Subgroups — C₃×ΓL(1,𝔽₈)⋊D₂₄₇ #44527862

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)⋊D₂₄₇	C₃	1	3	yes	maximal, direct product factor
3	C₃×D₁₇₂₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
5	C₃×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²×D₂₄₇	—	7	1	no	maximal, Hall
7	D₁₇₂₉	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	maximal, Hall
9	C₃²×C₁₃×C₁₉	—	7	1	no	Hall
10	C₇×C₁₃×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×ΓL(1,𝔽₈)⋊D₁₃	—	19	1	no	maximal, Hall
12	C₃×D₂₄₇	—	7	3	no	—
13	C₃×D₂₄₇	—	7	1	no	—
14	C₃×C₁₉×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented, Hall
15	C₃×C₁₃×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented, Hall
16	C₃×D₁₃₃	—	13	1	no	—
17	ΓL(1,𝔽₈)⋊D₁₉	—	13	3	no	—
18	C₃×C₁₃×C₁₉	—	7	3	no	—
19	C₃×C₁₃×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
20	C₃×D₉₁	—	19	1	no	—
21	ΓL(1,𝔽₈)⋊D₁₃	—	19	3	no	—
22	D₂₄₇	—	7	1	no	Hall
23	C₃×C₇×C₁₉	C₃×D₁₃	1	1	yes	characteristic, complemented
24	C₁₉×ΓL(1,𝔽₈)	C₃×D₁₃	1	3	yes	complemented
25	C₃²×D₁₉	—	91	1	no	Hall
26	C₃×C₇×C₁₃	C₃×D₁₉	1	1	yes	characteristic, complemented
27	C₁₃×ΓL(1,𝔽₈)	C₃×D₁₉	1	3	yes	complemented
28	D₁₃₃	—	13	1	no	Hall
29	C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
30	C₃²×D₁₃	—	133	1	no	Hall
31	D₉₁	—	19	1	no	Hall
32	C₃²×C₁₉	—	7	1	no	Hall
33	C₇×C₁₉	C₃²×D₁₃	1	1	yes	characteristic, complemented, Hall
34	C₃×AGL(1,𝔽₇)	—	247	1	no	Hall
35	C₃²×C₁₃	—	7	1	no	Hall
36	C₃×D₁₉	—	91	3	no	—
37	C₃×D₁₉	—	91	1	no	—
38	C₇×C₁₃	C₃²×D₁₉	1	1	yes	characteristic, complemented, Hall
39	C₃×D₁₃	—	133	3	no	—
40	C₃×D₁₃	—	133	1	no	—
41	C₃×ΓL(1,𝔽₈)	D₂₄₇	1	1	yes	characteristic, complemented, Hall
42	C₃×C₁₉	—	7	3	no	—
43	C₃×C₁₉	ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic, complemented
44	C₃×D₇	—	247	1	no	—
45	AGL(1,𝔽₇)	—	247	3	no	—
46	C₃×C₁₃	—	7	3	no	—
47	C₃×C₁₃	ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented
48	D₁₉	—	91	1	no	Hall
49	D₁₃	—	133	1	no	Hall
50	C₃×C₇	C₃×D₂₄₇	1	1	yes	characteristic, complemented
51	ΓL(1,𝔽₈)	C₃×D₂₄₇	1	3	yes	complemented
52	C₁₉	C₃×ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃²	—	1729	1	no	Hall
54	D₇	—	247	1	no	Hall
55	C₁₃	C₃×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented, Sylow
56	C₃²	—	7	1	no	Sylow
57	C₇	C₃²×D₂₄₇	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	—	1729	3	no	—
59	C₂×C₃	—	1729	1	no	—
60	C₃	—	7	3	no	—
61	C₃	ΓL(1,𝔽₈)⋊D₂₄₇	1	1	yes	characteristic, direct product factor
62	C₂	—	1729	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)⋊D₂₄₇
commutator, cosocle, hypocenter	C₇×C₁₃×C₁₉	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₃×C₁₉	C₂×C₃
LCS	G ▷ C₇×C₁₃×C₁₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. ΓL(1,𝔽₈)⋊D₂₄₇	C₃	direct (N × H)
3. C₃×D₁₇₂₉	C₃	semidirect
4. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
5. C₃×C₇×C₁₃×C₁₉	C₂×C₃	semidirect
7. D₁₇₂₉	C₃²	semidirect
10. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
14. C₃×C₁₉×ΓL(1,𝔽₈)	D₁₃	semidirect
15. C₃×C₁₃×ΓL(1,𝔽₈)	D₁₉	semidirect
19. C₃×C₁₃×C₁₉	AGL(1,𝔽₇)	semidirect
23. C₃×C₇×C₁₉	C₃×D₁₃	semidirect
24. C₁₉×ΓL(1,𝔽₈)	C₃×D₁₃	semidirect
26. C₃×C₇×C₁₃	C₃×D₁₉	semidirect
27. C₁₃×ΓL(1,𝔽₈)	C₃×D₁₉	semidirect
29. C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	semidirect
33. C₇×C₁₉	C₃²×D₁₃	semidirect
38. C₇×C₁₃	C₃²×D₁₉	semidirect
41. C₃×ΓL(1,𝔽₈)	D₂₄₇	semidirect
43. C₃×C₁₉	ΓL(1,𝔽₈)⋊D₁₃	semidirect
47. C₃×C₁₃	ΓL(1,𝔽₈)⋊D₁₉	semidirect
50. C₃×C₇	C₃×D₂₄₇	semidirect
51. ΓL(1,𝔽₈)	C₃×D₂₄₇	semidirect
52. C₁₉	C₃×ΓL(1,𝔽₈)⋊D₁₃	semidirect
55. C₁₃	C₃×ΓL(1,𝔽₈)⋊D₁₉	semidirect
57. C₇	C₃²×D₂₄₇	semidirect
61. C₃	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
C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)⋊D₂₄₇	3
C₃×D₁₇₂₉	3
C₃²×D₂₄₇	7
C₃×ΓL(1,𝔽₈)⋊D₁₉	13
C₃×ΓL(1,𝔽₈)⋊D₁₃	19

Subgroups — C3×ΓL(1,𝔽8)⋊D247 #44527862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)⋊D₂₄₇ #44527862