Subgroups — C₄×C₃⋊(C₇⋊AGL(1,𝔽₅)) #13132626

Order 1680.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))	C₂	1	2	yes	maximal
2	C₃×C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
4	C₂×C₃×C₇×D₅	C₂²	1	1	yes	characteristic
5	C₃⋊(C₇⋊AGL(1,𝔽₅))	C₄	1	4	yes	direct product factor
6	C₃×C₄×C₅×C₇	C₄	1	1	yes	characteristic, complemented
7	C₃×C₇×Dic₅	C₄	1	1	yes	characteristic, complemented
8	C₄×Dic₂₁	—	5	1	no	maximal, Hall
9	C₂×C₇⋊AGL(1,𝔽₅)	—	3	2	no	—
10	C₄×C₇×D₅	D₃	1	1	yes	characteristic
11	C₄×ΓL(1,𝔽₁₆)	—	7	1	no	maximal, Hall
12	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₃×C₇×D₅	C₂×C₄	1	2	yes	—
14	C₂×C₃×C₄×C₇	—	5	1	no	—
15	C₂×Dic₂₁	—	5	2	no	—
16	C₂×C₇×D₅	C₂×D₃	1	1	yes	characteristic
17	C₇⋊AGL(1,𝔽₅)	—	3	4	no	—
18	C₄×C₅×C₇	Dic₃	1	1	yes	characteristic, complemented
19	C₇×Dic₅	Dic₃	1	1	yes	characteristic, complemented
20	C₂×ΓL(1,𝔽₁₆)	—	7	2	no	—
21	C₃×C₄×D₅	D₇	1	1	yes	characteristic
22	C₄×Dic₇	—	15	1	no	Hall
23	C₃×C₅×C₇	C₄²	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₃×C₇	—	5	1	no	—
25	C₃×C₄×C₇	—	5	1	no	—
26	C₃×C₄×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
27	Dic₂₁	—	5	4	no	—
28	C₄×AGL(1,𝔽₅)	—	21	1	no	Hall
29	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
30	C₇×D₅	C₄×D₃	1	2	yes	—
31	C₂×C₃×D₅	C₂×D₇	1	1	yes	characteristic
32	ΓL(1,𝔽₁₆)	—	7	4	no	—
33	C₃×C₄×C₅	Dic₇	1	1	yes	characteristic, complemented
34	C₃×Dic₅	Dic₇	1	1	yes	characteristic, complemented
35	C₂×C₄×C₇	—	5	1	no	—
36	C₂×Dic₇	—	15	2	no	—
37	C₄×Dic₃	—	35	1	no	Hall
38	C₂×C₃×C₇	—	5	2	no	—
39	C₂×C₃×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₂×AGL(1,𝔽₅)	—	21	2	no	—
41	C₄×D₅	D₂₁	1	1	yes	characteristic
42	C₅×C₇	C₄×Dic₃	1	1	yes	characteristic, complemented, Hall
43	C₂×C₃×C₅	C₂×Dic₇	1	1	yes	characteristic
44	C₃×D₅	C₄×D₇	1	2	yes	—
45	C₂²×C₇	—	5	1	no	—
46	C₄×C₇	—	5	1	no	—
47	C₄×C₇	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
48	Dic₇	—	15	4	no	—
49	C₂×C₃×C₄	—	5	1	no	—
50	C₂×Dic₃	—	35	2	no	—
51	C₃×C₇	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
52	C₂×D₅	C₂×D₂₁	1	1	yes	characteristic
53	AGL(1,𝔽₅)	—	21	4	no	—
54	C₄×C₅	Dic₂₁	1	1	yes	characteristic, complemented
55	Dic₅	Dic₂₁	1	1	yes	characteristic, complemented
56	C₄²	—	105	1	no	Sylow
57	C₃×C₅	C₄×Dic₇	1	1	yes	characteristic, complemented, Hall
58	C₂×C₇	—	5	2	no	—
59	C₂×C₇	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
60	C₂²×C₃	—	5	1	no	—
61	C₃×C₄	—	5	1	no	—
62	C₃×C₄	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
63	Dic₃	—	35	4	no	—
64	C₂×C₅	C₂×Dic₂₁	1	1	yes	characteristic
65	D₅	C₄×D₂₁	1	2	yes	—
66	C₂×C₄	—	5	1	no	—
67	C₂×C₄	—	105	2	no	—
68	C₇	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₃	—	5	2	no	—
70	C₂×C₃	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₅	C₄×Dic₂₁	1	1	yes	characteristic, complemented, Sylow
72	C₂²	—	5	1	no	—
73	C₄	—	5	1	no	—
74	C₄	—	105	4	no	—
75	C₄	C₃⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic, direct product factor
76	C₃	C₄×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	5	2	no	—
78	C₂	C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₃⋊(C₇⋊AGL(1,𝔽₅))
commutator, hypocenter	C₃×C₅×C₇	C₄²
Frattini	C₂	C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))
Fitting	C₃×C₄×C₅×C₇	C₄
socle	C₂×C₃×C₅×C₇	C₂×C₄
prime radical, cosocle	C₂×C₃×C₇×D₅	C₂²
LCS	G ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
5. C₃⋊(C₇⋊AGL(1,𝔽₅))	C₄	direct (N × H)
6. C₃×C₄×C₅×C₇	C₄	semidirect
7. C₃×C₇×Dic₅	C₄	semidirect
18. C₄×C₅×C₇	Dic₃	semidirect
19. C₇×Dic₅	Dic₃	semidirect
23. C₃×C₅×C₇	C₄²	semidirect
26. C₃×C₄×C₇	AGL(1,𝔽₅)	semidirect
33. C₃×C₄×C₅	Dic₇	semidirect
34. C₃×Dic₅	Dic₇	semidirect
42. C₅×C₇	C₄×Dic₃	semidirect
47. C₄×C₇	ΓL(1,𝔽₁₆)	semidirect
51. C₃×C₇	C₄×AGL(1,𝔽₅)	semidirect
54. C₄×C₅	Dic₂₁	semidirect
55. Dic₅	Dic₂₁	semidirect
57. C₃×C₅	C₄×Dic₇	semidirect
62. C₃×C₄	C₇⋊AGL(1,𝔽₅)	semidirect
68. C₇	C₄×ΓL(1,𝔽₁₆)	semidirect
71. C₅	C₄×Dic₂₁	semidirect
75. C₄	C₃⋊(C₇⋊AGL(1,𝔽₅))	direct (N × H)
76. C₃	C₄×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
C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))	2
C₃×C₄×C₇×D₅	2
C₄×C₇⋊AGL(1,𝔽₅)	3
C₄×Dic₂₁	5
C₄×ΓL(1,𝔽₁₆)	7

Subgroups — C4×C3⋊(C7⋊AGL(1,𝔽5)) #13132626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃⋊(C₇⋊AGL(1,𝔽₅)) #13132626