Subgroups — C₇×A₄×AGL(1,𝔽₅) #13132927

Order 1680.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₅×A₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₇×A₄	C₄	1	1	yes	characteristic, complemented
4	C₃×C₇×AGL(1,𝔽₅)	—	4	1	no	maximal
5	C₄×C₇×A₄	—	5	1	no	maximal, Hall
6	C₂²×C₇×D₅	C₂×C₃	1	1	yes	characteristic
7	C₂×C₇×AGL(1,𝔽₅)	—	3	1	no	—
8	C₂×C₇×AGL(1,𝔽₅)	—	3	1	no	—
9	A₄×AGL(1,𝔽₅)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₇×D₅	—	4	1	no	—
11	C₂×C₇×A₄	—	5	1	no	—
12	C₂²×C₅×C₇	C₃×C₄	1	1	yes	characteristic, complemented
13	C₂×C₇×D₅	—	3	1	no	—
14	C₂×C₇×D₅	—	3	1	no	—
15	C₇×AGL(1,𝔽₅)	—	3	1	no	—
16	C₇×AGL(1,𝔽₅)	A₄	1	1	yes	characteristic, direct product factor
17	D₅×A₄	C₂×C₇	1	1	yes	characteristic
18	C₂²×C₄×C₇	—	5	1	no	Hall
19	C₃×C₅×C₇	—	4	1	no	Hall
20	C₇×A₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
21	C₃×C₄×C₇	—	20	1	no	—
22	C₂²×AGL(1,𝔽₅)	C₃×C₇	1	1	yes	characteristic, complemented, Hall
23	C₂×C₅×C₇	—	3	1	no	—
24	C₇×D₅	—	3	1	no	—
25	C₇×D₅	C₂×A₄	1	1	yes	characteristic
26	C₅×A₄	C₄×C₇	1	1	yes	characteristic, complemented
27	C₃×AGL(1,𝔽₅)	—	4	1	no	—
28	C₂³×C₇	—	5	1	no	—
29	C₂×C₄×C₇	—	15	1	no	—
30	C₂×C₄×C₇	—	15	1	no	—
31	C₄×A₄	—	5	1	no	Hall
32	C₂×C₃×C₇	—	20	1	no	—
33	C₂²×D₅	C₂×C₃×C₇	1	1	yes	characteristic
34	C₂×AGL(1,𝔽₅)	—	3	1	no	—
35	C₂×AGL(1,𝔽₅)	—	3	1	no	—
36	C₅×C₇	C₄×A₄	1	1	yes	characteristic, complemented, Hall
37	C₃×D₅	—	4	1	no	—
38	C₂²×C₇	—	15	1	no	—
39	C₂²×C₇	—	15	1	no	—
40	C₂²×C₇	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
41	C₄×C₇	—	5	1	no	—
42	C₄×C₇	—	15	1	no	—
43	C₂×A₄	—	5	1	no	—
44	C₃×C₇	—	4	1	no	Hall
45	C₂²×C₅	C₃×C₄×C₇	1	1	yes	characteristic, complemented
46	C₂×D₅	—	3	1	no	—
47	C₂×D₅	—	3	1	no	—
48	AGL(1,𝔽₅)	—	3	1	no	—
49	AGL(1,𝔽₅)	C₇×A₄	1	1	yes	characteristic, direct product factor
50	C₂²×C₄	—	5	1	no	Sylow
51	C₃×C₅	—	4	1	no	Hall
52	C₂×C₇	—	3	1	no	—
53	C₂×C₇	—	5	1	no	—
54	C₂×C₇	—	15	1	no	—
55	A₄	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
56	C₃×C₄	—	20	1	no	—
57	C₂×C₅	—	3	1	no	—
58	D₅	—	3	1	no	—
59	D₅	C₂×C₇×A₄	1	1	yes	characteristic
60	C₂³	—	5	1	no	—
61	C₂×C₄	—	15	1	no	—
62	C₂×C₄	—	15	1	no	—
63	C₇	A₄×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	—	20	1	no	—
65	C₅	C₄×C₇×A₄	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	15	1	no	—
67	C₂²	—	15	1	no	—
68	C₂²	C₃×C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
69	C₄	—	5	1	no	—
70	C₄	—	15	1	no	—
71	C₃	—	4	1	no	Sylow
72	C₂	—	3	1	no	—
73	C₂	—	5	1	no	—
74	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₇	A₄×AGL(1,𝔽₅)
commutator, hypocenter	C₂²×C₅	C₃×C₄×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇	C₃×C₄
prime radical	C₇×D₅×A₄	C₂
cosocle	C₂²×D₅	C₂×C₃×C₇
LCS	G ▷ C₂²×C₅
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×AGL(1,𝔽₅)	C₃	semidirect
3. C₅×C₇×A₄	C₄	semidirect
9. A₄×AGL(1,𝔽₅)	C₇	direct (N × H)
12. C₂²×C₅×C₇	C₃×C₄	semidirect
16. C₇×AGL(1,𝔽₅)	A₄	direct (N × H)
20. C₇×A₄	AGL(1,𝔽₅)	direct (N × H)
22. C₂²×AGL(1,𝔽₅)	C₃×C₇	semidirect
26. C₅×A₄	C₄×C₇	semidirect
36. C₅×C₇	C₄×A₄	semidirect
40. C₂²×C₇	C₃×AGL(1,𝔽₅)	semidirect
45. C₂²×C₅	C₃×C₄×C₇	semidirect
49. AGL(1,𝔽₅)	C₇×A₄	direct (N × H)
55. A₄	C₇×AGL(1,𝔽₅)	direct (N × H)
63. C₇	A₄×AGL(1,𝔽₅)	direct (N × H)
65. C₅	C₄×C₇×A₄	semidirect
68. C₂²	C₃×C₇×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₅×A₄	2
C₂²×C₇×AGL(1,𝔽₅)	3
C₃×C₇×AGL(1,𝔽₅)	4
C₄×C₇×A₄	5
A₄×AGL(1,𝔽₅)	7

Subgroups — C7×A4×AGL(1,𝔽5) #13132927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×A₄×AGL(1,𝔽₅) #13132927