Subgroups — C₂×ΓL(1,𝔽₈)×D₂₄₇ #35209111

Order 20748.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₂₄₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₂₄₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₂₄₇	C₂×C₃	1	2	yes	complemented
6	C₂×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×D₂₄₇	—	7	1	no	maximal, Hall
8	C₇×C₁₃×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×ΓL(1,𝔽₈)×D₁₉	—	13	1	no	maximal, Hall
10	C₂×C₃×C₁₃×C₁₉	—	7	1	no	—
11	C₃×D₂₄₇	—	7	2	no	—
12	C₂×ΓL(1,𝔽₈)×D₁₃	—	19	1	no	maximal, Hall
13	C₂×D₂₄₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₁₉×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented
15	ΓL(1,𝔽₈)×D₁₉	—	13	2	no	—
16	C₃×C₁₃×C₁₉	—	7	1	no	Hall
17	C₂×C₁₃×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×D₁₃	—	19	2	no	—
19	C₂×C₇×D₁₉	—	13	1	no	Hall
20	C₂×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	D₂₄₇	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
22	C₁₉×ΓL(1,𝔽₈)	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
23	C₂×C₇×D₁₃	—	19	1	no	Hall
24	C₁₃×ΓL(1,𝔽₈)	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
25	C₂×C₇×C₁₉	C₃×D₁₃	1	1	yes	characteristic, complemented
26	C₇×D₁₉	—	13	2	no	—
27	C₁₃×C₁₉	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃×D₁₉	—	91	1	no	Hall
29	C₂×C₇×C₁₃	C₃×D₁₉	1	1	yes	characteristic, complemented
30	C₇×D₁₃	—	19	2	no	—
31	C₂×C₃×D₁₃	—	133	1	no	Hall
32	C₇×C₁₉	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₁₉	—	7	1	no	—
34	C₃×D₁₉	—	91	2	no	—
35	C₇×C₁₃	C₂×C₃×D₁₉	1	1	yes	characteristic, complemented, Hall
36	C₂²×ΓL(1,𝔽₈)	—	247	1	no	Hall
37	C₂×C₃×C₁₃	—	7	1	no	—
38	C₃×D₁₃	—	133	2	no	—
39	C₂×D₁₉	—	13	1	no	Hall
40	C₃×C₁₉	—	7	1	no	Hall
41	C₂×D₁₃	—	19	1	no	Hall
42	C₂×ΓL(1,𝔽₈)	—	247	2	no	—
43	C₂×ΓL(1,𝔽₈)	D₂₄₇	1	1	yes	characteristic, direct product factor
44	C₃×C₁₃	—	7	1	no	Hall
45	C₂×C₁₉	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
46	D₁₉	—	13	2	no	—
47	C₂²×C₇	—	247	1	no	Hall
48	C₂×C₁₃	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented
49	D₁₃	—	19	2	no	—
50	ΓL(1,𝔽₈)	C₂×D₂₄₇	1	1	yes	characteristic, direct product factor, Hall
51	C₁₉	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₇	—	247	2	no	—
53	C₂×C₇	C₃×D₂₄₇	1	1	yes	characteristic, complemented
54	C₁₃	C₂×ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₃	—	1729	1	no	Hall
56	C₇	C₂×C₃×D₂₄₇	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	7	1	no	—
58	C₂×C₃	—	1729	2	no	—
59	C₂²	—	247	1	no	Sylow
60	C₃	—	7	1	no	Sylow
61	C₂	—	247	2	no	—
62	C₂	ΓL(1,𝔽₈)×D₂₄₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×D₂₄₇
commutator, cosocle, hypocenter	C₇×C₁₃×C₁₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₃×C₁₉	C₂×C₃
LCS	G ▷ C₇×C₁₃×C₁₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₂₄₇	C₂	direct (N × H)
2. C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×C₇×D₂₄₇	C₃	semidirect
4. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₇×D₂₄₇	C₂×C₃	semidirect
6. C₂×C₇×C₁₃×C₁₉	C₂×C₃	semidirect
8. C₇×C₁₃×C₁₉	C₂²×C₃	semidirect
13. C₂×D₂₄₇	ΓL(1,𝔽₈)	direct (N × H)
14. C₂×C₁₉×ΓL(1,𝔽₈)	D₁₃	semidirect
17. C₂×C₁₃×ΓL(1,𝔽₈)	D₁₉	semidirect
20. C₂×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
21. D₂₄₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
22. C₁₉×ΓL(1,𝔽₈)	C₂×D₁₃	semidirect
24. C₁₃×ΓL(1,𝔽₈)	C₂×D₁₉	semidirect
25. C₂×C₇×C₁₉	C₃×D₁₃	semidirect
27. C₁₃×C₁₉	C₂²×ΓL(1,𝔽₈)	semidirect
29. C₂×C₇×C₁₃	C₃×D₁₉	semidirect
32. C₇×C₁₉	C₂×C₃×D₁₃	semidirect
35. C₇×C₁₃	C₂×C₃×D₁₉	semidirect
43. C₂×ΓL(1,𝔽₈)	D₂₄₇	direct (N × H)
45. C₂×C₁₉	ΓL(1,𝔽₈)×D₁₃	semidirect
48. C₂×C₁₃	ΓL(1,𝔽₈)×D₁₉	semidirect
50. ΓL(1,𝔽₈)	C₂×D₂₄₇	direct (N × H)
51. C₁₉	C₂×ΓL(1,𝔽₈)×D₁₃	semidirect
53. C₂×C₇	C₃×D₂₄₇	semidirect
54. C₁₃	C₂×ΓL(1,𝔽₈)×D₁₉	semidirect
56. C₇	C₂×C₃×D₂₄₇	semidirect
62. C₂	ΓL(1,𝔽₈)×D₂₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₂₄₇	2
C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	2
C₂×C₇×D₂₄₇	3
C₂×C₃×D₂₄₇	7
C₂×ΓL(1,𝔽₈)×D₁₉	13
C₂×ΓL(1,𝔽₈)×D₁₃	19

Subgroups — C2×ΓL(1,𝔽8)×D247 #35209111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×D₂₄₇ #35209111