Subgroups — C₃×C₃₁×C₃⋊#1043 #42124919

Order 27900.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₃₁×C₃⋊#1043	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₃₁×#1043	—	3	1	no	maximal
4	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₃₁×ΓL(1,𝔽₁₆)	—	5	6	no	maximal
6	C₃×C₃₁×C₅⋊D₅	—	2	1	no	—
7	C₃×C₃₁×C₅⋊D₅	D₃	1	1	yes	characteristic
8	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
9	C₃₁×#1043	—	3	1	no	Hall
10	C₃²×D₅×C₃₁	—	5	6	no	—
11	C₃×C₅²×C₃₁	—	2	1	no	—
12	C₃×C₅²×C₃₁	Dic₃	1	1	yes	characteristic, complemented
13	C₃×C₅²×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
14	C₃₁×ΓL(1,𝔽₁₆)	—	5	6	no	—
15	C₃×AGL(1,𝔽₅)×C₃₁	—	15	6	no	—
16	C₃₁×C₅⋊D₅	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	6	yes	complemented
18	C₃×Dic₃×C₃₁	—	25	1	no	Hall
19	C₃×D₅×C₃₁	—	5	6	no	—
20	C₃×D₅×C₃₁	—	5	6	no	—
21	C₃×D₅×C₃₁	—	10	6	no	—
22	C₃×C₃⋊#1043	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅²×C₃₁	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
24	AGL(1,𝔽₅)×C₃₁	—	15	6	no	—
25	C₂×C₃²×C₃₁	—	25	1	no	—
26	C₃×C₅×C₃₁	—	2	6	no	—
27	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	6	yes	complemented
28	C₃×C₅×C₃₁	ΓL(1,𝔽₁₆)	1	6	yes	complemented
29	C₃²×C₅⋊D₅	C₂×C₃₁	1	1	yes	characteristic
30	C₃×C₄×C₃₁	—	75	1	no	—
31	Dic₃×C₃₁	—	25	1	no	—
32	D₅×C₃₁	—	5	6	no	—
33	C₃⋊#1043	C₃×C₃₁	1	1	yes	characteristic, direct product factor
34	C₃×#1043	—	3	1	no	—
35	C₃²×C₃₁	#1043	1	1	yes	characteristic, complemented, Hall
36	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₁	—	25	1	no	—
38	C₂×C₃×C₃₁	—	25	1	no	—
39	C₂×C₃×C₃₁	—	50	1	no	—
40	C₃×ΓL(1,𝔽₁₆)	—	5	6	no	—
41	C₅×C₃₁	C₃×ΓL(1,𝔽₁₆)	1	6	yes	complemented
42	C₃×C₅⋊D₅	—	2	1	no	—
43	C₃×C₅⋊D₅	D₃×C₃₁	1	1	yes	characteristic
44	C₃×C₅⋊D₅	C₂×C₃×C₃₁	1	1	yes	characteristic
45	C₄×C₃₁	—	75	1	no	Hall
46	#1043	—	3	1	no	Hall
47	C₃×C₃₁	—	2	1	no	—
48	C₃×C₃₁	C₃×#1043	1	1	yes	characteristic, complemented
49	C₃×C₃₁	C₃⋊#1043	1	1	yes	characteristic, direct product factor
50	C₃²×D₅	—	5	6	no	—
51	C₃×C₅²	—	2	1	no	—
52	C₃×C₅²	Dic₃×C₃₁	1	1	yes	characteristic, complemented
53	C₃×C₅²	C₃×C₄×C₃₁	1	1	yes	characteristic, complemented
54	C₂×C₃₁	—	25	1	no	—
55	ΓL(1,𝔽₁₆)	—	5	6	no	—
56	C₃×AGL(1,𝔽₅)	—	15	6	no	—
57	C₅⋊D₅	C₃×D₃×C₃₁	1	1	yes	characteristic
58	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	6	yes	complemented
59	C₃×Dic₃	—	25	1	no	Hall
60	C₃₁	C₃×C₃⋊#1043	1	1	yes	characteristic, direct product factor, Sylow
61	C₃×D₅	—	5	6	no	—
62	C₃×D₅	—	5	6	no	—
63	C₃×D₅	—	10	6	no	—
64	C₅²	C₃×Dic₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
65	AGL(1,𝔽₅)	—	15	6	no	—
66	C₂×C₃²	—	25	1	no	—
67	C₃×C₅	—	2	6	no	—
68	C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	1	6	yes	complemented
69	C₃×C₅	C₃₁×ΓL(1,𝔽₁₆)	1	6	yes	complemented
70	C₃×C₄	—	75	1	no	—
71	Dic₃	—	25	1	no	—
72	D₅	—	5	6	no	—
73	C₃²	C₃₁×#1043	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	25	1	no	—
75	C₂×C₃	—	25	1	no	—
76	C₂×C₃	—	50	1	no	—
77	C₅	C₃×C₃₁×ΓL(1,𝔽₁₆)	1	6	yes	complemented
78	C₄	—	75	1	no	Sylow
79	C₃	—	2	1	no	—
80	C₃	C₃×C₃₁×#1043	1	1	yes	characteristic, complemented
81	C₃	C₃₁×C₃⋊#1043	1	1	yes	characteristic, direct product factor
82	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₃₁	C₃⋊#1043
commutator, hypocenter	C₃×C₅²	C₃×C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₃²×C₃₁×C₅⋊D₅	C₂
cosocle	C₃×C₅⋊D₅	C₂×C₃×C₃₁
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₃×C₃₁

Complements

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

N	H ≅ G/N	type
2. C₃₁×C₃⋊#1043	C₃	direct (N × H)
4. C₃²×C₅²×C₃₁	C₄	semidirect
12. C₃×C₅²×C₃₁	Dic₃	semidirect
13. C₃×C₅²×C₃₁	C₃×C₄	semidirect
17. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
22. C₃×C₃⋊#1043	C₃₁	direct (N × H)
23. C₅²×C₃₁	C₃×Dic₃	semidirect
27. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
28. C₃×C₅×C₃₁	ΓL(1,𝔽₁₆)	semidirect
33. C₃⋊#1043	C₃×C₃₁	direct (N × H)
35. C₃²×C₃₁	#1043	semidirect
36. C₃²×C₅²	C₄×C₃₁	semidirect
41. C₅×C₃₁	C₃×ΓL(1,𝔽₁₆)	semidirect
48. C₃×C₃₁	C₃×#1043	semidirect
49. C₃×C₃₁	C₃⋊#1043	direct (N × H)
52. C₃×C₅²	Dic₃×C₃₁	semidirect
53. C₃×C₅²	C₃×C₄×C₃₁	semidirect
58. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
60. C₃₁	C₃×C₃⋊#1043	direct (N × H)
64. C₅²	C₃×Dic₃×C₃₁	semidirect
68. C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	semidirect
69. C₃×C₅	C₃₁×ΓL(1,𝔽₁₆)	semidirect
73. C₃²	C₃₁×#1043	semidirect
77. C₅	C₃×C₃₁×ΓL(1,𝔽₁₆)	semidirect
80. C₃	C₃×C₃₁×#1043	semidirect
81. C₃	C₃₁×C₃⋊#1043	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₃₁×C₅⋊D₅	2
C₃₁×C₃⋊#1043	3
C₃×C₃₁×#1043	3
C₃×C₃₁×ΓL(1,𝔽₁₆)	5
C₃×C₃⋊#1043	31

Subgroups — C3×C31×C3⋊#1043 #42124919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₁×C₃⋊#1043 #42124919