Subgroups — C₃₁×C₃⋊(C₃⋊#1035) #42124432

Order 27900.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁×D₂₅	C₂	1	1	yes	maximal, characteristic
2	C₃₁×C₃⋊#1035	—	3	4	no	maximal
3	C₃²×C₂₅×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃₁×C₃⋊ΓL(1,𝔽₁₆)	—	5	1	no	maximal
5	C₃×C₃₁×D₂₅	D₃	1	4	yes	—
6	C₃₁×#1035	—	9	1	no	Hall
7	C₃²×D₅×C₃₁	—	5	1	no	—
8	C₃×C₂₅×C₃₁	Dic₃	1	4	yes	complemented
9	C₃₁×ΓL(1,𝔽₁₆)	—	15	4	no	—
10	C₃₁×D₂₅	C₃⋊D₃	1	1	yes	characteristic
11	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	1	yes	characteristic
12	C₃₁×C₃⋊Dic₃	—	25	1	no	Hall
13	C₃×D₅×C₃₁	—	5	4	no	—
14	C₃⋊(C₃⋊#1035)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂₅×C₃₁	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
16	AGL(1,𝔽₅)×C₃₁	—	45	1	no	—
17	C₂×C₃²×C₃₁	—	25	1	no	—
18	C₃×C₅×C₃₁	ΓL(1,𝔽₁₆)	1	4	yes	—
19	C₃²×D₂₅	C₂×C₃₁	1	1	yes	characteristic
20	Dic₃×C₃₁	—	75	4	no	—
21	D₅×C₃₁	—	5	1	no	—
22	C₃⋊#1035	—	3	4	no	—
23	C₃²×C₃₁	#1035	1	1	yes	characteristic, complemented, Hall
24	C₃²×C₂₅	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₃₁	—	25	4	no	—
26	C₃⋊ΓL(1,𝔽₁₆)	—	5	1	no	—
27	C₅×C₃₁	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
28	C₃×D₂₅	D₃×C₃₁	1	4	yes	—
29	C₄×C₃₁	—	225	1	no	Hall
30	#1035	—	9	1	no	Hall
31	C₃×C₃₁	C₃⋊#1035	1	4	yes	complemented
32	C₃²×D₅	—	5	1	no	—
33	C₃×C₂₅	Dic₃×C₃₁	1	4	yes	complemented
34	C₂×C₃₁	—	25	1	no	—
35	ΓL(1,𝔽₁₆)	—	15	4	no	—
36	D₂₅	C₃⋊D₃×C₃₁	1	1	yes	characteristic
37	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	1	yes	characteristic
38	C₃⋊Dic₃	—	25	1	no	Hall
39	C₃₁	C₃⋊(C₃⋊#1035)	1	1	yes	characteristic, direct product factor, Sylow
40	C₃×D₅	—	5	4	no	—
41	C₂₅	C₃₁×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
42	AGL(1,𝔽₅)	—	45	1	no	—
43	C₂×C₃²	—	25	1	no	—
44	C₃×C₅	C₃₁×ΓL(1,𝔽₁₆)	1	4	yes	—
45	Dic₃	—	75	4	no	—
46	D₅	—	5	1	no	—
47	C₃²	C₃₁×#1035	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	—	25	4	no	—
49	C₅	C₃₁×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
50	C₄	—	225	1	no	Sylow
51	C₃	C₃₁×C₃⋊#1035	1	4	yes	complemented
52	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁	C₃⋊(C₃⋊#1035)
commutator, hypocenter	C₃²×C₂₅	C₄×C₃₁
Frattini	C₅	C₃₁×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₃²×C₂₅×C₃₁	C₄
socle	C₃²×C₅×C₃₁	AGL(1,𝔽₅)
prime radical	C₃²×C₃₁×D₂₅	C₂
cosocle	C₃²×D₂₅	C₂×C₃₁
LCS	G ▷ C₃²×C₂₅
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
3. C₃²×C₂₅×C₃₁	C₄	semidirect
8. C₃×C₂₅×C₃₁	Dic₃	semidirect
14. C₃⋊(C₃⋊#1035)	C₃₁	direct (N × H)
15. C₂₅×C₃₁	C₃⋊Dic₃	semidirect
23. C₃²×C₃₁	#1035	semidirect
24. C₃²×C₂₅	C₄×C₃₁	semidirect
31. C₃×C₃₁	C₃⋊#1035	semidirect
33. C₃×C₂₅	Dic₃×C₃₁	semidirect
39. C₃₁	C₃⋊(C₃⋊#1035)	direct (N × H)
41. C₂₅	C₃₁×C₃⋊Dic₃	semidirect
47. C₃²	C₃₁×#1035	semidirect
51. C₃	C₃₁×C₃⋊#1035	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃₁×D₂₅	2
C₃₁×C₃⋊#1035	3
C₃₁×C₃⋊ΓL(1,𝔽₁₆)	5
C₃⋊(C₃⋊#1035)	31

Subgroups — C31×C3⋊(C3⋊#1035) #42124432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×C₃⋊(C₃⋊#1035) #42124432