Subgroups — C₂×C₅×Q₈×ΓL(1,𝔽₈) #13132822

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₅	C₂²×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₇	C₂³×C₃×C₅
Frattini	C₂	C₂³×C₅×ΓL(1,𝔽₈)
Fitting	C₂×C₅×C₇×Q₈	C₃
socle	C₂²×C₅×C₇	C₂²×C₃
prime radical	C₂²×C₅×ΓL(1,𝔽₈)	C₂²
hypercenter	C₂×C₅×Q₈	ΓL(1,𝔽₈)
hypocenter	C₇	C₂×C₃×C₅×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×ΓL(1,𝔽₈)	C₂	direct (N × H)
3. C₂×C₅×C₇×Q₈	C₃	semidirect
6. C₂×Q₈×ΓL(1,𝔽₈)	C₅	direct (N × H)
7. C₅×C₇×Q₈	C₂×C₃	semidirect
10. C₂×C₅×ΓL(1,𝔽₈)	Q₈	direct (N × H)
12. Q₈×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
18. C₂×C₇×Q₈	C₃×C₅	semidirect
19. C₅×ΓL(1,𝔽₈)	C₂×Q₈	direct (N × H)
22. C₂×C₅×Q₈	ΓL(1,𝔽₈)	direct (N × H)
23. C₂×C₅×C₇	C₃×Q₈	semidirect
27. C₇×Q₈	C₂×C₃×C₅	semidirect
30. C₂×ΓL(1,𝔽₈)	C₅×Q₈	direct (N × H)
32. C₅×Q₈	C₂×ΓL(1,𝔽₈)	direct (N × H)
34. C₅×C₇	C₂×C₃×Q₈	semidirect
41. ΓL(1,𝔽₈)	C₂×C₅×Q₈	direct (N × H)
44. C₂×Q₈	C₅×ΓL(1,𝔽₈)	direct (N × H)
46. C₂×C₇	C₃×C₅×Q₈	semidirect
50. C₂×C₅	Q₈×ΓL(1,𝔽₈)	direct (N × H)
52. Q₈	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
54. C₇	C₂×C₃×C₅×Q₈	semidirect
57. C₅	C₂×Q₈×ΓL(1,𝔽₈)	direct (N × H)
61. C₂	C₅×Q₈×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
C₂×C₄×C₅×ΓL(1,𝔽₈)	2
C₂×C₅×C₇×Q₈	3
C₂×Q₈×ΓL(1,𝔽₈)	5
C₂×C₃×C₅×Q₈	7

Subgroups — C2×C5×Q8×ΓL(1,𝔽8) #13132822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×Q₈×ΓL(1,𝔽₈) #13132822