Subgroups — C₂×D₇⋊A₄ #66749

Order 336.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇⋊A₄	C₂	1	2	yes	maximal, direct product factor
3	C₂³×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
5	C₂×AGL(1,𝔽₇)	—	4	1	no	maximal
6	C₂³×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×D₇	—	3	1	no	—
8	C₂²×D₇	—	3	1	no	—
9	C₂²×D₇	—	3	2	no	—
10	C₂²×D₇	C₂×C₃	1	2	yes	complemented
11	C₂²×A₄	—	7	1	no	maximal, Hall
12	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
13	AGL(1,𝔽₇)	—	4	2	no	—
14	C₂²×C₇	—	3	1	no	—
15	C₂²×C₇	—	3	1	no	—
16	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
17	C₂×D₇	—	3	2	no	—
18	C₂×D₇	—	3	1	no	—
19	C₂×D₇	—	3	2	no	—
20	C₂×D₇	—	3	2	no	—
21	C₂×D₇	—	3	2	no	—
22	C₂×D₇	A₄	1	1	yes	characteristic, complemented
23	C₂×A₄	—	7	2	no	—
24	C₂×A₄	—	7	1	no	—
25	ΓL(1,𝔽₈)	—	4	1	no	Hall
26	C₂⁴	—	7	1	no	Sylow
27	C₂×C₇	—	3	1	no	—
28	C₂×C₇	—	3	1	no	—
29	C₂×C₇	C₂×A₄	1	1	yes	characteristic, complemented
30	D₇	—	3	2	no	—
31	D₇	C₂×A₄	1	2	yes	complemented
32	C₂²×C₃	—	28	1	no	—
33	A₄	—	7	1	no	—
34	C₂³	—	7	2	no	—
35	C₂³	—	21	1	no	—
36	C₂³	—	21	2	no	—
37	C₂³	—	21	1	no	—
38	C₂³	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
39	C₇	C₂²×A₄	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃	—	28	2	no	—
41	C₂×C₃	—	28	1	no	—
42	C₂²	—	3	1	no	—
43	C₂²	—	3	1	no	—
44	C₂²	—	7	1	no	—
45	C₂²	—	21	2	no	—
46	C₂²	—	21	2	no	—
47	C₂²	—	21	2	no	—
48	C₂²	—	21	2	no	—
49	C₂²	—	21	1	no	—
50	C₂²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
51	C₃	—	28	1	no	Sylow
52	C₂	—	3	1	no	—
53	C₂	—	3	1	no	—
54	C₂	—	7	2	no	—
55	C₂	—	21	2	no	—
56	C₂	D₇⋊A₄	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₇⋊A₄
commutator, cosocle, hypocenter	C₂²×C₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊ΓL(1,𝔽₈)	C₂	semidirect
2. D₇⋊A₄	C₂	direct (N × H)
3. C₂³×D₇	C₃	semidirect
4. C₂²⋊ΓL(1,𝔽₈)	C₂²	semidirect
6. C₂³×C₇	C₂×C₃	semidirect
10. C₂²×D₇	C₂×C₃	semidirect
16. C₂²×C₇	C₂²×C₃	semidirect
22. C₂×D₇	A₄	semidirect
29. C₂×C₇	C₂×A₄	semidirect
31. D₇	C₂×A₄	semidirect
38. C₂³	AGL(1,𝔽₇)	semidirect
39. C₇	C₂²×A₄	semidirect
50. C₂²	C₂×AGL(1,𝔽₇)	semidirect
56. C₂	D₇⋊A₄	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×C₂²⋊ΓL(1,𝔽₈)	2
D₇⋊A₄	2
C₂³×D₇	3
C₂×AGL(1,𝔽₇)	4
C₂²×A₄	7

Subgroups — C2×D7⋊A4 #66749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇⋊A₄ #66749