Subgroups — C₅×AGL(1,𝔽₇)×D₇₃ #44501293

Order 30660.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×ΓL(1,𝔽₈)⋊D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×AGL(1,𝔽₇)×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×ΓL(1,𝔽₈)×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₇×D₇₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅×ΓL(1,𝔽₈)×C₇₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	AGL(1,𝔽₇)×D₇₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₅₁₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×D₇×C₇₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×C₇×D₇₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₅×D₇₃	—	7	1	no	maximal, Hall
11	ΓL(1,𝔽₈)⋊D₇₃	C₂×C₅	1	1	yes	characteristic, complemented
12	AGL(1,𝔽₇)×C₇₃	C₂×C₅	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)×D₇₃	C₂×C₅	1	1	yes	characteristic, complemented
14	C₅×C₇×C₇₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
15	C₃×C₅×D₇₃	—	7	1	no	—
16	C₃×C₅×D₇₃	—	7	1	no	—
17	C₂×C₃×C₅×C₇₃	—	7	1	no	—
18	D₇×D₇₃	C₃×C₅	1	1	yes	characteristic, complemented, Hall
19	ΓL(1,𝔽₈)×C₇₃	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×D₇₃	—	7	1	no	Hall
21	C₃×C₅×C₇₃	—	7	1	no	Hall
22	D₅₁₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
23	C₇×D₇₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
24	D₇×C₇₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
25	C₂×C₃×D₇₃	—	7	1	no	Hall
26	C₂×C₅×C₇₃	—	7	1	no	—
27	C₅×D₇₃	—	7	1	no	—
28	C₅×D₇₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
29	C₇×C₇₃	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₇₃	—	7	1	no	—
31	C₃×D₇₃	—	7	1	no	—
32	C₃×D₇₃	—	7	1	no	—
33	C₂×C₅×AGL(1,𝔽₇)	—	73	1	no	maximal, Hall
34	C₅×C₇₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
35	C₂×D₇₃	—	7	1	no	Hall
36	C₃×C₇₃	—	7	1	no	Hall
37	C₂×C₅×ΓL(1,𝔽₈)	—	73	1	no	—
38	C₅×AGL(1,𝔽₇)	—	73	1	no	—
39	C₅×AGL(1,𝔽₇)	D₇₃	1	1	yes	characteristic, direct product factor
40	C₂×C₇₃	—	7	1	no	—
41	D₇₃	—	7	1	no	—
42	D₇₃	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
43	C₂×C₅×D₇	—	73	1	no	Hall
44	C₅×ΓL(1,𝔽₈)	C₂×D₇₃	1	1	yes	characteristic, complemented, Hall
45	C₂×AGL(1,𝔽₇)	—	73	1	no	Hall
46	C₇₃	C₂×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₅×C₇	—	73	1	no	—
48	C₅×D₇	—	73	1	no	—
49	C₅×D₇	C₃×D₇₃	1	1	yes	characteristic, complemented
50	C₂²×C₃×C₅	—	511	1	no	Hall
51	C₂×ΓL(1,𝔽₈)	—	73	1	no	—
52	AGL(1,𝔽₇)	—	73	1	no	—
53	AGL(1,𝔽₇)	C₅×D₇₃	1	1	yes	characteristic, direct product factor
54	C₅×C₇	C₂×C₃×D₇₃	1	1	yes	characteristic, complemented, Hall
55	C₂×C₃×C₅	—	7	1	no	—
56	C₂×C₃×C₅	—	511	1	no	—
57	C₂×C₃×C₅	—	511	1	no	—
58	C₂×D₇	—	73	1	no	Hall
59	ΓL(1,𝔽₈)	C₂×C₅×D₇₃	1	1	yes	characteristic, complemented, Hall
60	C₂²×C₅	—	511	1	no	Hall
61	C₃×C₅	—	7	1	no	Hall
62	C₂×C₇	—	73	1	no	—
63	D₇	—	73	1	no	—
64	D₇	C₃×C₅×D₇₃	1	1	yes	characteristic, complemented
65	C₂²×C₃	—	511	1	no	Hall
66	C₂×C₅	—	7	1	no	—
67	C₂×C₅	—	73	1	no	—
68	C₂×C₅	—	511	1	no	—
69	C₇	C₂×C₃×C₅×D₇₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	1	no	—
71	C₂×C₃	—	511	1	no	—
72	C₂×C₃	—	511	1	no	—
73	C₅	AGL(1,𝔽₇)×D₇₃	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	511	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	7	1	no	—
77	C₂	—	73	1	no	—
78	C₂	—	511	1	no	—

