Subgroups — C₃₁×C₃⋊(C₃⋊#1042) #42124955

Order 27900.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×D₅×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×C₃⋊#1042	—	3	4	no	maximal
3	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃₁×C₃⋊ΓL(1,𝔽₁₆)	—	5	1	no	maximal
5	C₃₁×C₃⋊Dic₁₅	—	5	1	no	maximal
6	C₃×C₅×D₅×C₃₁	D₃	1	4	yes	—
7	C₃₁×#1042	—	9	1	no	Hall
8	C₃²×D₅×C₃₁	D₅	1	1	yes	characteristic
9	C₂×C₃²×C₅×C₃₁	—	5	1	no	—
10	C₃×C₅²×C₃₁	Dic₃	1	4	yes	complemented
11	C₃₁×ΓL(1,𝔽₁₆)	—	15	4	no	—
12	C₃₁×Dic₁₅	—	15	4	no	—
13	C₅×D₅×C₃₁	C₃⋊D₃	1	1	yes	characteristic
14	C₃²×C₅×C₃₁	—	4	1	no	—
15	C₃²×C₅×C₃₁	Dic₅	1	1	yes	characteristic, complemented
16	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
17	C₃₁×C₃⋊Dic₃	—	25	1	no	Hall
18	C₂×C₃×C₅×C₃₁	—	5	4	no	—
19	C₃×D₅×C₃₁	D₁₅	1	4	yes	—
20	C₃⋊(C₃⋊#1042)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅²×C₃₁	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
22	AGL(1,𝔽₅)×C₃₁	—	45	1	no	—
23	Dic₅×C₃₁	—	45	1	no	—
24	C₂×C₃²×C₃₁	—	5	1	no	—
25	C₃×C₅×C₃₁	—	4	4	no	—
26	C₃×C₅×C₃₁	Dic₁₅	1	4	yes	complemented
27	C₃×C₅×C₃₁	ΓL(1,𝔽₁₆)	1	4	yes	complemented
28	C₃²×C₅×D₅	C₂×C₃₁	1	1	yes	characteristic
29	Dic₃×C₃₁	—	75	4	no	—
30	C₂×C₅×C₃₁	—	5	1	no	—
31	D₅×C₃₁	C₃⋊D₁₅	1	1	yes	characteristic
32	C₃⋊#1042	—	3	4	no	—
33	C₃²×C₃₁	#1042	1	1	yes	characteristic, complemented, Hall
34	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₃₁	—	5	4	no	—
36	C₃⋊ΓL(1,𝔽₁₆)	—	5	1	no	—
37	C₃⋊Dic₁₅	—	5	1	no	—
38	C₅×C₃₁	—	4	1	no	—
39	C₅×C₃₁	C₃⋊Dic₁₅	1	1	yes	characteristic, complemented
40	C₅×C₃₁	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
41	C₃×C₅×D₅	D₃×C₃₁	1	4	yes	—
42	C₄×C₃₁	—	225	1	no	Hall
43	#1042	—	9	1	no	Hall
44	C₃×C₃₁	C₃⋊#1042	1	4	yes	complemented
45	C₂×C₃²×C₅	—	5	1	no	—
46	C₃²×D₅	D₅×C₃₁	1	1	yes	characteristic
47	C₃×C₅²	Dic₃×C₃₁	1	4	yes	complemented
48	C₂×C₃₁	—	5	1	no	—
49	ΓL(1,𝔽₁₆)	—	15	4	no	—
50	Dic₁₅	—	15	4	no	—
51	C₅×D₅	C₃⋊D₃×C₃₁	1	1	yes	characteristic
52	C₃²×C₅	—	4	1	no	—
53	C₃²×C₅	Dic₅×C₃₁	1	1	yes	characteristic, complemented
54	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	1	yes	characteristic, complemented
55	C₃⋊Dic₃	—	25	1	no	Hall
56	C₃₁	C₃⋊(C₃⋊#1042)	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×C₅	—	5	4	no	—
58	C₃×D₅	D₁₅×C₃₁	1	4	yes	—
59	C₅²	C₃₁×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
60	AGL(1,𝔽₅)	—	45	1	no	—
61	Dic₅	—	45	1	no	—
62	C₂×C₃²	—	5	1	no	—
63	C₃×C₅	—	4	4	no	—
64	C₃×C₅	C₃₁×Dic₁₅	1	4	yes	complemented
65	C₃×C₅	C₃₁×ΓL(1,𝔽₁₆)	1	4	yes	complemented
66	Dic₃	—	75	4	no	—
67	C₂×C₅	—	5	1	no	—
68	D₅	C₃₁×C₃⋊D₁₅	1	1	yes	characteristic
69	C₃²	C₃₁×#1042	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	5	4	no	—
71	C₅	—	4	1	no	—
72	C₅	C₃₁×C₃⋊Dic₁₅	1	1	yes	characteristic, complemented
73	C₅	C₃₁×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
74	C₄	—	225	1	no	Sylow
75	C₃	C₃₁×C₃⋊#1042	1	4	yes	complemented
76	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁	C₃⋊(C₃⋊#1042)
commutator, hypocenter	C₃²×C₅²	C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₃²×C₅×D₅×C₃₁	C₂
cosocle	C₃²×C₅×D₅	C₂×C₃₁
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅²×C₃₁	C₄	semidirect
10. C₃×C₅²×C₃₁	Dic₃	semidirect
15. C₃²×C₅×C₃₁	Dic₅	semidirect
16. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
20. C₃⋊(C₃⋊#1042)	C₃₁	direct (N × H)
21. C₅²×C₃₁	C₃⋊Dic₃	semidirect
26. C₃×C₅×C₃₁	Dic₁₅	semidirect
27. C₃×C₅×C₃₁	ΓL(1,𝔽₁₆)	semidirect
33. C₃²×C₃₁	#1042	semidirect
34. C₃²×C₅²	C₄×C₃₁	semidirect
39. C₅×C₃₁	C₃⋊Dic₁₅	semidirect
40. C₅×C₃₁	C₃⋊ΓL(1,𝔽₁₆)	semidirect
44. C₃×C₃₁	C₃⋊#1042	semidirect
47. C₃×C₅²	Dic₃×C₃₁	semidirect
53. C₃²×C₅	Dic₅×C₃₁	semidirect
54. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
56. C₃₁	C₃⋊(C₃⋊#1042)	direct (N × H)
59. C₅²	C₃₁×C₃⋊Dic₃	semidirect
64. C₃×C₅	C₃₁×Dic₁₅	semidirect
65. C₃×C₅	C₃₁×ΓL(1,𝔽₁₆)	semidirect
69. C₃²	C₃₁×#1042	semidirect
72. C₅	C₃₁×C₃⋊Dic₁₅	semidirect
73. C₅	C₃₁×C₃⋊ΓL(1,𝔽₁₆)	semidirect
75. C₃	C₃₁×C₃⋊#1042	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×D₅×C₃₁	2
C₃₁×C₃⋊#1042	3
C₃₁×C₃⋊ΓL(1,𝔽₁₆)	5
C₃₁×C₃⋊Dic₁₅	5
C₃⋊(C₃⋊#1042)	31

Subgroups — C31×C3⋊(C3⋊#1042) #42124955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×C₃⋊(C₃⋊#1042) #42124955