Subgroups — C₃×(C₂²⋊D₇)⋊C₃ #92653

Order 504.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₂²⋊D₇)⋊C₃	C₃	1	3	yes	maximal, direct product factor
6	C₂×C₃×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₂×C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₃×AGL(1,𝔽₇)	—	2	1	no	—
9	C₂²×C₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂²×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
12	C₂×AGL(1,𝔽₇)	C₂×C₃	1	3	yes	complemented
13	C₃×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
14	ΓL(1,𝔽₈)⋊C₄	C₂×C₃	1	3	yes	complemented
15	C₃²×D₄	—	7	1	no	maximal, Hall
16	C₃×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic, complemented, Hall
17	C₂²⋊D₇	C₃²	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₇	—	2	1	no	—
19	C₂×C₃×C₇	C₂²×C₃	1	1	yes	characteristic
20	C₃×D₇	—	2	1	no	—
21	C₂×ΓL(1,𝔽₈)	—	2	3	no	—
22	C₂×ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	—
23	AGL(1,𝔽₇)	—	2	3	no	—
24	C₂²×C₃²	—	7	1	no	—
25	C₂²×C₃²	—	7	1	no	—
26	C₃²×C₄	—	7	1	no	—
27	C₂²×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
28	C₂×D₇	C₂×C₃²	1	1	yes	characteristic, complemented
29	Dic₇	C₂×C₃²	1	1	yes	characteristic, complemented
30	C₃×D₄	—	7	3	no	—
31	C₃×D₄	—	7	1	no	—
32	C₃×C₇	C₃×D₄	1	1	yes	characteristic, complemented
33	ΓL(1,𝔽₈)	C₃×D₄	1	3	yes	complemented
34	C₂×C₃²	—	7	1	no	—
35	C₂×C₃²	—	14	1	no	—
36	C₂×C₃²	—	14	1	no	—
37	C₂×C₇	—	2	1	no	—
38	C₂×C₇	C₂²×C₃²	1	1	yes	characteristic
39	D₇	—	2	1	no	—
40	C₂²×C₃	—	7	3	no	—
41	C₂²×C₃	—	7	1	no	—
42	C₂²×C₃	—	7	3	no	—
43	C₂²×C₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
44	C₃×C₄	—	7	3	no	—
45	C₃×C₄	—	7	1	no	—
46	C₃²	—	7	1	no	Sylow
47	D₄	—	7	1	no	Sylow
48	C₇	C₃²×D₄	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	2	1	no	—
50	C₂×C₃	—	7	3	no	—
51	C₂×C₃	—	14	3	no	—
52	C₂×C₃	—	14	3	no	—
53	C₂×C₃	—	14	1	no	—
54	C₂×C₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
55	C₂²	—	7	1	no	—
56	C₂²	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
57	C₄	—	7	1	no	—
58	C₃	—	7	3	no	—
59	C₃	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic, direct product factor
60	C₂	—	2	1	no	—
61	C₂	—	14	1	no	—
62	C₂	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₇	C₂²×C₃²
Frattini	C₂	C₂×C₃×AGL(1,𝔽₇)
Fitting	C₂²×C₃×C₇	C₂×C₃
socle	C₂×C₃×C₇	C₂²×C₃
hypercenter	C₂²×C₃	AGL(1,𝔽₇)
hypocenter	C₇	C₃²×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂×C₃×AGL(1,𝔽₇)	C₂	semidirect
3. C₃×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
4. C₃×C₂²⋊D₇	C₃	semidirect
5. (C₂²⋊D₇)⋊C₃	C₃	direct (N × H)
9. C₂²×C₃×C₇	C₂×C₃	semidirect
10. C₂×C₃×D₇	C₂×C₃	semidirect
11. C₂²×ΓL(1,𝔽₈)	C₂×C₃	semidirect
12. C₂×AGL(1,𝔽₇)	C₂×C₃	semidirect
13. C₃×Dic₇	C₂×C₃	semidirect
14. ΓL(1,𝔽₈)⋊C₄	C₂×C₃	semidirect
16. C₃×ΓL(1,𝔽₈)	D₄	semidirect
17. C₂²⋊D₇	C₃²	semidirect
27. C₂²×C₇	C₂×C₃²	semidirect
28. C₂×D₇	C₂×C₃²	semidirect
29. Dic₇	C₂×C₃²	semidirect
32. C₃×C₇	C₃×D₄	semidirect
33. ΓL(1,𝔽₈)	C₃×D₄	semidirect
43. C₂²×C₃	AGL(1,𝔽₇)	semidirect
48. C₇	C₃²×D₄	semidirect
56. C₂²	C₃×AGL(1,𝔽₇)	semidirect
59. C₃	(C₂²⋊D₇)⋊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
C₂²×C₃×ΓL(1,𝔽₈)	2
C₂×C₃×AGL(1,𝔽₇)	2
C₃×ΓL(1,𝔽₈)⋊C₄	2
C₃×C₂²⋊D₇	3
(C₂²⋊D₇)⋊C₃	3
C₃²×D₄	7

Subgroups — C3×(C22⋊D7)⋊C3 #92653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₂²⋊D₇)⋊C₃ #92653