Subgroups — C₇×C₃⋊(C₇⋊AGL(1,𝔽₅)) #15668831

Order 2940.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
3	C₃×C₅×C₇²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×Dic₂₁	—	5	1	no	maximal, Hall
5	C₇²×D₅	D₃	1	1	yes	characteristic
6	C₃⋊(C₇⋊AGL(1,𝔽₅))	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₇×ΓL(1,𝔽₁₆)	—	7	1	no	maximal
8	C₂×C₃×C₇²	—	5	1	no	—
9	C₅×C₇²	Dic₃	1	1	yes	characteristic, complemented, Hall
10	C₃×C₇×D₅	—	2	3	no	—
11	C₃×C₇×D₅	D₇	1	1	yes	characteristic
12	C₃×C₇×D₅	C₂×C₇	1	1	yes	characteristic
13	C₇×Dic₇	—	15	1	no	Hall
14	C₃×C₇²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
15	C₇⋊AGL(1,𝔽₅)	—	3	1	no	—
16	C₇×AGL(1,𝔽₅)	—	21	1	no	—
17	C₃×C₅×C₇	—	2	3	no	—
18	C₃×C₅×C₇	Dic₇	1	1	yes	characteristic, complemented
19	C₃×C₅×C₇	C₄×C₇	1	1	yes	characteristic, complemented
20	C₂×C₇²	—	5	1	no	—
21	Dic₂₁	—	5	1	no	—
22	C₇×Dic₃	—	35	1	no	—
23	C₇×D₅	—	2	3	no	—
24	C₇×D₅	D₃×C₇	1	1	yes	characteristic
25	C₇×D₅	D₂₁	1	1	yes	characteristic
26	ΓL(1,𝔽₁₆)	—	7	1	no	Hall
27	C₇²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₇	—	5	1	no	—
29	C₂×C₃×C₇	—	5	1	no	—
30	C₂×C₃×C₇	—	10	3	no	—
31	C₅×C₇	—	2	3	no	—
32	C₅×C₇	C₇×Dic₃	1	1	yes	characteristic, complemented
33	C₅×C₇	Dic₂₁	1	1	yes	characteristic, complemented
34	C₃×D₅	C₇×D₇	1	1	yes	characteristic
35	C₄×C₇	—	105	1	no	—
36	Dic₇	—	15	1	no	—
37	C₃×C₇	—	2	3	no	—
38	C₃×C₇	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₃×C₇	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
40	AGL(1,𝔽₅)	—	21	1	no	Hall
41	C₃×C₅	C₇×Dic₇	1	1	yes	characteristic, complemented, Hall
42	C₂×C₇	—	5	1	no	—
43	C₂×C₇	—	5	1	no	—
44	C₂×C₇	—	10	3	no	—
45	Dic₃	—	35	1	no	Hall
46	D₅	C₇×D₂₁	1	1	yes	characteristic
47	C₇	—	2	3	no	—
48	C₇	C₇×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
49	C₇	C₃⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic, direct product factor
50	C₂×C₃	—	5	1	no	—
51	C₅	C₇×Dic₂₁	1	1	yes	characteristic, complemented, Sylow
52	C₄	—	105	1	no	Sylow
53	C₃	C₇×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇	C₃⋊(C₇⋊AGL(1,𝔽₅))
commutator, hypocenter	C₃×C₅×C₇	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇²	C₄
prime radical	C₃×C₇²×D₅	C₂
cosocle	C₃×C₇×D₅	C₂×C₇
LCS	G ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₇²	C₄	semidirect
6. C₃⋊(C₇⋊AGL(1,𝔽₅))	C₇	direct (N × H)
9. C₅×C₇²	Dic₃	semidirect
14. C₃×C₇²	AGL(1,𝔽₅)	semidirect
18. C₃×C₅×C₇	Dic₇	semidirect
19. C₃×C₅×C₇	C₄×C₇	semidirect
27. C₇²	ΓL(1,𝔽₁₆)	semidirect
32. C₅×C₇	C₇×Dic₃	semidirect
33. C₅×C₇	Dic₂₁	semidirect
38. C₃×C₇	C₇×AGL(1,𝔽₅)	semidirect
39. C₃×C₇	C₇⋊AGL(1,𝔽₅)	semidirect
41. C₃×C₅	C₇×Dic₇	semidirect
48. C₇	C₇×ΓL(1,𝔽₁₆)	semidirect
49. C₇	C₃⋊(C₇⋊AGL(1,𝔽₅))	direct (N × H)
51. C₅	C₇×Dic₂₁	semidirect
53. C₃	C₇×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
C₃×C₇²×D₅	2
C₇×C₇⋊AGL(1,𝔽₅)	3
C₇×Dic₂₁	5
C₃⋊(C₇⋊AGL(1,𝔽₅))	7
C₇×ΓL(1,𝔽₁₆)	7

Subgroups — C7×C3⋊(C7⋊AGL(1,𝔽5)) #15668831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊(C₇⋊AGL(1,𝔽₅)) #15668831