Subgroups — C₁₇×#13049959 #36349010

Order 22134.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#10620253	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₇×C₁₇×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₇×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₁₇×C₃₁⋊C₃	—	7	1	no	maximal, Hall
5	C₁₇×C₃₁⋊C₃	—	7	1	no	Hall
6	#13049959	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₁₇×C₃₁	—	7	1	no	Hall
8	C₁₇×AGL(1,𝔽₇)	—	31	1	no	maximal, Hall
9	#10620253	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
10	C₁₇×C₃₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
11	D₇×C₃₁	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
12	C₁₇×ΓL(1,𝔽₈)	—	31	1	no	Hall
13	D₇×C₁₇	C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
14	C₇×C₃₁	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃₁⋊C₃	—	7	1	no	Hall
16	C₇×C₁₇	C₂×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₁₇	—	217	1	no	Hall
18	C₃₁⋊C₃	—	7	1	no	Hall
19	C₂×C₃₁	—	7	1	no	Hall
20	C₃×C₁₇	—	217	1	no	Hall
21	AGL(1,𝔽₇)	—	31	1	no	Hall
22	C₂×C₁₇	—	7	1	no	Hall
23	C₃₁	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
24	ΓL(1,𝔽₈)	—	31	1	no	Hall
25	C₁₇	#13049959	1	1	yes	characteristic, direct product factor, Sylow
26	D₇	C₁₇×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
27	C₇	C₂×C₁₇×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	217	1	no	Hall
29	C₃	—	217	1	no	Sylow
30	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₇	#13049959
commutator, cosocle, hypocenter	C₇×C₃₁	C₂×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₃₁	C₂×C₃
LCS	G ▷ C₇×C₃₁
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₁₇×#10620253	C₂	semidirect
2. D₇×C₁₇×C₃₁	C₃	semidirect
3. C₇×C₁₇×C₃₁	C₂×C₃	semidirect
6. #13049959	C₁₇	direct (N × H)
9. #10620253	C₂×C₁₇	semidirect
10. C₁₇×C₃₁	AGL(1,𝔽₇)	semidirect
11. D₇×C₃₁	C₃×C₁₇	semidirect
13. D₇×C₁₇	C₃₁⋊C₃	semidirect
14. C₇×C₃₁	C₂×C₃×C₁₇	semidirect
16. C₇×C₁₇	C₂×C₃₁⋊C₃	semidirect
23. C₃₁	C₁₇×AGL(1,𝔽₇)	semidirect
25. C₁₇	#13049959	direct (N × H)
26. D₇	C₁₇×C₃₁⋊C₃	semidirect
27. C₇	C₂×C₁₇×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×#10620253	2
D₇×C₁₇×C₃₁	3
C₂×C₁₇×C₃₁⋊C₃	7
#13049959	17
C₁₇×AGL(1,𝔽₇)	31

Subgroups — C17×#13049959 #36349010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13049959 #36349010