	subgroup	G/H
center, hypercenter	C₅	AGL(1,𝔽₇)×D₇₃
commutator, cosocle, hypocenter	C₇×C₇₃	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₇₃	C₂²×C₃
LCS	G ▷ C₇×C₇₃
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×ΓL(1,𝔽₈)⋊D₇₃	C₂	semidirect
2. C₅×AGL(1,𝔽₇)×C₇₃	C₂	semidirect
3. C₅×ΓL(1,𝔽₈)×D₇₃	C₂	semidirect
4. C₅×D₇×D₇₃	C₃	semidirect
5. C₅×ΓL(1,𝔽₈)×C₇₃	C₂²	semidirect
6. AGL(1,𝔽₇)×D₇₃	C₅	direct (N × H)
7. C₅×D₅₁₁	C₂×C₃	semidirect
8. C₅×D₇×C₇₃	C₂×C₃	semidirect
9. C₅×C₇×D₇₃	C₂×C₃	semidirect
11. ΓL(1,𝔽₈)⋊D₇₃	C₂×C₅	semidirect
12. AGL(1,𝔽₇)×C₇₃	C₂×C₅	semidirect
13. ΓL(1,𝔽₈)×D₇₃	C₂×C₅	semidirect
14. C₅×C₇×C₇₃	C₂²×C₃	semidirect
18. D₇×D₇₃	C₃×C₅	semidirect
19. ΓL(1,𝔽₈)×C₇₃	C₂²×C₅	semidirect
22. D₅₁₁	C₂×C₃×C₅	semidirect
23. C₇×D₇₃	C₂×C₃×C₅	semidirect
24. D₇×C₇₃	C₂×C₃×C₅	semidirect
28. C₅×D₇₃	AGL(1,𝔽₇)	direct (N × H)
29. C₇×C₇₃	C₂²×C₃×C₅	semidirect
34. C₅×C₇₃	C₂×AGL(1,𝔽₇)	semidirect
39. C₅×AGL(1,𝔽₇)	D₇₃	direct (N × H)
42. D₇₃	C₅×AGL(1,𝔽₇)	direct (N × H)
44. C₅×ΓL(1,𝔽₈)	C₂×D₇₃	semidirect
46. C₇₃	C₂×C₅×AGL(1,𝔽₇)	semidirect
49. C₅×D₇	C₃×D₇₃	semidirect
53. AGL(1,𝔽₇)	C₅×D₇₃	direct (N × H)
54. C₅×C₇	C₂×C₃×D₇₃	semidirect
59. ΓL(1,𝔽₈)	C₂×C₅×D₇₃	semidirect
64. D₇	C₃×C₅×D₇₃	semidirect
69. C₇	C₂×C₃×C₅×D₇₃	semidirect
73. C₅	AGL(1,𝔽₇)×D₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×ΓL(1,𝔽₈)⋊D₇₃	2
C₅×AGL(1,𝔽₇)×C₇₃	2
C₅×ΓL(1,𝔽₈)×D₇₃	2
C₅×D₇×D₇₃	3
AGL(1,𝔽₇)×D₇₃	5
C₂×C₃×C₅×D₇₃	7
C₂×C₅×AGL(1,𝔽₇)	73

Subgroups — C5×AGL(1,𝔽7)×D73 #44501293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×AGL(1,𝔽₇)×D₇₃ #44501293