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	C₂²×C₅×ΓL(1,𝔽₈)	1	3	yes	—
74	C₄	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
75	C₄	C₂²×C₅×ΓL(1,𝔽₈)	1	3	yes	—
76	C₃	—	7	1	no	Sylow
77	C₂	—	2	3	no	—
78	C₂	C₂³×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₇	C₂³×C₃×C₅
Frattini	C₂	C₂³×C₅×ΓL(1,𝔽₈)
Fitting	C₅×C₇×#41	C₃
socle	C₂×C₅×C₇	C₂³×C₃
hypercenter	C₅×#41	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×C₅×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
2. C₅×D₄×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	semidirect
4. C₅×C₇×#41	C₃	semidirect
8. #41×ΓL(1,𝔽₈)	C₅	direct (N × H)
9. C₅×C₇×Q₈	C₂×C₃	semidirect
10. C₅×C₇×D₄	C₂×C₃	semidirect
11. C₂×C₄×C₅×C₇	C₂×C₃	semidirect
15. Q₈×ΓL(1,𝔽₈)	C₂×C₅	semidirect
16. D₄×ΓL(1,𝔽₈)	C₂×C₅	semidirect
17. C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	semidirect
24. C₇×#41	C₃×C₅	semidirect
25. C₅×ΓL(1,𝔽₈)	#41	direct (N × H)
29. C₅×#41	ΓL(1,𝔽₈)	direct (N × H)
35. C₇×Q₈	C₂×C₃×C₅	semidirect
36. C₇×D₄	C₂×C₃×C₅	semidirect
37. C₂×C₄×C₇	C₂×C₃×C₅	semidirect
41. C₅×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
42. C₅×D₄	C₂×ΓL(1,𝔽₈)	semidirect
43. C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	semidirect
44. C₅×C₇	C₃×#41	semidirect
53. ΓL(1,𝔽₈)	C₅×#41	direct (N × H)
57. #41	C₅×ΓL(1,𝔽₈)	direct (N × H)
66. Q₈	C₂×C₅×ΓL(1,𝔽₈)	semidirect
67. D₄	C₂×C₅×ΓL(1,𝔽₈)	semidirect
68. C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	semidirect
69. C₇	C₃×C₅×#41	semidirect
72. C₅	#41×ΓL(1,𝔽₈)	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×Q₈×ΓL(1,𝔽₈)	2
C₅×D₄×ΓL(1,𝔽₈)	2
C₂×C₄×C₅×ΓL(1,𝔽₈)	2
C₅×C₇×#41	3
#41×ΓL(1,𝔽₈)	5
C₃×C₅×#41	7

Subgroups — C5×#41×ΓL(1,𝔽8) #13132823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#41×ΓL(1,𝔽₈) #13132823