Subgroups — (C₂×AGL(1,𝔽₇))⋊C₄ #66548

Order 336.

68 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	(C₂×D₇)⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₂×AGL(1,𝔽₇)	—	2	1	no	—
7	C₂×AGL(1,𝔽₇)	—	2	1	no	—
8	C₂×AGL(1,𝔽₇)	C₄	1	2	yes	complemented
9	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
10	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
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	C₃×C₂²⋊C₄	—	7	1	no	maximal, Hall
15	C₂×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
16	C₂×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
17	C₂×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
18	AGL(1,𝔽₇)	—	2	2	no	—
19	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
20	C₂×D₇	—	2	1	no	—
21	C₂×D₇	—	2	1	no	—
22	C₂×D₇	C₃×C₄	1	2	yes	complemented
23	C₄×C₇	—	2	1	no	—
24	Dic₇	—	2	1	no	—
25	C₂³×C₃	—	7	1	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	1	yes	characteristic
31	C₂×C₇	C₃×D₄	1	1	yes	characteristic
32	C₂×C₇	C₃×D₄	1	1	yes	characteristic
33	D₇	—	2	2	no	—
34	C₂²×C₃	—	7	2	no	—
35	C₂²×C₃	—	7	1	no	—
36	C₂²×C₃	—	14	1	no	—
37	C₂²×C₃	—	14	1	no	—
38	C₃×C₄	—	14	1	no	—
39	C₃×C₄	—	14	1	no	—
40	C₂³	—	7	1	no	—
41	C₂×C₄	—	7	1	no	—
42	C₂×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
43	C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃	—	7	1	no	—
45	C₂×C₃	—	7	1	no	—
46	C₂×C₃	—	7	1	no	—
47	C₂×C₃	—	14	2	no	—
48	C₂²	—	7	2	no	—
49	C₂²	—	14	1	no	—
50	C₂²	—	14	1	no	—
51	C₂²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	14	1	no	—
54	C₃	—	7	1	no	Sylow
55	C₂	—	14	2	no	—
56	C₂	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
57	C₂	C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic
58	C₂	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂×C₃×C₄
Fitting	C₂×C₄×C₇	C₂×C₃
socle, cosocle	C₂²×C₇	C₂²×C₃
prime radical	C₂²×AGL(1,𝔽₇)	C₂
hypercenter	C₂×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ 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. (C₂×D₇)⋊C₄	C₃	semidirect
8. C₂×AGL(1,𝔽₇)	C₄	semidirect
12. C₂×C₄×C₇	C₂×C₃	semidirect
13. C₂×Dic₇	C₂×C₃	semidirect
22. C₂×D₇	C₃×C₄	semidirect
28. ΓL(1,𝔽₈)	C₂²⋊C₄	semidirect
42. C₂×C₄	AGL(1,𝔽₇)	semidirect
43. C₇	C₃×C₂²⋊C₄	semidirect

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
(C₂×D₇)⋊C₄	3
C₃×C₂²⋊C₄	7

Subgroups — (C2×AGL(1,𝔽7))⋊C4 #66548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₂×AGL(1,𝔽₇))⋊C₄ #66548