Subgroups — C₂×D₃×AGL(1,𝔽₇) #92736

Order 504.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×AGL(1,𝔽₇)	C₂	1	4	yes	maximal, direct product factor
5	C₂×D₃×D₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×AGL(1,𝔽₇)	—	3	1	no	maximal
7	C₂×C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
8	C₃⋊AGL(1,𝔽₇)	C₂²	1	2	yes	complemented
9	D₃×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
10	C₃×AGL(1,𝔽₇)	C₂²	1	2	yes	complemented
11	C₂×D₂₁	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×D₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₇	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂²×ΓL(1,𝔽₈)	—	3	1	no	—
15	D₃×D₇	C₂×C₃	1	4	yes	complemented
16	C₂×AGL(1,𝔽₇)	—	2	1	no	—
17	C₂×AGL(1,𝔽₇)	—	3	1	no	—
18	C₂×AGL(1,𝔽₇)	—	3	4	no	—
19	C₂×AGL(1,𝔽₇)	D₃	1	1	yes	characteristic, direct product factor
20	C₂²×C₃×D₃	—	7	1	no	maximal, Hall
21	C₃×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic, complemented, Hall
22	C₂²×D₇	—	3	1	no	Hall
23	C₂×C₃×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
24	D₂₁	C₂²×C₃	1	2	yes	complemented
25	C₃×D₇	C₂²×C₃	1	2	yes	complemented
26	D₃×C₇	C₂²×C₃	1	2	yes	complemented
27	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
28	C₂×ΓL(1,𝔽₈)	—	3	2	no	—
29	C₂×ΓL(1,𝔽₈)	C₂×D₃	1	1	yes	characteristic, complemented
30	AGL(1,𝔽₇)	—	2	2	no	—
31	AGL(1,𝔽₇)	—	3	2	no	—
32	AGL(1,𝔽₇)	C₂×D₃	1	2	yes	direct product factor
33	C₂²×C₃²	—	7	1	no	—
34	C₂×C₃×D₃	—	7	4	no	—
35	C₂×C₃×D₃	—	7	1	no	—
36	C₂×C₃×D₃	—	7	1	no	—
37	C₂²×C₇	—	3	1	no	—
38	C₂×D₇	—	3	1	no	—
39	C₂×D₇	—	3	4	no	—
40	C₂×D₇	C₃×D₃	1	1	yes	characteristic, complemented
41	C₂³×C₃	—	21	1	no	—
42	C₂²×D₃	—	7	1	no	—
43	C₃×C₇	C₂³×C₃	1	1	yes	characteristic, complemented
44	ΓL(1,𝔽₈)	—	2	1	no	—
45	ΓL(1,𝔽₈)	C₂²×D₃	1	1	yes	characteristic, complemented
46	C₂×C₃²	—	7	2	no	—
47	C₂×C₃²	—	7	1	no	—
48	C₃×D₃	—	7	2	no	—
49	C₃×D₃	—	7	2	no	—
50	C₂×C₇	—	3	2	no	—
51	C₂×C₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented
52	D₇	—	3	2	no	—
53	D₇	C₂×C₃×D₃	1	2	yes	complemented
54	C₂²×C₃	—	7	1	no	—
55	C₂²×C₃	—	7	1	no	—
56	C₂²×C₃	—	14	1	no	—
57	C₂²×C₃	—	21	1	no	—
58	C₂²×C₃	—	21	4	no	—
59	C₂²×C₃	—	21	1	no	—
60	C₂×D₃	—	7	1	no	—
61	C₂×D₃	—	7	4	no	—
62	C₂×D₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
63	C₃²	—	7	1	no	Sylow
64	C₂³	—	21	1	no	Sylow
65	C₇	C₂²×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	7	2	no	—
67	C₂×C₃	—	7	2	no	—
68	C₂×C₃	—	7	1	no	—
69	C₂×C₃	—	14	2	no	—
70	C₂×C₃	—	14	1	no	—
71	C₂×C₃	—	21	2	no	—
72	C₂×C₃	—	21	2	no	—
73	C₂×C₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
74	D₃	—	7	2	no	—
75	D₃	C₂×AGL(1,𝔽₇)	1	2	yes	direct product factor
76	C₂²	—	3	1	no	—
77	C₂²	—	7	1	no	—
78	C₂²	—	21	4	no	—
79	C₂²	—	21	1	no	—
80	C₃	—	7	1	no	—
81	C₃	—	14	1	no	—
82	C₃	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
83	C₂	—	3	2	no	—
84	C₂	—	7	2	no	—
85	C₂	—	21	2	no	—
86	C₂	D₃×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₃×C₇	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇	C₂²×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⋊AGL(1,𝔽₇)	C₂	semidirect
2. C₂×D₃×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×C₃×AGL(1,𝔽₇)	C₂	semidirect
4. D₃×AGL(1,𝔽₇)	C₂	direct (N × H)
5. C₂×D₃×D₇	C₃	semidirect
7. C₂×C₃×ΓL(1,𝔽₈)	C₂²	semidirect
8. C₃⋊AGL(1,𝔽₇)	C₂²	semidirect
9. D₃×ΓL(1,𝔽₈)	C₂²	semidirect
10. C₃×AGL(1,𝔽₇)	C₂²	semidirect
11. C₂×D₂₁	C₂×C₃	semidirect
12. C₂×D₃×C₇	C₂×C₃	semidirect
13. C₂×C₃×D₇	C₂×C₃	semidirect
15. D₃×D₇	C₂×C₃	semidirect
19. C₂×AGL(1,𝔽₇)	D₃	direct (N × H)
21. C₃×ΓL(1,𝔽₈)	C₂³	semidirect
23. C₂×C₃×C₇	C₂²×C₃	semidirect
24. D₂₁	C₂²×C₃	semidirect
25. C₃×D₇	C₂²×C₃	semidirect
26. D₃×C₇	C₂²×C₃	semidirect
29. C₂×ΓL(1,𝔽₈)	C₂×D₃	semidirect
32. AGL(1,𝔽₇)	C₂×D₃	direct (N × H)
40. C₂×D₇	C₃×D₃	semidirect
43. C₃×C₇	C₂³×C₃	semidirect
45. ΓL(1,𝔽₈)	C₂²×D₃	semidirect
51. C₂×C₇	C₂×C₃×D₃	semidirect
53. D₇	C₂×C₃×D₃	semidirect
62. C₂×D₃	AGL(1,𝔽₇)	direct (N × H)
65. C₇	C₂²×C₃×D₃	semidirect
73. C₂×C₃	C₂×AGL(1,𝔽₇)	semidirect
75. D₃	C₂×AGL(1,𝔽₇)	direct (N × H)
82. C₃	C₂²×AGL(1,𝔽₇)	semidirect
86. C₂	D₃×AGL(1,𝔽₇)	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×C₃⋊AGL(1,𝔽₇)	2
C₂×D₃×ΓL(1,𝔽₈)	2
C₂×C₃×AGL(1,𝔽₇)	2
D₃×AGL(1,𝔽₇)	2
C₂×D₃×D₇	3
C₂²×AGL(1,𝔽₇)	3
C₂²×C₃×D₃	7

Subgroups — C2×D3×AGL(1,𝔽7) #92736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×AGL(1,𝔽₇) #92736