Subgroups — C₃²×C₃₁×#1044 #42124895

Order 27900.

192 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	C₃×C₃₁×#1044	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×AGL(1,𝔽₅)×C₃₁	—	5	2	no	maximal
5	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	4	yes	—
6	C₃₁×#1044	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×D₅×C₃₁	—	5	2	no	—
8	C₃²×D₅×C₃₁	—	10	2	no	—
9	C₃×C₅²×C₃₁	C₃×C₄	1	4	yes	complemented
10	C₃×AGL(1,𝔽₅)×C₃₁	—	5	8	no	—
11	C₃₁×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic
12	C₃²×C₅×C₃₁	—	2	2	no	—
13	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	2	yes	complemented
14	C₃²×C₄×C₃₁	—	25	1	no	Hall
15	C₃×D₅×C₃₁	—	5	8	no	—
16	C₃×D₅×C₃₁	—	10	8	no	—
17	C₃²×#1044	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅²×C₃₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
19	AGL(1,𝔽₅)×C₃₁	—	5	2	no	—
20	C₂×C₃²×C₃₁	—	25	1	no	—
21	C₃×C₅×C₃₁	—	2	8	no	—
22	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	8	yes	complemented
23	C₃²×C₅⋊D₅	C₂×C₃₁	1	1	yes	characteristic
24	C₃×C₄×C₃₁	—	25	4	no	—
25	D₅×C₃₁	—	5	2	no	—
26	D₅×C₃₁	—	10	2	no	—
27	C₃×#1044	C₃×C₃₁	1	4	yes	direct product factor
28	C₃²×C₃₁	#1044	1	1	yes	characteristic, direct product factor, Hall
29	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₃₁	—	25	4	no	—
31	C₃²×AGL(1,𝔽₅)	—	5	2	no	—
32	C₅×C₃₁	—	2	2	no	—
33	C₅×C₃₁	C₃²×AGL(1,𝔽₅)	1	2	yes	complemented
34	C₃×C₅⋊D₅	C₂×C₃×C₃₁	1	4	yes	—
35	C₄×C₃₁	—	25	1	no	Hall
36	#1044	C₃²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
37	C₃×C₃₁	C₃×#1044	1	4	yes	direct product factor
38	C₃²×D₅	—	5	2	no	—
39	C₃²×D₅	—	10	2	no	—
40	C₃×C₅²	C₃×C₄×C₃₁	1	4	yes	complemented
41	C₂×C₃₁	—	25	1	no	—
42	C₃×AGL(1,𝔽₅)	—	5	8	no	—
43	C₅⋊D₅	C₂×C₃²×C₃₁	1	1	yes	characteristic
44	C₃²×C₅	—	2	2	no	—
45	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	2	yes	complemented
46	C₃²×C₄	—	25	1	no	Hall
47	C₃₁	C₃²×#1044	1	1	yes	characteristic, direct product factor, Sylow
48	C₃×D₅	—	5	8	no	—
49	C₃×D₅	—	10	8	no	—
50	C₅²	C₃²×C₄×C₃₁	1	1	yes	characteristic, complemented, Sylow
51	AGL(1,𝔽₅)	—	5	2	no	—
52	C₂×C₃²	—	25	1	no	—
53	C₃×C₅	—	2	8	no	—
54	C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	1	8	yes	complemented
55	C₃×C₄	—	25	4	no	—
56	D₅	—	5	2	no	—
57	D₅	—	10	2	no	—
58	C₃²	C₃₁×#1044	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃	—	25	4	no	—
60	C₅	—	2	2	no	—
61	C₅	C₃²×AGL(1,𝔽₅)×C₃₁	1	2	yes	complemented
62	C₄	—	25	1	no	Sylow
63	C₃	C₃×C₃₁×#1044	1	4	yes	direct product factor
64	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₃₁	#1044
commutator, hypocenter	C₅²	C₃²×C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₃²×C₃₁×C₅⋊D₅	C₂
cosocle	C₅⋊D₅	C₂×C₃²×C₃₁
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃²×C₃₁

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₁×#1044	C₃	direct (N × H)
3. C₃²×C₅²×C₃₁	C₄	semidirect
6. C₃₁×#1044	C₃²	direct (N × H)
9. C₃×C₅²×C₃₁	C₃×C₄	semidirect
13. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
17. C₃²×#1044	C₃₁	direct (N × H)
18. C₅²×C₃₁	C₃²×C₄	semidirect
22. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
27. C₃×#1044	C₃×C₃₁	direct (N × H)
28. C₃²×C₃₁	#1044	direct (N × H)
29. C₃²×C₅²	C₄×C₃₁	semidirect
33. C₅×C₃₁	C₃²×AGL(1,𝔽₅)	semidirect
36. #1044	C₃²×C₃₁	direct (N × H)
37. C₃×C₃₁	C₃×#1044	direct (N × H)
40. C₃×C₅²	C₃×C₄×C₃₁	semidirect
45. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
47. C₃₁	C₃²×#1044	direct (N × H)
50. C₅²	C₃²×C₄×C₃₁	semidirect
54. C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	semidirect
58. C₃²	C₃₁×#1044	direct (N × H)
61. C₅	C₃²×AGL(1,𝔽₅)×C₃₁	semidirect
63. C₃	C₃×C₃₁×#1044	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
C₃×C₃₁×#1044	3
C₃²×AGL(1,𝔽₅)×C₃₁	5
C₃²×#1044	31

Subgroups — C32×C31×#1044 #42124895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₁×#1044 #42124895