Subgroups — C₃²×C₃₁×#1042 #42124887

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	C₃×C₃₁×#1042	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	1	no	maximal
5	C₃²×Dic₅×C₃₁	—	5	1	no	maximal
6	C₃×C₅×D₅×C₃₁	C₂×C₃	1	4	yes	—
7	C₃₁×#1042	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×D₅×C₃₁	D₅	1	1	yes	characteristic
9	C₂×C₃²×C₅×C₃₁	—	5	1	no	—
10	C₃×C₅²×C₃₁	C₃×C₄	1	4	yes	complemented
11	C₃×AGL(1,𝔽₅)×C₃₁	—	5	4	no	—
12	C₃×Dic₅×C₃₁	—	5	4	no	—
13	C₅×D₅×C₃₁	C₂×C₃²	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	C₃²×C₄×C₃₁	—	25	1	no	Hall
18	C₂×C₃×C₅×C₃₁	—	5	4	no	—
19	C₃×D₅×C₃₁	C₃×D₅	1	4	yes	—
20	C₃²×#1042	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅²×C₃₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
22	AGL(1,𝔽₅)×C₃₁	—	5	1	no	—
23	Dic₅×C₃₁	—	5	1	no	—
24	C₂×C₃²×C₃₁	—	5	1	no	—
25	C₃×C₅×C₃₁	—	4	4	no	—
26	C₃×C₅×C₃₁	C₃×Dic₅	1	4	yes	complemented
27	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	4	yes	complemented
28	C₃²×C₅×D₅	C₂×C₃₁	1	1	yes	characteristic
29	C₃×C₄×C₃₁	—	25	4	no	—
30	C₂×C₅×C₃₁	—	5	1	no	—
31	D₅×C₃₁	C₃²×D₅	1	1	yes	characteristic
32	C₃×#1042	C₃×C₃₁	1	4	yes	direct product factor
33	C₃²×C₃₁	#1042	1	1	yes	characteristic, direct product factor, Hall
34	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₃₁	—	5	4	no	—
36	C₃²×AGL(1,𝔽₅)	—	5	1	no	—
37	C₃²×Dic₅	—	5	1	no	—
38	C₅×C₃₁	—	4	1	no	—
39	C₅×C₃₁	C₃²×Dic₅	1	1	yes	characteristic, complemented
40	C₅×C₃₁	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
41	C₃×C₅×D₅	C₂×C₃×C₃₁	1	4	yes	—
42	C₄×C₃₁	—	25	1	no	Hall
43	#1042	C₃²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
44	C₃×C₃₁	C₃×#1042	1	4	yes	direct product factor
45	C₂×C₃²×C₅	—	5	1	no	—
46	C₃²×D₅	D₅×C₃₁	1	1	yes	characteristic
47	C₃×C₅²	C₃×C₄×C₃₁	1	4	yes	complemented
48	C₂×C₃₁	—	5	1	no	—
49	C₃×AGL(1,𝔽₅)	—	5	4	no	—
50	C₃×Dic₅	—	5	4	no	—
51	C₅×D₅	C₂×C₃²×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	C₃²×C₄	—	25	1	no	Hall
56	C₃₁	C₃²×#1042	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×C₅	—	5	4	no	—
58	C₃×D₅	C₃×D₅×C₃₁	1	4	yes	—
59	C₅²	C₃²×C₄×C₃₁	1	1	yes	characteristic, complemented, Sylow
60	AGL(1,𝔽₅)	—	5	1	no	—
61	Dic₅	—	5	1	no	—
62	C₂×C₃²	—	5	1	no	—
63	C₃×C₅	—	4	4	no	—
64	C₃×C₅	C₃×Dic₅×C₃₁	1	4	yes	complemented
65	C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	1	4	yes	complemented
66	C₃×C₄	—	25	4	no	—
67	C₂×C₅	—	5	1	no	—
68	D₅	C₃²×D₅×C₃₁	1	1	yes	characteristic
69	C₃²	C₃₁×#1042	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	—	5	4	no	—
71	C₅	—	4	1	no	—
72	C₅	C₃²×Dic₅×C₃₁	1	1	yes	characteristic, complemented
73	C₅	C₃²×AGL(1,𝔽₅)×C₃₁	1	1	yes	characteristic, complemented
74	C₄	—	25	1	no	Sylow
75	C₃	C₃×C₃₁×#1042	1	4	yes	direct product factor
76	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₃₁	#1042
commutator, hypocenter	C₅²	C₃²×C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₃²×C₅×D₅×C₃₁	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×#1042	C₃	direct (N × H)
3. C₃²×C₅²×C₃₁	C₄	semidirect
7. C₃₁×#1042	C₃²	direct (N × H)
10. C₃×C₅²×C₃₁	C₃×C₄	semidirect
15. C₃²×C₅×C₃₁	Dic₅	semidirect
16. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
20. C₃²×#1042	C₃₁	direct (N × H)
21. C₅²×C₃₁	C₃²×C₄	semidirect
26. C₃×C₅×C₃₁	C₃×Dic₅	semidirect
27. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
32. C₃×#1042	C₃×C₃₁	direct (N × H)
33. C₃²×C₃₁	#1042	direct (N × H)
34. C₃²×C₅²	C₄×C₃₁	semidirect
39. C₅×C₃₁	C₃²×Dic₅	semidirect
40. C₅×C₃₁	C₃²×AGL(1,𝔽₅)	semidirect
43. #1042	C₃²×C₃₁	direct (N × H)
44. C₃×C₃₁	C₃×#1042	direct (N × H)
47. C₃×C₅²	C₃×C₄×C₃₁	semidirect
53. C₃²×C₅	Dic₅×C₃₁	semidirect
54. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
56. C₃₁	C₃²×#1042	direct (N × H)
59. C₅²	C₃²×C₄×C₃₁	semidirect
64. C₃×C₅	C₃×Dic₅×C₃₁	semidirect
65. C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	semidirect
69. C₃²	C₃₁×#1042	direct (N × H)
72. C₅	C₃²×Dic₅×C₃₁	semidirect
73. C₅	C₃²×AGL(1,𝔽₅)×C₃₁	semidirect
75. C₃	C₃×C₃₁×#1042	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×C₅×D₅×C₃₁	2
C₃×C₃₁×#1042	3
C₃²×AGL(1,𝔽₅)×C₃₁	5
C₃²×Dic₅×C₃₁	5
C₃²×#1042	31

Subgroups — C32×C31×#1042 #42124887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₁×#1042 #42124887