Subgroups — D₇×C₄₁×AGL(1,𝔽₇) #37563085

Order 24108.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×ΓL(1,𝔽₈)×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×ΓL(1,𝔽₈)⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₄₁×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇²×C₄₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₇×ΓL(1,𝔽₈)×C₄₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₄₁×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₇×C₄₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₇×C₄₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×D₇×C₄₁	—	7	1	no	maximal
10	C₂×C₄₁×AGL(1,𝔽₇)	—	7	1	no	maximal
11	C₇²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₇×C₄₁	—	7	1	no	—
13	C₃×D₇×C₄₁	—	7	1	no	—
14	C₃×D₇×C₄₁	—	7	1	no	—
15	C₂×ΓL(1,𝔽₈)×C₄₁	—	7	1	no	—
16	C₄₁×AGL(1,𝔽₇)	—	7	1	no	—
17	C₄₁×AGL(1,𝔽₇)	D₇	1	1	yes	characteristic, direct product factor
18	C₂×D₇×C₄₁	—	7	1	no	—
19	C₂×D₇×C₄₁	—	7	1	no	—
20	C₃×C₇×C₄₁	—	7	1	no	—
21	ΓL(1,𝔽₈)×C₄₁	C₂×D₇	1	1	yes	characteristic, complemented
22	D₇×AGL(1,𝔽₇)	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₇×C₄₁	—	7	1	no	—
24	C₂×C₇×C₄₁	—	7	1	no	—
25	D₇×C₄₁	—	7	1	no	—
26	D₇×C₄₁	—	7	1	no	—
27	D₇×C₄₁	—	42	1	no	—
28	D₇×C₄₁	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
29	D₇×C₄₁	C₃×D₇	1	1	yes	characteristic, complemented
30	C₂²×C₃×C₄₁	—	49	1	no	Hall
31	ΓL(1,𝔽₈)⋊D₇	C₂×C₄₁	1	1	yes	characteristic, complemented
32	D₇×ΓL(1,𝔽₈)	C₂×C₄₁	1	1	yes	characteristic, complemented
33	C₇×AGL(1,𝔽₇)	C₂×C₄₁	1	1	yes	characteristic, complemented
34	C₇×C₄₁	—	6	1	no	—
35	C₇×C₄₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
36	C₇×C₄₁	C₂×C₃×D₇	1	1	yes	characteristic, complemented
37	C₂×C₃×C₄₁	—	7	1	no	—
38	C₂×C₃×C₄₁	—	49	1	no	—
39	C₂×C₃×C₄₁	—	49	1	no	—
40	D₇²	C₃×C₄₁	1	1	yes	characteristic, complemented, Hall
41	C₂²×C₄₁	—	49	1	no	Hall
42	C₇×ΓL(1,𝔽₈)	C₂²×C₄₁	1	1	yes	characteristic, complemented, Hall
43	C₃×C₄₁	—	7	1	no	Hall
44	C₇⋊D₇	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
45	C₇×D₇	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
46	C₇×D₇	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
47	C₂×C₃×D₇	—	7	1	no	—
48	C₂×AGL(1,𝔽₇)	—	7	1	no	—
49	C₂×C₄₁	—	7	1	no	—
50	C₂×C₄₁	—	7	1	no	—
51	C₂×C₄₁	—	49	1	no	—
52	C₇²	C₂²×C₃×C₄₁	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₇	—	7	1	no	—
54	C₃×D₇	—	7	1	no	—
55	C₃×D₇	—	7	1	no	—
56	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
57	AGL(1,𝔽₇)	—	7	1	no	—
58	AGL(1,𝔽₇)	D₇×C₄₁	1	1	yes	characteristic, direct product factor
59	C₄₁	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×D₇	—	7	1	no	—
61	C₂×D₇	—	7	1	no	—
62	C₃×C₇	—	7	1	no	—
63	ΓL(1,𝔽₈)	C₂×D₇×C₄₁	1	1	yes	characteristic, complemented
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	7	1	no	—
66	D₇	—	7	1	no	—
67	D₇	—	7	1	no	—
68	D₇	—	42	1	no	—
69	D₇	C₄₁×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
70	D₇	C₃×D₇×C₄₁	1	1	yes	characteristic, complemented
71	C₂²×C₃	—	49	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	C₂×C₄₁×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
74	C₇	C₂×C₃×D₇×C₄₁	1	1	yes	characteristic, complemented
75	C₂×C₃	—	7	1	no	—
76	C₂×C₃	—	49	1	no	—
77	C₂×C₃	—	49	1	no	—
78	C₂²	—	49	1	no	Sylow
79	C₃	—	7	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	7	1	no	—
82	C₂	—	49	1	no	—

