Subgroups — C₃²×C₇×#10596815 #49141540

Order 36540.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×D₁₄₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#10596815	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₇×C₂₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₇×#1181	—	5	1	no	maximal, Hall
5	C₃×C₇×D₁₄₅	C₂×C₃	1	4	yes	—
6	C₃²×#10596815	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₇×#10596815	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×C₇×D₂₉	—	5	1	no	—
9	C₃×C₅×C₇×C₂₉	C₃×C₄	1	4	yes	complemented
10	C₃²×D₁₄₅	C₂×C₇	1	1	yes	characteristic
11	C₃×C₇×#1181	—	5	4	no	—
12	C₇×D₁₄₅	C₂×C₃²	1	1	yes	characteristic
13	C₃²×C₇×C₂₉	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
14	C₃×#10596815	C₃×C₇	1	4	yes	direct product factor
15	C₃²×C₅×C₂₉	C₄×C₇	1	1	yes	characteristic, complemented, Hall
16	C₃²×C₇×AGL(1,𝔽₅)	—	29	1	no	maximal, Hall
17	C₃×C₇×D₂₉	—	5	4	no	—
18	C₃²×#1181	—	5	1	no	Hall
19	C₅×C₇×C₂₉	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
20	C₃×D₁₄₅	C₂×C₃×C₇	1	4	yes	—
21	C₇×#1181	—	5	1	no	Hall
22	C₃²×C₇×D₅	—	29	1	no	—
23	C₃×C₇×C₂₉	C₃×AGL(1,𝔽₅)	1	4	yes	complemented
24	#10596815	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
25	C₃²×D₂₉	—	5	1	no	—
26	C₃×C₅×C₂₉	C₃×C₄×C₇	1	4	yes	complemented
27	C₃×C₇×AGL(1,𝔽₅)	—	29	4	no	—
28	C₇×D₂₉	—	5	1	no	—
29	C₃×#1181	—	5	4	no	—
30	C₃²×C₅×C₇	#1181	1	1	yes	characteristic, complemented, Hall
31	D₁₄₅	C₂×C₃²×C₇	1	1	yes	characteristic
32	C₃²×C₂₉	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
33	C₃²×C₄×C₇	—	145	1	no	Hall
34	C₃×C₇×D₅	—	29	4	no	—
35	C₇×C₂₉	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
36	C₃²×AGL(1,𝔽₅)	—	29	1	no	Hall
37	C₃×D₂₉	—	5	4	no	—
38	C₅×C₂₉	C₃²×C₄×C₇	1	1	yes	characteristic, complemented, Hall
39	C₇×AGL(1,𝔽₅)	—	29	1	no	Hall
40	C₂×C₃²×C₇	—	145	1	no	—
41	#1181	—	5	1	no	Hall
42	C₃×C₅×C₇	C₃×#1181	1	4	yes	complemented
43	C₃²×D₅	—	29	1	no	—
44	C₃×C₂₉	C₃×C₇×AGL(1,𝔽₅)	1	4	yes	complemented
45	C₃×C₄×C₇	—	145	4	no	—
46	C₇×D₅	—	29	1	no	—
47	C₃²×C₇	#10596815	1	1	yes	characteristic, direct product factor, Hall
48	C₃×AGL(1,𝔽₅)	—	29	4	no	—
49	D₂₉	—	5	1	no	—
50	C₃²×C₅	C₇×#1181	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₇	—	145	4	no	—
52	C₃²×C₄	—	145	1	no	Hall
53	C₅×C₇	C₃²×#1181	1	1	yes	characteristic, complemented, Hall
54	C₃×D₅	—	29	4	no	—
55	C₂₉	C₃²×C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
56	C₄×C₇	—	145	1	no	Hall
57	C₃×C₇	C₃×#10596815	1	4	yes	direct product factor
58	AGL(1,𝔽₅)	—	29	1	no	Hall
59	C₂×C₃²	—	145	1	no	—
60	C₃×C₅	C₃×C₇×#1181	1	4	yes	complemented
61	C₂×C₇	—	145	1	no	—
62	C₃×C₄	—	145	4	no	—
63	D₅	—	29	1	no	—
64	C₃²	C₇×#10596815	1	1	yes	characteristic, direct product factor, Sylow
65	C₇	C₃²×#10596815	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	—	145	4	no	—
67	C₅	C₃²×C₇×#1181	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	145	1	no	Sylow
69	C₃	C₃×C₇×#10596815	1	4	yes	direct product factor
70	C₂	—	145	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₇	#10596815
commutator, hypocenter	C₅×C₂₉	C₃²×C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₂₉	C₄
prime radical	C₃²×C₇×D₁₄₅	C₂
cosocle	D₁₄₅	C₂×C₃²×C₇
LCS	G ▷ C₅×C₂₉
UCS	C₁ ◁ C₃²×C₇

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#10596815	C₃	direct (N × H)
3. C₃²×C₅×C₇×C₂₉	C₄	semidirect
6. C₃²×#10596815	C₇	direct (N × H)
7. C₇×#10596815	C₃²	direct (N × H)
9. C₃×C₅×C₇×C₂₉	C₃×C₄	semidirect
13. C₃²×C₇×C₂₉	AGL(1,𝔽₅)	semidirect
14. C₃×#10596815	C₃×C₇	direct (N × H)
15. C₃²×C₅×C₂₉	C₄×C₇	semidirect
19. C₅×C₇×C₂₉	C₃²×C₄	semidirect
23. C₃×C₇×C₂₉	C₃×AGL(1,𝔽₅)	semidirect
24. #10596815	C₃²×C₇	direct (N × H)
26. C₃×C₅×C₂₉	C₃×C₄×C₇	semidirect
30. C₃²×C₅×C₇	#1181	semidirect
32. C₃²×C₂₉	C₇×AGL(1,𝔽₅)	semidirect
35. C₇×C₂₉	C₃²×AGL(1,𝔽₅)	semidirect
38. C₅×C₂₉	C₃²×C₄×C₇	semidirect
42. C₃×C₅×C₇	C₃×#1181	semidirect
44. C₃×C₂₉	C₃×C₇×AGL(1,𝔽₅)	semidirect
47. C₃²×C₇	#10596815	direct (N × H)
50. C₃²×C₅	C₇×#1181	semidirect
53. C₅×C₇	C₃²×#1181	semidirect
55. C₂₉	C₃²×C₇×AGL(1,𝔽₅)	semidirect
57. C₃×C₇	C₃×#10596815	direct (N × H)
60. C₃×C₅	C₃×C₇×#1181	semidirect
64. C₃²	C₇×#10596815	direct (N × H)
65. C₇	C₃²×#10596815	direct (N × H)
67. C₅	C₃²×C₇×#1181	semidirect
69. C₃	C₃×C₇×#10596815	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×C₇×D₁₄₅	2
C₃×C₇×#10596815	3
C₃²×C₇×#1181	5
C₃²×#10596815	7
C₃²×C₇×AGL(1,𝔽₅)	29

Subgroups — C32×C7×#10596815 #49141540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₇×#10596815 #49141540