Subgroups — C₂×ΓL(1,𝔽₈)×D₃₀₁ #38760504

Order 25284.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×ΓL(1,𝔽₈)×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×D₃₀₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₃₀₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×ΓL(1,𝔽₈)×C₄₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₃₀₁	C₂×C₃	1	2	yes	complemented
6	C₂×C₇²×C₄₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×D₃₀₁	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)×D₄₃	—	7	1	no	maximal
9	C₇²×C₄₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₇×C₄₃	—	7	1	no	—
11	C₃×D₃₀₁	—	7	2	no	—
12	C₂×ΓL(1,𝔽₈)×C₄₃	D₇	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)×D₄₃	—	7	2	no	—
14	C₂×D₃₀₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₂×C₇×D₄₃	—	7	1	no	—
16	C₃×C₇×C₄₃	—	7	1	no	—
17	ΓL(1,𝔽₈)×C₄₃	C₂×D₇	1	1	yes	characteristic, complemented
18	C₂×C₇×C₄₃	—	6	1	no	—
19	C₂×C₇×C₄₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	C₂×C₇×C₄₃	C₃×D₇	1	1	yes	characteristic, complemented
21	D₃₀₁	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
22	C₇×D₄₃	—	7	2	no	—
23	C₂×D₇×ΓL(1,𝔽₈)	—	43	1	no	maximal, Hall
24	C₂×C₃×D₄₃	—	49	1	no	Hall
25	C₇×C₄₃	—	6	1	no	—
26	C₇×C₄₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	C₇×C₄₃	C₂×C₃×D₇	1	1	yes	characteristic, complemented
28	C₂×C₇×ΓL(1,𝔽₈)	D₄₃	1	1	yes	characteristic, complemented
29	D₇×ΓL(1,𝔽₈)	—	43	2	no	—
30	C₂×C₃×C₄₃	—	7	1	no	—
31	C₃×D₄₃	—	49	2	no	—
32	C₂×C₇×D₇	—	43	1	no	Hall
33	C₂×D₄₃	—	7	1	no	Hall
34	C₇×ΓL(1,𝔽₈)	C₂×D₄₃	1	1	yes	characteristic, complemented, Hall
35	C₃×C₄₃	—	7	1	no	Hall
36	C₂×C₇²	C₃×D₄₃	1	1	yes	characteristic, complemented
37	C₇×D₇	—	43	2	no	—
38	C₂×C₄₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
39	D₄₃	—	7	2	no	—
40	C₂×C₃×D₇	—	301	1	no	—
41	C₂²×ΓL(1,𝔽₈)	—	301	1	no	—
42	C₇²	C₂×C₃×D₄₃	1	1	yes	characteristic, complemented, Sylow
43	C₄₃	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₇	—	7	1	no	—
45	C₃×D₇	—	301	2	no	—
46	C₂×ΓL(1,𝔽₈)	—	301	2	no	—
47	C₂×ΓL(1,𝔽₈)	D₃₀₁	1	1	yes	characteristic, direct product factor
48	C₂²×C₇	—	301	1	no	—
49	C₂×D₇	—	43	1	no	—
50	C₃×C₇	—	7	1	no	—
51	ΓL(1,𝔽₈)	C₂×D₃₀₁	1	1	yes	characteristic, direct product factor
52	C₂×C₇	—	6	1	no	—
53	C₂×C₇	—	301	2	no	—
54	C₂×C₇	ΓL(1,𝔽₈)×D₄₃	1	1	yes	characteristic, complemented
55	C₂×C₇	C₃×D₃₀₁	1	1	yes	characteristic, complemented
56	D₇	—	43	2	no	—
57	C₂²×C₃	—	2107	1	no	Hall
58	C₇	—	6	1	no	—
59	C₇	C₂×ΓL(1,𝔽₈)×D₄₃	1	1	yes	characteristic, complemented
60	C₇	C₂×C₃×D₃₀₁	1	1	yes	characteristic, complemented
61	C₂×C₃	—	7	1	no	—
62	C₂×C₃	—	2107	2	no	—
63	C₂²	—	301	1	no	Sylow
64	C₃	—	7	1	no	Sylow
65	C₂	—	301	2	no	—
66	C₂	ΓL(1,𝔽₈)×D₃₀₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×D₃₀₁
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×ΓL(1,𝔽₈)×C₄₃	C₂	semidirect
2. ΓL(1,𝔽₈)×D₃₀₁	C₂	direct (N × H)
3. C₂×C₇×D₃₀₁	C₃	semidirect
4. C₇×ΓL(1,𝔽₈)×C₄₃	C₂²	semidirect
5. C₇×D₃₀₁	C₂×C₃	semidirect
6. C₂×C₇²×C₄₃	C₂×C₃	semidirect
9. C₇²×C₄₃	C₂²×C₃	semidirect
12. C₂×ΓL(1,𝔽₈)×C₄₃	D₇	semidirect
14. C₂×D₃₀₁	ΓL(1,𝔽₈)	direct (N × H)
17. ΓL(1,𝔽₈)×C₄₃	C₂×D₇	semidirect
19. C₂×C₇×C₄₃	C₂×ΓL(1,𝔽₈)	semidirect
20. C₂×C₇×C₄₃	C₃×D₇	semidirect
21. D₃₀₁	C₂×ΓL(1,𝔽₈)	direct (N × H)
26. C₇×C₄₃	C₂²×ΓL(1,𝔽₈)	semidirect
27. C₇×C₄₃	C₂×C₃×D₇	semidirect
28. C₂×C₇×ΓL(1,𝔽₈)	D₄₃	semidirect
34. C₇×ΓL(1,𝔽₈)	C₂×D₄₃	semidirect
36. C₂×C₇²	C₃×D₄₃	semidirect
38. C₂×C₄₃	D₇×ΓL(1,𝔽₈)	semidirect
42. C₇²	C₂×C₃×D₄₃	semidirect
43. C₄₃	C₂×D₇×ΓL(1,𝔽₈)	semidirect
47. C₂×ΓL(1,𝔽₈)	D₃₀₁	direct (N × H)
51. ΓL(1,𝔽₈)	C₂×D₃₀₁	direct (N × H)
54. C₂×C₇	ΓL(1,𝔽₈)×D₄₃	semidirect
55. C₂×C₇	C₃×D₃₀₁	semidirect
59. C₇	C₂×ΓL(1,𝔽₈)×D₄₃	semidirect
60. C₇	C₂×C₃×D₃₀₁	semidirect
66. C₂	ΓL(1,𝔽₈)×D₃₀₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×ΓL(1,𝔽₈)×C₄₃	2
ΓL(1,𝔽₈)×D₃₀₁	2
C₂×C₇×D₃₀₁	3
C₂×C₃×D₃₀₁	7
C₂×ΓL(1,𝔽₈)×D₄₃	7
C₂×D₇×ΓL(1,𝔽₈)	43

Subgroups — C2×ΓL(1,𝔽8)×D301 #38760504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×D₃₀₁ #38760504