Subgroups — D₅×C₅⋊AGL(1,𝔽₇) #14523600

Order 2100.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊(C₅⋊AGL(1,𝔽₇))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₅⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×D₃₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₅⋊AGL(1,𝔽₇)	—	5	1	no	maximal
7	D₅×AGL(1,𝔽₇)	—	5	1	no	maximal
8	C₅⋊D₃₅	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×D₃₅	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×C₇×D₅	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×D₅²	—	7	1	no	maximal, Hall
12	C₂×C₅×ΓL(1,𝔽₈)	—	5	1	no	—
13	C₅⋊AGL(1,𝔽₇)	—	5	1	no	—
14	C₅⋊AGL(1,𝔽₇)	—	5	1	no	—
15	C₅⋊AGL(1,𝔽₇)	—	10	2	no	—
16	C₅⋊AGL(1,𝔽₇)	D₅	1	1	yes	characteristic, direct product factor
17	D₅×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented
18	C₅×AGL(1,𝔽₇)	—	5	1	no	—
19	C₅²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
20	C₃×C₅⋊D₅	—	7	1	no	—
21	C₃×C₅×D₅	—	7	1	no	—
22	C₃×C₅×D₅	—	7	1	no	—
23	C₂×D₃₅	—	5	1	no	—
24	D₅×D₇	—	5	1	no	—
25	C₅×ΓL(1,𝔽₈)	—	2	2	no	—
26	C₅×ΓL(1,𝔽₈)	C₂×D₅	1	1	yes	characteristic, complemented
27	C₅×ΓL(1,𝔽₈)	C₂×D₅	1	1	yes	characteristic, complemented
28	D₅²	—	7	1	no	Hall
29	C₂×AGL(1,𝔽₇)	—	25	1	no	Hall
30	C₃×C₅²	—	7	1	no	Hall
31	C₂×C₅×C₇	—	5	1	no	—
32	D₃₅	—	5	1	no	—
33	D₃₅	—	5	1	no	—
34	D₃₅	—	10	2	no	—
35	D₃₅	C₃×D₅	1	1	yes	characteristic, complemented
36	C₅×D₇	—	5	1	no	—
37	C₇×D₅	C₃×D₅	1	1	yes	characteristic, complemented
38	C₂×C₃×D₅	—	35	1	no	—
39	C₂×C₃×D₅	—	35	1	no	—
40	C₅⋊D₅	—	7	1	no	—
41	C₅×D₅	—	7	1	no	—
42	C₅×D₅	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
43	C₂×ΓL(1,𝔽₈)	—	5	1	no	—
44	AGL(1,𝔽₇)	—	5	1	no	—
45	AGL(1,𝔽₇)	—	25	1	no	—
46	C₅×C₇	—	2	2	no	—
47	C₅×C₇	C₂×C₃×D₅	1	1	yes	characteristic, complemented
48	C₅×C₇	C₂×C₃×D₅	1	1	yes	characteristic, complemented
49	C₂×C₃×C₅	—	35	1	no	—
50	C₂×C₃×C₅	—	35	1	no	—
51	C₃×D₅	—	7	1	no	—
52	C₃×D₅	—	7	1	no	—
53	C₃×D₅	—	35	1	no	—
54	C₃×D₅	—	35	1	no	—
55	C₃×D₅	—	70	2	no	—
56	C₂×D₇	—	25	1	no	Hall
57	C₅²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
58	ΓL(1,𝔽₈)	D₅²	1	1	yes	characteristic, complemented, Hall
59	C₂×D₅	—	35	1	no	—
60	C₂×D₅	—	35	1	no	—
61	C₃×C₅	—	7	1	no	—
62	C₃×C₅	—	7	1	no	—
63	C₃×C₅	—	14	2	no	—
64	C₂×C₇	—	5	1	no	—
65	D₇	—	5	1	no	—
66	D₇	—	25	1	no	—
67	C₂²×C₃	—	175	1	no	Hall
68	C₂×C₅	—	5	1	no	—
69	C₂×C₅	—	35	1	no	—
70	D₅	—	7	1	no	—
71	D₅	—	35	1	no	—
72	D₅	—	35	1	no	—
73	D₅	—	70	2	no	—
74	D₅	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
75	C₇	C₃×D₅²	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃	—	35	1	no	—
77	C₂×C₃	—	35	1	no	—
78	C₂×C₃	—	175	1	no	—
79	C₅	—	2	2	no	—
80	C₅	D₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
81	C₅	C₂×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
82	C₂²	—	175	1	no	Sylow
83	C₃	—	7	1	no	Sylow
84	C₂	—	5	1	no	—
85	C₂	—	35	1	no	—
86	C₂	—	175	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²×C₇	C₂²×C₃
LCS	G ▷ C₅²×C₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅⋊(C₅⋊AGL(1,𝔽₇))	C₂	semidirect
2. C₅×C₅⋊AGL(1,𝔽₇)	C₂	semidirect
3. C₅×D₅×ΓL(1,𝔽₈)	C₂	semidirect
4. D₅×D₃₅	C₃	semidirect
5. C₅²×ΓL(1,𝔽₈)	C₂²	semidirect
8. C₅⋊D₃₅	C₂×C₃	semidirect
9. C₅×D₃₅	C₂×C₃	semidirect
10. C₅×C₇×D₅	C₂×C₃	semidirect
16. C₅⋊AGL(1,𝔽₇)	D₅	direct (N × H)
17. D₅×ΓL(1,𝔽₈)	D₅	semidirect
19. C₅²×C₇	C₂²×C₃	semidirect
26. C₅×ΓL(1,𝔽₈)	C₂×D₅	semidirect
27. C₅×ΓL(1,𝔽₈)	C₂×D₅	semidirect
35. D₃₅	C₃×D₅	semidirect
37. C₇×D₅	C₃×D₅	semidirect
42. C₅×D₅	AGL(1,𝔽₇)	semidirect
47. C₅×C₇	C₂×C₃×D₅	semidirect
48. C₅×C₇	C₂×C₃×D₅	semidirect
57. C₅²	C₂×AGL(1,𝔽₇)	semidirect
58. ΓL(1,𝔽₈)	D₅²	semidirect
74. D₅	C₅⋊AGL(1,𝔽₇)	direct (N × H)
75. C₇	C₃×D₅²	semidirect
80. C₅	D₅×AGL(1,𝔽₇)	semidirect
81. C₅	C₂×C₅⋊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
C₅⋊(C₅⋊AGL(1,𝔽₇))	2
C₅×C₅⋊AGL(1,𝔽₇)	2
C₅×D₅×ΓL(1,𝔽₈)	2
D₅×D₃₅	3
C₂×C₅⋊AGL(1,𝔽₇)	5
D₅×AGL(1,𝔽₇)	5
C₃×D₅²	7

Subgroups — D5×C5⋊AGL(1,𝔽7) #14523600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×C₅⋊AGL(1,𝔽₇) #14523600