Subgroups — C₂×C₇×S₄ #66745

Order 336.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×S₄	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₄	—	3	1	no	maximal, Hall
4	C₂×D₃×C₇	—	4	1	no	maximal
5	C₇×A₄	C₂²	1	1	yes	characteristic, complemented
6	C₂³×C₇	—	3	1	no	—
7	C₂³×C₇	D₃	1	1	yes	characteristic, complemented
8	C₇×D₄	—	3	2	no	—
9	C₇×D₄	—	3	2	no	—
10	C₂×C₄×C₇	—	3	1	no	—
11	C₂×S₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₇	—	4	1	no	—
13	D₃×C₇	—	4	2	no	—
14	C₂²×C₇	—	3	1	no	—
15	C₂²×C₇	—	3	1	no	—
16	C₂²×C₇	—	3	2	no	—
17	C₂²×C₇	—	6	1	no	—
18	C₂²×C₇	—	6	1	no	—
19	C₂²×C₇	C₂×D₃	1	1	yes	characteristic, complemented
20	C₄×C₇	—	3	2	no	—
21	C₂×A₄	C₂×C₇	1	1	yes	characteristic, complemented
22	S₄	C₂×C₇	1	2	yes	direct product factor
23	C₃×C₇	—	4	1	no	Hall
24	C₂×D₄	—	3	1	no	Sylow
25	C₂×C₇	—	3	1	no	—
26	C₂×C₇	—	3	1	no	—
27	C₂×C₇	—	6	2	no	—
28	C₂×C₇	S₄	1	1	yes	characteristic, direct product factor
29	C₂×D₃	—	4	1	no	—
30	A₄	C₂²×C₇	1	1	yes	characteristic, complemented
31	C₂³	—	3	1	no	—
32	C₂³	D₃×C₇	1	1	yes	characteristic, complemented
33	D₄	—	3	2	no	—
34	D₄	—	3	2	no	—
35	C₂×C₄	—	3	1	no	—
36	C₇	C₂×S₄	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₃	—	4	1	no	—
38	D₃	—	4	2	no	—
39	C₂²	—	3	1	no	—
40	C₂²	—	3	1	no	—
41	C₂²	—	3	2	no	—
42	C₂²	—	6	1	no	—
43	C₂²	—	6	1	no	—
44	C₂²	C₂×D₃×C₇	1	1	yes	characteristic, complemented
45	C₄	—	3	2	no	—
46	C₃	—	4	1	no	Sylow
47	C₂	—	3	1	no	—
48	C₂	—	3	1	no	—
49	C₂	—	6	2	no	—
50	C₂	C₇×S₄	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇	S₄
commutator, cosocle, hypocenter	A₄	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂³×C₇	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×A₄	C₂	semidirect
2. C₇×S₄	C₂	direct (N × H)
5. C₇×A₄	C₂²	semidirect
7. C₂³×C₇	D₃	semidirect
11. C₂×S₄	C₇	direct (N × H)
19. C₂²×C₇	C₂×D₃	semidirect
21. C₂×A₄	C₂×C₇	semidirect
22. S₄	C₂×C₇	direct (N × H)
28. C₂×C₇	S₄	direct (N × H)
30. A₄	C₂²×C₇	semidirect
32. C₂³	D₃×C₇	semidirect
36. C₇	C₂×S₄	direct (N × H)
44. C₂²	C₂×D₃×C₇	semidirect
50. C₂	C₇×S₄	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×C₇×A₄	2
C₇×S₄	2
C₂×C₇×D₄	3
C₂×D₃×C₇	4
C₂×S₄	7

Subgroups — C2×C7×S4 #66745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×S₄ #66745