Subgroups — C₂×C₉×AGL(1,𝔽₇) #10625439

Order 756.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×AGL(1,𝔽₇)	C₃	1	1	yes	maximal, characteristic
4	C₂×C₉×D₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇⋊C₉	C₃	1	2	yes	maximal, complemented
6	C₉×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic
8	C₃×AGL(1,𝔽₇)	C₂×C₃	1	2	yes	—
9	C₂×C₇×C₉	C₂×C₃	1	1	yes	characteristic, complemented
10	C₉×D₇	C₂×C₃	1	2	yes	complemented
11	C₂×C₇⋊C₉	C₂×C₃	1	2	yes	complemented
12	D₇⋊C₉	C₂×C₃	1	4	yes	complemented
13	C₂²×C₃×C₉	—	7	1	no	maximal, Hall
14	C₂×C₃×D₇	C₃²	1	1	yes	characteristic
15	C₂×AGL(1,𝔽₇)	C₉	1	3	yes	direct product factor
16	C₃×ΓL(1,𝔽₈)	C₂²×C₃	1	1	yes	characteristic
17	C₇×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₇⋊C₉	C₂²×C₃	1	2	yes	complemented
19	C₂×C₃×C₉	—	7	2	no	—
20	C₂×C₃×C₉	—	7	1	no	—
21	C₂×C₃×C₇	C₂×C₃²	1	1	yes	characteristic
22	C₃×D₇	C₂×C₃²	1	2	yes	—
23	C₂×ΓL(1,𝔽₈)	C₂×C₉	1	3	yes	complemented
24	AGL(1,𝔽₇)	C₂×C₉	1	6	yes	direct product factor
25	C₂²×C₃²	—	7	1	no	—
26	C₂²×C₉	—	7	2	no	—
27	C₂²×C₉	—	7	1	no	—
28	C₂×D₇	C₃×C₉	1	1	yes	characteristic, complemented, Hall
29	C₃×C₉	—	7	1	no	Sylow
30	C₃×C₇	C₂²×C₃²	1	1	yes	characteristic
31	ΓL(1,𝔽₈)	C₂²×C₉	1	3	yes	complemented
32	C₂×C₃²	—	7	2	no	—
33	C₂×C₃²	—	7	1	no	—
34	C₂×C₉	—	7	4	no	—
35	C₂×C₉	—	7	2	no	—
36	C₂×C₉	—	7	2	no	—
37	C₂×C₉	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
38	C₂×C₇	C₂×C₃×C₉	1	1	yes	characteristic, complemented
39	D₇	C₂×C₃×C₉	1	2	yes	complemented
40	C₂²×C₃	—	7	3	no	—
41	C₂²×C₃	—	7	1	no	—
42	C₃²	—	7	1	no	—
43	C₉	—	7	2	no	—
44	C₉	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
45	C₇	C₂²×C₃×C₉	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃	—	7	6	no	—
47	C₂×C₃	—	7	2	no	—
48	C₂×C₃	—	7	3	no	—
49	C₂×C₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
50	C₂²	—	7	1	no	Sylow
51	C₃	—	7	3	no	—
52	C₃	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
53	C₂	—	7	2	no	—
54	C₂	C₉×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₉	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂²×C₃×C₉
Frattini	C₃	C₂×C₃×AGL(1,𝔽₇)
Fitting	C₂×C₇×C₉	C₂×C₃
socle	C₂×C₃×C₇	C₂×C₃²
prime radical	C₂×C₃×AGL(1,𝔽₇)	C₃
cosocle	C₃×C₇	C₂²×C₃²
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₉

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×ΓL(1,𝔽₈)	C₂	semidirect
2. C₉×AGL(1,𝔽₇)	C₂	direct (N × H)
4. C₂×C₉×D₇	C₃	semidirect
5. C₂×D₇⋊C₉	C₃	semidirect
6. C₉×ΓL(1,𝔽₈)	C₂²	semidirect
9. C₂×C₇×C₉	C₂×C₃	semidirect
10. C₉×D₇	C₂×C₃	semidirect
11. C₂×C₇⋊C₉	C₂×C₃	semidirect
12. D₇⋊C₉	C₂×C₃	semidirect
15. C₂×AGL(1,𝔽₇)	C₉	direct (N × H)
17. C₇×C₉	C₂²×C₃	semidirect
18. C₇⋊C₉	C₂²×C₃	semidirect
23. C₂×ΓL(1,𝔽₈)	C₂×C₉	semidirect
24. AGL(1,𝔽₇)	C₂×C₉	direct (N × H)
28. C₂×D₇	C₃×C₉	semidirect
31. ΓL(1,𝔽₈)	C₂²×C₉	semidirect
37. C₂×C₉	AGL(1,𝔽₇)	direct (N × H)
38. C₂×C₇	C₂×C₃×C₉	semidirect
39. D₇	C₂×C₃×C₉	semidirect
44. C₉	C₂×AGL(1,𝔽₇)	direct (N × H)
45. C₇	C₂²×C₃×C₉	semidirect
54. C₂	C₉×AGL(1,𝔽₇)	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×C₉×ΓL(1,𝔽₈)	2
C₉×AGL(1,𝔽₇)	2
C₂×C₃×AGL(1,𝔽₇)	3
C₂×C₉×D₇	3
C₂×D₇⋊C₉	3
C₂²×C₃×C₉	7

Subgroups — C2×C9×AGL(1,𝔽7) #10625439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×AGL(1,𝔽₇) #10625439