Subgroups — C₂×C₅×AGL(1,𝔽₁₇) #72244751

Order 2720.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#3633	C₂	1	1	yes	maximal, characteristic
2	C₅×AGL(1,𝔽₁₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×#595	C₄	1	1	yes	characteristic
4	C₅×#3633	C₄	1	1	yes	characteristic
5	C₅×#3633	C₂²	1	1	yes	characteristic
6	C₂×AGL(1,𝔽₁₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₅×D₁₇	C₈	1	1	yes	characteristic
8	C₅×#595	C₈	1	1	yes	characteristic
9	C₅×#595	C₂×C₄	1	1	yes	characteristic
10	C₂×#3633	C₂×C₅	1	1	yes	characteristic
11	AGL(1,𝔽₁₇)	C₂×C₅	1	2	yes	direct product factor
12	C₂×C₅×C₁₇	C₁₆	1	1	yes	characteristic, complemented
13	C₅×D₁₇	C₁₆	1	1	yes	characteristic, complemented
14	C₅×D₁₇	C₂×C₈	1	1	yes	characteristic
15	C₂×C₅×C₁₆	—	17	1	no	maximal, Hall
16	C₂×#595	C₄×C₅	1	1	yes	characteristic
17	#3633	C₄×C₅	1	1	yes	characteristic
18	#3633	C₂²×C₅	1	1	yes	characteristic
19	C₅×C₁₇	C₂×C₁₆	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×C₈	—	17	1	no	—
21	C₅×C₁₆	—	17	2	no	—
22	C₂×D₁₇	C₅×C₈	1	1	yes	characteristic
23	#595	C₅×C₈	1	1	yes	characteristic
24	#595	C₂×C₄×C₅	1	1	yes	characteristic
25	C₂×C₄×C₅	—	17	1	no	—
26	C₅×C₈	—	17	1	no	—
27	C₅×C₈	—	17	1	no	—
28	C₂×C₁₇	C₅×C₁₆	1	1	yes	characteristic, complemented
29	D₁₇	C₅×C₁₆	1	1	yes	characteristic, complemented
30	D₁₇	C₂×C₅×C₈	1	1	yes	characteristic
31	C₂×C₁₆	—	17	1	no	Sylow
32	C₂²×C₅	—	17	1	no	—
33	C₄×C₅	—	17	1	no	—
34	C₄×C₅	—	17	1	no	—
35	C₁₇	C₂×C₅×C₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₈	—	17	1	no	—
37	C₁₆	—	17	2	no	—
38	C₂×C₅	—	17	1	no	—
39	C₂×C₅	—	17	1	no	—
40	C₂×C₅	AGL(1,𝔽₁₇)	1	1	yes	characteristic, direct product factor
41	C₂×C₄	—	17	1	no	—
42	C₈	—	17	1	no	—
43	C₈	—	17	1	no	—
44	C₅	C₂×AGL(1,𝔽₁₇)	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²	—	17	1	no	—
46	C₄	—	17	1	no	—
47	C₄	—	17	1	no	—
48	C₂	—	17	1	no	—
49	C₂	—	17	1	no	—
50	C₂	C₅×AGL(1,𝔽₁₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	AGL(1,𝔽₁₇)
commutator, hypocenter	C₁₇	C₂×C₅×C₁₆
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₇	C₁₆
prime radical	C₂×C₅×D₁₇	C₈
cosocle	#3633	C₂²×C₅
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×AGL(1,𝔽₁₇)	C₂	direct (N × H)
6. C₂×AGL(1,𝔽₁₇)	C₅	direct (N × H)
11. AGL(1,𝔽₁₇)	C₂×C₅	direct (N × H)
12. C₂×C₅×C₁₇	C₁₆	semidirect
13. C₅×D₁₇	C₁₆	semidirect
19. C₅×C₁₇	C₂×C₁₆	semidirect
28. C₂×C₁₇	C₅×C₁₆	semidirect
29. D₁₇	C₅×C₁₆	semidirect
35. C₁₇	C₂×C₅×C₁₆	semidirect
40. C₂×C₅	AGL(1,𝔽₁₇)	direct (N × H)
44. C₅	C₂×AGL(1,𝔽₁₇)	direct (N × H)
50. C₂	C₅×AGL(1,𝔽₁₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#3633	2
C₅×AGL(1,𝔽₁₇)	2
C₂×AGL(1,𝔽₁₇)	5
C₂×C₅×C₁₆	17

Subgroups — C2×C5×AGL(1,𝔽17) #72244751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×AGL(1,𝔽₁₇) #72244751