Subgroups — D₇×S₄ #66743

Order 336.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇⋊S₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×S₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×D₇	—	3	1	no	maximal, Hall
5	C₇×A₄	C₂²	1	1	yes	characteristic, complemented
6	D₃×D₇	—	4	1	no	maximal
7	C₂²×D₇	—	3	1	no	—
8	C₂²×D₇	D₃	1	1	yes	characteristic, complemented
9	C₇×D₄	—	3	1	no	—
10	C₂²⋊D₇	—	3	1	no	—
11	C₂²⋊D₇	—	3	1	no	—
12	D₂₈	—	3	1	no	—
13	C₄×D₇	—	3	1	no	—
14	C₂×S₄	—	7	1	no	maximal, Hall
15	D₂₁	—	4	1	no	—
16	C₃×D₇	—	4	1	no	—
17	D₃×C₇	—	4	1	no	—
18	C₂²×C₇	—	3	1	no	—
19	C₂²×C₇	C₂×D₃	1	1	yes	characteristic, complemented
20	C₂×D₇	—	3	1	no	—
21	C₂×D₇	—	3	1	no	—
22	C₂×D₇	—	3	1	no	—
23	C₂×D₇	—	6	1	no	—
24	C₂×D₇	—	6	1	no	—
25	C₄×C₇	—	3	1	no	—
26	Dic₇	—	3	1	no	—
27	C₂×A₄	—	7	1	no	—
28	S₄	—	7	1	no	—
29	S₄	D₇	1	1	yes	characteristic, direct product factor
30	C₃×C₇	—	4	1	no	Hall
31	C₂×D₄	—	21	1	no	Sylow
32	C₂×C₇	—	3	1	no	—
33	C₂×C₇	—	6	1	no	—
34	D₇	—	3	1	no	—
35	D₇	—	6	1	no	—
36	D₇	S₄	1	1	yes	characteristic, direct product factor
37	C₂×D₃	—	28	1	no	—
38	A₄	C₂×D₇	1	1	yes	characteristic, complemented
39	C₂³	—	7	1	no	—
40	C₂³	—	21	1	no	—
41	D₄	—	3	1	no	—
42	D₄	—	21	1	no	—
43	D₄	—	21	1	no	—
44	D₄	—	21	1	no	—
45	C₂×C₄	—	21	1	no	—
46	C₇	C₂×S₄	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	28	1	no	—
48	D₃	—	4	1	no	—
49	D₃	—	28	1	no	—
50	C₂²	—	3	1	no	—
51	C₂²	—	21	1	no	—
52	C₂²	—	21	1	no	—
53	C₂²	—	21	1	no	—
54	C₂²	—	42	1	no	—
55	C₂²	—	42	1	no	—
56	C₂²	D₃×D₇	1	1	yes	characteristic, complemented
57	C₄	—	3	1	no	—
58	C₄	—	21	1	no	—
59	C₃	—	4	1	no	Sylow
60	C₂	—	3	1	no	—
61	C₂	—	6	1	no	—
62	C₂	—	7	1	no	—
63	C₂	—	21	1	no	—
64	C₂	—	42	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. A₄×D₇	C₂	semidirect
2. C₇⋊S₄	C₂	semidirect
3. C₇×S₄	C₂	semidirect
5. C₇×A₄	C₂²	semidirect
8. C₂²×D₇	D₃	semidirect
19. C₂²×C₇	C₂×D₃	semidirect
29. S₄	D₇	direct (N × H)
36. D₇	S₄	direct (N × H)
38. A₄	C₂×D₇	semidirect
46. C₇	C₂×S₄	semidirect
56. C₂²	D₃×D₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
A₄×D₇	2
C₇⋊S₄	2
C₇×S₄	2
D₄×D₇	3
D₃×D₇	4
C₂×S₄	7

Subgroups — D7×S4 #66743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×S₄ #66743