Subgroups — AGL(1,𝔽₇)×D₃₇₇ #44539845

Order 31668.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)⋊D₃₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×C₂₉×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×D₃₇₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₃₇₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₁₃×ΓL(1,𝔽₈)×C₂₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₂₆₃₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₁₃×D₇×C₂₉	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₃₇₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×D₃₇₇	—	7	1	no	maximal, Hall
10	C₇×C₁₃×C₂₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	AGL(1,𝔽₇)×D₂₉	—	13	1	no	maximal, Hall
12	C₃×D₃₇₇	—	7	1	no	—
13	C₃×D₃₇₇	—	7	1	no	—
14	C₂×C₃×C₁₃×C₂₉	—	7	1	no	—
15	C₂×D₃₇₇	—	7	1	no	Hall
16	ΓL(1,𝔽₈)⋊D₂₉	—	13	1	no	—
17	C₂₉×AGL(1,𝔽₇)	D₁₃	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×D₂₉	—	13	1	no	—
19	C₃×C₁₃×C₂₉	—	7	1	no	Hall
20	D₁₃×AGL(1,𝔽₇)	—	29	1	no	maximal, Hall
21	D₇×D₂₉	—	13	1	no	Hall
22	C₂×C₁₃×C₂₉	—	7	1	no	—
23	D₃₇₇	—	7	1	no	—
24	D₃₇₇	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
25	ΓL(1,𝔽₈)×C₂₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
26	ΓL(1,𝔽₈)⋊D₁₃	—	29	1	no	—
27	C₁₃×AGL(1,𝔽₇)	D₂₉	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₈)×D₁₃	—	29	1	no	—
29	D₂₀₃	—	13	1	no	—
30	C₇×D₂₉	—	13	1	no	—
31	D₇×C₂₉	C₃×D₁₃	1	1	yes	characteristic, complemented
32	C₁₃×C₂₉	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
33	D₇×D₁₃	—	29	1	no	Hall
34	C₂×C₃×D₂₉	—	91	1	no	Hall
35	C₁₃×ΓL(1,𝔽₈)	C₂×D₂₉	1	1	yes	characteristic, complemented, Hall
36	C₇×C₂₉	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
37	D₉₁	—	29	1	no	—
38	C₇×D₁₃	—	29	1	no	—
39	C₁₃×D₇	C₃×D₂₉	1	1	yes	characteristic, complemented
40	C₂×C₃×C₂₉	—	7	1	no	—
41	C₃×D₂₉	—	91	1	no	—
42	C₃×D₂₉	—	91	1	no	—
43	C₂×C₃×D₁₃	—	203	1	no	Hall
44	C₂×D₂₉	—	91	1	no	Hall
45	C₇×C₁₃	C₂×C₃×D₂₉	1	1	yes	characteristic, complemented, Hall
46	C₃×C₂₉	—	7	1	no	Hall
47	C₂×AGL(1,𝔽₇)	—	377	1	no	Hall
48	C₂×C₃×C₁₃	—	7	1	no	—
49	C₃×D₁₃	—	203	1	no	—
50	C₃×D₁₃	—	203	1	no	—
51	C₂×C₂₉	—	7	1	no	—
52	D₂₉	—	13	1	no	—
53	D₂₉	—	91	1	no	—
54	C₂×D₁₃	—	203	1	no	Hall
55	C₂×ΓL(1,𝔽₈)	—	377	1	no	—
56	AGL(1,𝔽₇)	—	377	1	no	—
57	AGL(1,𝔽₇)	D₃₇₇	1	1	yes	characteristic, direct product factor
58	C₃×C₁₃	—	7	1	no	Hall
59	C₂₉	D₁₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₇	—	377	1	no	Hall
61	C₂×C₁₃	—	7	1	no	—
62	D₁₃	—	29	1	no	—
63	D₁₃	—	203	1	no	—
64	ΓL(1,𝔽₈)	C₂×D₃₇₇	1	1	yes	characteristic, complemented, Hall
65	C₂×C₇	—	377	1	no	—
66	D₇	—	377	1	no	—
67	D₇	C₃×D₃₇₇	1	1	yes	characteristic, complemented
68	C₁₃	AGL(1,𝔽₇)×D₂₉	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₃	—	2639	1	no	Hall
70	C₇	C₂×C₃×D₃₇₇	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	7	1	no	—
72	C₂×C₃	—	2639	1	no	—
73	C₂×C₃	—	2639	1	no	—
74	C₂²	—	2639	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	7	1	no	—
77	C₂	—	2639	1	no	—
78	C₂	—	377	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₂₉	C₂²×C₃
LCS	G ▷ C₇×C₁₃×C₂₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)⋊D₃₇₇	C₂	semidirect
2. C₁₃×C₂₉×AGL(1,𝔽₇)	C₂	semidirect
3. ΓL(1,𝔽₈)×D₃₇₇	C₂	semidirect
4. D₇×D₃₇₇	C₃	semidirect
5. C₁₃×ΓL(1,𝔽₈)×C₂₉	C₂²	semidirect
6. D₂₆₃₉	C₂×C₃	semidirect
7. C₁₃×D₇×C₂₉	C₂×C₃	semidirect
8. C₇×D₃₇₇	C₂×C₃	semidirect
10. C₇×C₁₃×C₂₉	C₂²×C₃	semidirect
17. C₂₉×AGL(1,𝔽₇)	D₁₃	semidirect
24. D₃₇₇	AGL(1,𝔽₇)	direct (N × H)
25. ΓL(1,𝔽₈)×C₂₉	C₂×D₁₃	semidirect
27. C₁₃×AGL(1,𝔽₇)	D₂₉	semidirect
31. D₇×C₂₉	C₃×D₁₃	semidirect
32. C₁₃×C₂₉	C₂×AGL(1,𝔽₇)	semidirect
35. C₁₃×ΓL(1,𝔽₈)	C₂×D₂₉	semidirect
36. C₇×C₂₉	C₂×C₃×D₁₃	semidirect
39. C₁₃×D₇	C₃×D₂₉	semidirect
45. C₇×C₁₃	C₂×C₃×D₂₉	semidirect
57. AGL(1,𝔽₇)	D₃₇₇	direct (N × H)
59. C₂₉	D₁₃×AGL(1,𝔽₇)	semidirect
64. ΓL(1,𝔽₈)	C₂×D₃₇₇	semidirect
67. D₇	C₃×D₃₇₇	semidirect
68. C₁₃	AGL(1,𝔽₇)×D₂₉	semidirect
70. C₇	C₂×C₃×D₃₇₇	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)⋊D₃₇₇	2
C₁₃×C₂₉×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)×D₃₇₇	2
D₇×D₃₇₇	3
C₂×C₃×D₃₇₇	7
AGL(1,𝔽₇)×D₂₉	13
D₁₃×AGL(1,𝔽₇)	29

Subgroups — AGL(1,𝔽7)×D377 #44539845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₇)×D₃₇₇ #44539845