Subgroups — C₄²×AGL(1,𝔽₇) #10620707

Order 672.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×AGL(1,𝔽₇)	C₂	1	3	yes	maximal
2	C₄²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₄²×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
6	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
7	C₂×ΓL(1,𝔽₈)⋊C₄	C₂²	1	3	yes	—
8	C₄×AGL(1,𝔽₇)	C₄	1	12	yes	direct product factor
9	C₂×C₄×D₇	C₂×C₃	1	3	yes	—
10	C₄²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₄×Dic₇	C₂×C₃	1	3	yes	complemented
12	C₂×C₃×C₄²	—	7	1	no	maximal, Hall
13	C₂²×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
14	C₂×AGL(1,𝔽₇)	C₂×C₄	1	6	yes	—
15	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	6	yes	complemented
16	ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	6	yes	complemented
17	C₂²×D₇	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂²×C₃	1	3	yes	—
19	C₂×Dic₇	C₂²×C₃	1	3	yes	—
20	C₄×D₇	C₃×C₄	1	12	yes	complemented
21	C₂²×C₃×C₄	—	7	3	no	—
22	C₃×C₄²	—	7	3	no	—
23	C₃×C₄²	—	7	1	no	—
24	C₂×ΓL(1,𝔽₈)	C₂²×C₄	1	3	yes	—
25	AGL(1,𝔽₇)	C₄²	1	4	yes	direct product factor
26	C₂×C₄²	—	7	1	no	Sylow
27	C₂²×C₇	C₂³×C₃	1	1	yes	characteristic
28	C₂×D₇	C₂×C₃×C₄	1	6	yes	—
29	C₄×C₇	C₂×C₃×C₄	1	6	yes	complemented
30	Dic₇	C₂×C₃×C₄	1	6	yes	complemented
31	C₂³×C₃	—	7	1	no	—
32	C₂×C₃×C₄	—	7	12	no	—
33	C₂×C₃×C₄	—	7	3	no	—
34	C₂×C₃×C₄	—	7	3	no	—
35	ΓL(1,𝔽₈)	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₄	—	7	3	no	—
37	C₄²	—	7	3	no	—
38	C₄²	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
39	C₂×C₇	C₂²×C₃×C₄	1	3	yes	—
40	D₇	C₃×C₄²	1	4	yes	complemented
41	C₂²×C₃	—	7	6	no	—
42	C₂²×C₃	—	7	1	no	—
43	C₃×C₄	—	7	6	no	—
44	C₃×C₄	—	7	6	no	—
45	C₂³	—	7	1	no	—
46	C₂×C₄	—	7	3	no	—
47	C₂×C₄	—	7	12	no	—
48	C₂×C₄	C₂×AGL(1,𝔽₇)	1	3	yes	—
49	C₇	C₂×C₃×C₄²	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	7	4	no	—
51	C₂×C₃	—	7	3	no	—
52	C₂²	—	7	6	no	—
53	C₂²	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
54	C₄	—	7	6	no	—
55	C₄	C₄×AGL(1,𝔽₇)	1	6	yes	direct product factor
56	C₃	—	7	1	no	Sylow
57	C₂	—	7	4	no	—
58	C₂	C₂×C₄×AGL(1,𝔽₇)	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₃×C₄²
Frattini	C₂²	C₂²×AGL(1,𝔽₇)
Fitting	C₄²×C₇	C₂×C₃
socle, cosocle	C₂²×C₇	C₂³×C₃
prime radical	C₂²×AGL(1,𝔽₇)	C₂²
LCS	G ▷ C₇
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×ΓL(1,𝔽₈)	C₂	semidirect
3. C₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
4. C₄²×D₇	C₃	semidirect
8. C₄×AGL(1,𝔽₇)	C₄	direct (N × H)
10. C₄²×C₇	C₂×C₃	semidirect
11. C₄×Dic₇	C₂×C₃	semidirect
15. C₄×ΓL(1,𝔽₈)	C₂×C₄	semidirect
16. ΓL(1,𝔽₈)⋊C₄	C₂×C₄	semidirect
20. C₄×D₇	C₃×C₄	semidirect
25. AGL(1,𝔽₇)	C₄²	direct (N × H)
29. C₄×C₇	C₂×C₃×C₄	semidirect
30. Dic₇	C₂×C₃×C₄	semidirect
35. ΓL(1,𝔽₈)	C₂×C₄²	semidirect
38. C₄²	AGL(1,𝔽₇)	direct (N × H)
40. D₇	C₃×C₄²	semidirect
49. C₇	C₂×C₃×C₄²	semidirect
55. C₄	C₄×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×C₄×AGL(1,𝔽₇)	2
C₄²×ΓL(1,𝔽₈)	2
C₄×ΓL(1,𝔽₈)⋊C₄	2
C₄²×D₇	3
C₂×C₃×C₄²	7

Subgroups — C42×AGL(1,𝔽7) #10620707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×AGL(1,𝔽₇) #10620707