Subgroups — C₇×#504⋊C₃ #13055058

Order 1344.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₇×#504	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×#41⋊C₃	C₄	1	1	yes	characteristic
4	C₃×C₇×C₁₆	—	4	1	no	maximal
5	C₇×#131	C₂×C₃	1	1	yes	characteristic
6	C₇×M₃₂	—	3	1	no	—
7	C₂×C₇×C₁₆	—	3	1	no	—
8	#504⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×SL(2,𝔽₃)	C₈	1	1	yes	characteristic
10	C₃×C₇×C₈	—	4	1	no	—
11	C₇×#41	C₃×C₄	1	1	yes	characteristic
12	C₇×M₁₆	—	3	1	no	—
13	C₂×C₇×C₈	—	3	1	no	—
14	C₇×C₁₆	—	3	1	no	—
15	C₇×C₁₆	A₄	1	1	yes	characteristic
16	#131⋊C₃	C₂×C₇	1	1	yes	characteristic
17	C₃×C₄×C₇	—	4	1	no	—
18	#504	C₃×C₇	1	1	yes	characteristic, complemented, Sylow
19	C₇×Q₈	C₃×C₈	1	1	yes	characteristic
20	C₇×D₄	—	3	1	no	—
21	C₂×C₄×C₇	—	3	1	no	—
22	C₇×C₈	—	3	1	no	—
23	C₇×C₈	C₂×A₄	1	1	yes	characteristic
24	#41⋊C₃	C₄×C₇	1	1	yes	characteristic
25	C₃×C₁₆	—	4	1	no	—
26	C₂×C₃×C₇	—	4	1	no	—
27	#131	C₂×C₃×C₇	1	1	yes	characteristic
28	M₃₂	—	3	1	no	—
29	C₂×C₁₆	—	3	1	no	—
30	C₂²×C₇	—	3	1	no	—
31	C₄×C₇	—	3	1	no	—
32	C₄×C₇	C₄×A₄	1	1	yes	characteristic
33	SL(2,𝔽₃)	C₇×C₈	1	1	yes	characteristic
34	C₃×C₈	—	4	1	no	—
35	C₃×C₇	—	4	1	no	Hall
36	#41	C₃×C₄×C₇	1	1	yes	characteristic
37	M₁₆	—	3	1	no	—
38	C₂×C₈	—	3	1	no	—
39	C₁₆	—	3	1	no	—
40	C₁₆	C₇×A₄	1	1	yes	characteristic
41	C₂×C₇	—	6	1	no	—
42	C₂×C₇	C₈×A₄	1	1	yes	characteristic
43	C₃×C₄	—	4	1	no	—
44	Q₈	C₃×C₇×C₈	1	1	yes	characteristic
45	D₄	—	3	1	no	—
46	C₂×C₄	—	3	1	no	—
47	C₈	—	3	1	no	—
48	C₈	C₂×C₇×A₄	1	1	yes	characteristic
49	C₇	#504⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₃	—	4	1	no	—
51	C₂²	—	3	1	no	—
52	C₄	—	3	1	no	—
53	C₄	C₄×C₇×A₄	1	1	yes	characteristic
54	C₃	—	4	1	no	Sylow
55	C₂	—	6	1	no	—
56	C₂	C₇×C₈×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₇×C₁₆	A₄
commutator, hypocenter	Q₈	C₃×C₇×C₈
Frattini	C₈	C₂×C₇×A₄
Fitting	C₇×#504	C₃
socle	C₂×C₇	C₈×A₄
prime radical	C₇×#41⋊C₃	C₄
cosocle	#131	C₂×C₃×C₇
LCS	G ▷ Q₈
UCS	C₁ ◁ C₇×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×#504	C₃	semidirect
8. #504⋊C₃	C₇	direct (N × H)
18. #504	C₃×C₇	semidirect
49. C₇	#504⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#131⋊C₃	2
C₇×#504	3
C₃×C₇×C₁₆	4
#504⋊C₃	7

Subgroups — C7×#504⋊C3 #13055058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#504⋊C₃ #13055058