Subgroups — C₃×D₃×AGL(1,𝔽₇) #10625521

Order 756.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₃×D₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×AGL(1,𝔽₇)	—	3	1	no	maximal
6	D₃×AGL(1,𝔽₇)	C₃	1	3	yes	maximal, direct product factor
7	C₃²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
8	C₃×D₂₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×ΓL(1,𝔽₈)	—	3	1	no	—
12	C₃⋊AGL(1,𝔽₇)	C₂×C₃	1	3	yes	complemented
13	D₃×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
14	C₃×AGL(1,𝔽₇)	—	2	1	no	—
15	C₃×AGL(1,𝔽₇)	—	2	3	no	—
16	C₃×AGL(1,𝔽₇)	—	3	1	no	—
17	C₃×AGL(1,𝔽₇)	D₃	1	1	yes	characteristic, direct product factor
18	C₃×AGL(1,𝔽₇)	C₂×C₃	1	3	yes	complemented
19	C₂×C₃²×D₃	—	7	1	no	maximal, Hall
20	C₂×C₃×D₇	—	3	1	no	—
21	D₃×D₇	C₃²	1	1	yes	characteristic, complemented
22	C₂×AGL(1,𝔽₇)	—	3	3	no	—
23	C₃²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
24	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
25	C₃×ΓL(1,𝔽₈)	—	2	3	no	—
26	C₃×ΓL(1,𝔽₈)	C₂×D₃	1	1	yes	characteristic, complemented
27	C₃×ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	complemented
28	C₂×C₃³	—	7	1	no	—
29	C₃²×D₃	—	7	1	no	—
30	C₃²×D₃	—	7	1	no	—
31	C₂×C₃×C₇	—	3	1	no	—
32	D₂₁	C₂×C₃²	1	1	yes	characteristic, complemented
33	C₃×D₇	—	2	1	no	—
34	C₃×D₇	—	3	1	no	—
35	C₃×D₇	C₃×D₃	1	1	yes	characteristic, complemented
36	C₃×D₇	C₂×C₃²	1	1	yes	characteristic, complemented
37	D₃×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
38	C₂×ΓL(1,𝔽₈)	—	3	3	no	—
39	AGL(1,𝔽₇)	—	2	3	no	—
40	AGL(1,𝔽₇)	—	3	3	no	—
41	AGL(1,𝔽₇)	C₃×D₃	1	3	yes	direct product factor
42	C₂²×C₃²	—	21	1	no	—
43	C₂×C₃×D₃	—	7	3	no	—
44	C₂×C₃×D₃	—	7	1	no	—
45	C₂×D₇	—	3	1	no	Hall
46	C₃³	—	7	1	no	Sylow
47	C₃×C₇	—	2	1	no	—
48	C₃×C₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented
49	C₃×C₇	C₂²×C₃²	1	1	yes	characteristic, complemented
50	ΓL(1,𝔽₈)	—	2	3	no	—
51	ΓL(1,𝔽₈)	C₂×C₃×D₃	1	3	yes	complemented
52	C₂×C₃²	—	7	1	no	—
53	C₂×C₃²	—	7	3	no	—
54	C₂×C₃²	—	7	1	no	—
55	C₂×C₃²	—	14	3	no	—
56	C₂×C₃²	—	14	1	no	—
57	C₂×C₃²	—	21	1	no	—
58	C₂×C₃²	—	21	1	no	—
59	C₃×D₃	—	7	3	no	—
60	C₃×D₃	—	7	1	no	—
61	C₃×D₃	—	7	3	no	—
62	C₃×D₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
63	C₂×C₇	—	3	1	no	—
64	D₇	—	3	1	no	—
65	D₇	C₃²×D₃	1	1	yes	characteristic, complemented
66	C₂²×C₃	—	21	1	no	—
67	C₂²×C₃	—	21	3	no	—
68	C₂×D₃	—	7	1	no	—
69	C₃²	—	7	1	no	—
70	C₃²	—	7	3	no	—
71	C₃²	—	14	3	no	—
72	C₃²	—	14	1	no	—
73	C₃²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
74	C₇	C₂×C₃²×D₃	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	3	1	no	—
76	C₂×C₃	—	7	3	no	—
77	C₂×C₃	—	7	1	no	—
78	C₂×C₃	—	7	1	no	—
79	C₂×C₃	—	14	1	no	—
80	C₂×C₃	—	14	3	no	—
81	C₂×C₃	—	21	3	no	—
82	C₂×C₃	—	21	3	no	—
83	C₂×C₃	—	21	1	no	—
84	D₃	—	7	1	no	—
85	D₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
86	C₂²	—	21	1	no	Sylow
87	C₃	—	2	1	no	—
88	C₃	—	7	3	no	—
89	C₃	—	14	3	no	—
90	C₃	D₃×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
91	C₃	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
92	C₂	—	3	1	no	—
93	C₂	—	7	1	no	—
94	C₂	—	21	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₃×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₃×C₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂²×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₃⋊AGL(1,𝔽₇)	C₂	semidirect
2. C₃×D₃×ΓL(1,𝔽₈)	C₂	semidirect
3. C₃²×AGL(1,𝔽₇)	C₂	semidirect
4. C₃×D₃×D₇	C₃	semidirect
6. D₃×AGL(1,𝔽₇)	C₃	direct (N × H)
7. C₃²×ΓL(1,𝔽₈)	C₂²	semidirect
8. C₃×D₂₁	C₂×C₃	semidirect
9. C₃×D₃×C₇	C₂×C₃	semidirect
10. C₃²×D₇	C₂×C₃	semidirect
12. C₃⋊AGL(1,𝔽₇)	C₂×C₃	semidirect
13. D₃×ΓL(1,𝔽₈)	C₂×C₃	semidirect
17. C₃×AGL(1,𝔽₇)	D₃	direct (N × H)
18. C₃×AGL(1,𝔽₇)	C₂×C₃	semidirect
21. D₃×D₇	C₃²	semidirect
23. C₃²×C₇	C₂²×C₃	semidirect
26. C₃×ΓL(1,𝔽₈)	C₂×D₃	semidirect
27. C₃×ΓL(1,𝔽₈)	C₂²×C₃	semidirect
32. D₂₁	C₂×C₃²	semidirect
35. C₃×D₇	C₃×D₃	semidirect
36. C₃×D₇	C₂×C₃²	semidirect
37. D₃×C₇	C₂×C₃²	semidirect
41. AGL(1,𝔽₇)	C₃×D₃	direct (N × H)
48. C₃×C₇	C₂×C₃×D₃	semidirect
49. C₃×C₇	C₂²×C₃²	semidirect
51. ΓL(1,𝔽₈)	C₂×C₃×D₃	semidirect
62. C₃×D₃	AGL(1,𝔽₇)	direct (N × H)
65. D₇	C₃²×D₃	semidirect
73. C₃²	C₂×AGL(1,𝔽₇)	semidirect
74. C₇	C₂×C₃²×D₃	semidirect
85. D₃	C₃×AGL(1,𝔽₇)	direct (N × H)
90. C₃	D₃×AGL(1,𝔽₇)	direct (N × H)
91. C₃	C₂×C₃×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃⋊AGL(1,𝔽₇)	2
C₃×D₃×ΓL(1,𝔽₈)	2
C₃²×AGL(1,𝔽₇)	2
C₃×D₃×D₇	3
C₂×C₃×AGL(1,𝔽₇)	3
D₃×AGL(1,𝔽₇)	3
C₂×C₃²×D₃	7

Subgroups — C3×D3×AGL(1,𝔽7) #10625521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×AGL(1,𝔽₇) #10625521