	subgroup	G/H
center, hypercenter	C₄₁	D₇×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₇²×C₄₁	C₂²×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄₁

Complements

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

N	H ≅ G/N	type
1. D₇×ΓL(1,𝔽₈)×C₄₁	C₂	semidirect
2. C₄₁×ΓL(1,𝔽₈)⋊D₇	C₂	semidirect
3. C₇×C₄₁×AGL(1,𝔽₇)	C₂	semidirect
4. D₇²×C₄₁	C₃	semidirect
5. C₇×ΓL(1,𝔽₈)×C₄₁	C₂²	semidirect
6. C₄₁×C₇⋊D₇	C₂×C₃	semidirect
7. C₇×D₇×C₄₁	C₂×C₃	semidirect
8. C₇×D₇×C₄₁	C₂×C₃	semidirect
11. C₇²×C₄₁	C₂²×C₃	semidirect
17. C₄₁×AGL(1,𝔽₇)	D₇	direct (N × H)
21. ΓL(1,𝔽₈)×C₄₁	C₂×D₇	semidirect
22. D₇×AGL(1,𝔽₇)	C₄₁	direct (N × H)
28. D₇×C₄₁	AGL(1,𝔽₇)	direct (N × H)
29. D₇×C₄₁	C₃×D₇	semidirect
31. ΓL(1,𝔽₈)⋊D₇	C₂×C₄₁	semidirect
32. D₇×ΓL(1,𝔽₈)	C₂×C₄₁	semidirect
33. C₇×AGL(1,𝔽₇)	C₂×C₄₁	semidirect
35. C₇×C₄₁	C₂×AGL(1,𝔽₇)	semidirect
36. C₇×C₄₁	C₂×C₃×D₇	semidirect
40. D₇²	C₃×C₄₁	semidirect
42. C₇×ΓL(1,𝔽₈)	C₂²×C₄₁	semidirect
44. C₇⋊D₇	C₂×C₃×C₄₁	semidirect
45. C₇×D₇	C₂×C₃×C₄₁	semidirect
46. C₇×D₇	C₂×C₃×C₄₁	semidirect
52. C₇²	C₂²×C₃×C₄₁	semidirect
58. AGL(1,𝔽₇)	D₇×C₄₁	direct (N × H)
59. C₄₁	D₇×AGL(1,𝔽₇)	direct (N × H)
63. ΓL(1,𝔽₈)	C₂×D₇×C₄₁	semidirect
69. D₇	C₄₁×AGL(1,𝔽₇)	direct (N × H)
70. D₇	C₃×D₇×C₄₁	semidirect
73. C₇	C₂×C₄₁×AGL(1,𝔽₇)	semidirect
74. C₇	C₂×C₃×D₇×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
D₇×ΓL(1,𝔽₈)×C₄₁	2
C₄₁×ΓL(1,𝔽₈)⋊D₇	2
C₇×C₄₁×AGL(1,𝔽₇)	2
D₇²×C₄₁	3
C₂×C₃×D₇×C₄₁	7
C₂×C₄₁×AGL(1,𝔽₇)	7
D₇×AGL(1,𝔽₇)	41

Subgroups — D7×C41×AGL(1,𝔽7) #37563085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₄₁×AGL(1,𝔽₇) #37563085