Subgroups — C₁₇×C₂²⋊((C₃₁⋊C₃)⋊C₅) #44539298

Order 31620.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₇×ΓL(1,𝔽₃₂)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₇×(C₃₁⋊C₃)⋊C₅	—	4	1	no	maximal, Hall
3	C₁₇×C₂²⋊(C₃₁⋊C₃)	C₅	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₁₇×ΓL(1,𝔽₃₂)	—	3	1	no	—
5	C₁₇×ΓL(1,𝔽₃₂)	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₁₇×C₃₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
7	C₂²⋊((C₃₁⋊C₃)⋊C₅)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₁₇×C₃₁⋊C₃	—	4	1	no	Hall
9	C₂×C₁₇×C₃₁	—	3	1	no	—
10	C₅×A₄×C₁₇	—	31	1	no	maximal, Hall
11	C₂²×ΓL(1,𝔽₃₂)	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
12	C₁₇×C₃₁	C₅×A₄	1	1	yes	characteristic, complemented, Hall
13	(C₃₁⋊C₃)⋊C₅	—	4	1	no	Hall
14	C₂²⋊(C₃₁⋊C₃)	C₅×C₁₇	1	1	yes	characteristic, complemented, Hall
15	C₂²×C₅×C₁₇	—	31	1	no	Hall
16	C₂×ΓL(1,𝔽₃₂)	—	3	1	no	—
17	C₃×C₅×C₁₇	—	124	1	no	Hall
18	A₄×C₁₇	—	31	1	no	Hall
19	C₂×C₅×C₁₇	—	93	1	no	—
20	ΓL(1,𝔽₃₂)	A₄×C₁₇	1	1	yes	characteristic, complemented, Hall
21	C₂²×C₃₁	C₃×C₅×C₁₇	1	1	yes	characteristic, complemented, Hall
22	C₃₁⋊C₃	—	4	1	no	Hall
23	C₅×C₁₇	—	31	1	no	Hall
24	C₂²×C₁₇	(C₃₁⋊C₃)⋊C₅	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃₁	—	3	1	no	—
26	C₅×A₄	—	31	1	no	Hall
27	C₃×C₁₇	—	124	1	no	Hall
28	C₂×C₁₇	—	3	1	no	—
29	C₃₁	C₅×A₄×C₁₇	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₅	—	31	1	no	Hall
31	C₁₇	C₂²⋊((C₃₁⋊C₃)⋊C₅)	1	1	yes	characteristic, direct product factor, Sylow
32	C₃×C₅	—	124	1	no	Hall
33	A₄	—	31	1	no	Hall
34	C₂×C₅	—	93	1	no	—
35	C₅	—	31	1	no	Sylow
36	C₂²	C₁₇×(C₃₁⋊C₃)⋊C₅	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	124	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇	C₂²⋊((C₃₁⋊C₃)⋊C₅)
commutator, cosocle, hypocenter	C₂²×C₃₁	C₃×C₅×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₁₇×C₃₁	C₃×C₅
LCS	G ▷ C₂²×C₃₁
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₇×ΓL(1,𝔽₃₂)	C₃	semidirect
3. C₁₇×C₂²⋊(C₃₁⋊C₃)	C₅	semidirect
5. C₁₇×ΓL(1,𝔽₃₂)	A₄	semidirect
6. C₂²×C₁₇×C₃₁	C₃×C₅	semidirect
7. C₂²⋊((C₃₁⋊C₃)⋊C₅)	C₁₇	direct (N × H)
11. C₂²×ΓL(1,𝔽₃₂)	C₃×C₁₇	semidirect
12. C₁₇×C₃₁	C₅×A₄	semidirect
14. C₂²⋊(C₃₁⋊C₃)	C₅×C₁₇	semidirect
20. ΓL(1,𝔽₃₂)	A₄×C₁₇	semidirect
21. C₂²×C₃₁	C₃×C₅×C₁₇	semidirect
24. C₂²×C₁₇	(C₃₁⋊C₃)⋊C₅	semidirect
29. C₃₁	C₅×A₄×C₁₇	semidirect
31. C₁₇	C₂²⋊((C₃₁⋊C₃)⋊C₅)	direct (N × H)
36. C₂²	C₁₇×(C₃₁⋊C₃)⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₇×ΓL(1,𝔽₃₂)	3
C₁₇×(C₃₁⋊C₃)⋊C₅	4
C₁₇×C₂²⋊(C₃₁⋊C₃)	5
C₂²⋊((C₃₁⋊C₃)⋊C₅)	17
C₅×A₄×C₁₇	31

Subgroups — C17×C22⋊((C31⋊C3)⋊C5) #44539298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×C₂²⋊((C₃₁⋊C₃)⋊C₅) #44539298