Subgroups — C₂×C₄×AGL(1,𝔽₇) #66653

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂²×C₃×C₄
Frattini	C₂	C₂²×AGL(1,𝔽₇)
Fitting	C₂×C₄×C₇	C₂×C₃
socle	C₂²×C₇	C₂²×C₃
prime radical	C₂²×AGL(1,𝔽₇)	C₂
cosocle	C₂×C₇	C₂³×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
4. C₄×AGL(1,𝔽₇)	C₂	direct (N × H)
5. C₂×C₄×D₇	C₃	semidirect
7. C₂×AGL(1,𝔽₇)	C₄	direct (N × H)
9. C₄×ΓL(1,𝔽₈)	C₂²	semidirect
10. ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
12. C₂×C₄×C₇	C₂×C₃	semidirect
13. C₂×Dic₇	C₂×C₃	semidirect
14. C₄×D₇	C₂×C₃	semidirect
16. C₂×ΓL(1,𝔽₈)	C₂×C₄	semidirect
18. AGL(1,𝔽₇)	C₂×C₄	direct (N × H)
20. C₂×D₇	C₃×C₄	semidirect
22. C₄×C₇	C₂²×C₃	semidirect
23. Dic₇	C₂²×C₃	semidirect
28. ΓL(1,𝔽₈)	C₂²×C₄	semidirect
30. C₂×C₇	C₂×C₃×C₄	semidirect
32. D₇	C₂×C₃×C₄	semidirect
41. C₂×C₄	AGL(1,𝔽₇)	direct (N × H)
42. C₇	C₂²×C₃×C₄	semidirect
50. C₄	C₂×AGL(1,𝔽₇)	direct (N × H)
53. C₂	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
C₂²×AGL(1,𝔽₇)	2
C₂×C₄×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊C₄	2
C₄×AGL(1,𝔽₇)	2
C₂×C₄×D₇	3
C₂²×C₃×C₄	7

Subgroups — C2×C4×AGL(1,𝔽7) #66653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×AGL(1,𝔽₇) #66653