Subgroups — C₃×C₇²×#63121 #50304353

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×D₆₅	C₂	1	1	yes	maximal, characteristic
2	C₇²×#63121	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₇²×C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇²×#261	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	C₃×C₇×#63121	C₇	1	8	yes	maximal, direct product factor
7	C₃×C₇²×D₁₃	—	5	1	no	—
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇²×AGL(1,𝔽₅)	—	13	1	no	maximal, Hall
10	C₃×C₇×D₆₅	C₂×C₇	1	8	yes	—
11	C₇²×#261	—	5	1	no	Hall
12	C₃×C₇²×C₁₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₇×#63121	C₃×C₇	1	8	yes	direct product factor
14	C₃×C₇²×D₅	—	13	1	no	—
15	C₃×C₅×C₇×C₁₃	C₄×C₇	1	8	yes	complemented
16	C₇²×D₁₃	—	5	1	no	—
17	C₃×C₇×#261	—	5	8	no	—
18	C₇²×AGL(1,𝔽₅)	—	13	1	no	Hall
19	C₇×D₆₅	C₂×C₃×C₇	1	8	yes	—
20	C₃×#63121	C₇²	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₅×C₇²	#261	1	1	yes	characteristic, complemented, Hall
22	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
23	C₃×C₄×C₇²	—	65	1	no	Hall
24	C₃×C₇×D₁₃	—	5	8	no	—
25	C₇²×D₅	—	13	1	no	—
26	C₅×C₇×C₁₃	C₃×C₄×C₇	1	8	yes	complemented
27	C₃×C₇×AGL(1,𝔽₅)	—	13	8	no	—
28	C₃×D₆₅	C₂×C₇²	1	1	yes	characteristic
29	C₇×#261	—	5	8	no	—
30	C₂×C₃×C₇²	—	65	1	no	—
31	C₃×C₇×C₁₃	C₇×AGL(1,𝔽₅)	1	8	yes	complemented
32	#63121	C₃×C₇²	1	1	yes	characteristic, direct product factor, Hall
33	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
34	C₃×C₇×D₅	—	13	8	no	—
35	C₄×C₇²	—	65	1	no	Hall
36	C₃×C₅×C₁₃	C₄×C₇²	1	1	yes	characteristic, complemented, Hall
37	C₇×D₁₃	—	5	8	no	—
38	C₃×#261	—	5	1	no	Hall
39	C₃×C₇²	#63121	1	1	yes	characteristic, direct product factor, Hall
40	C₇×AGL(1,𝔽₅)	—	13	8	no	—
41	D₆₅	C₂×C₃×C₇²	1	1	yes	characteristic
42	C₃×C₅×C₇	C₇×#261	1	8	yes	complemented
43	C₂×C₇²	—	65	1	no	—
44	C₇×C₁₃	C₃×C₇×AGL(1,𝔽₅)	1	8	yes	complemented
45	C₃×C₄×C₇	—	65	8	no	—
46	C₃×D₁₃	—	5	1	no	—
47	C₇×D₅	—	13	8	no	—
48	C₅×C₁₃	C₃×C₄×C₇²	1	1	yes	characteristic, complemented, Hall
49	C₃×AGL(1,𝔽₅)	—	13	1	no	Hall
50	#261	—	5	1	no	Hall
51	C₇²	C₃×#63121	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃×C₇	—	65	8	no	—
53	C₃×C₁₃	C₇²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
54	C₅×C₇	C₃×C₇×#261	1	8	yes	complemented
55	C₃×D₅	—	13	1	no	—
56	C₄×C₇	—	65	8	no	—
57	D₁₃	—	5	1	no	—
58	C₃×C₇	C₇×#63121	1	8	yes	direct product factor
59	AGL(1,𝔽₅)	—	13	1	no	Hall
60	C₃×C₅	C₇²×#261	1	1	yes	characteristic, complemented, Hall
61	C₂×C₇	—	65	8	no	—
62	C₁₃	C₃×C₇²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	65	1	no	Hall
64	D₅	—	13	1	no	—
65	C₇	C₃×C₇×#63121	1	8	yes	direct product factor
66	C₂×C₃	—	65	1	no	—
67	C₅	C₃×C₇²×#261	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	65	1	no	Sylow
69	C₃	C₇²×#63121	1	1	yes	characteristic, direct product factor, Sylow
70	C₂	—	65	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₇²	#63121
commutator, hypocenter	C₅×C₁₃	C₃×C₄×C₇²
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇²×C₁₃	C₄
prime radical	C₃×C₇²×D₆₅	C₂
cosocle	D₆₅	C₂×C₃×C₇²
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₃×C₇²

Complements

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

N	H ≅ G/N	type
2. C₇²×#63121	C₃	direct (N × H)
3. C₃×C₅×C₇²×C₁₃	C₄	semidirect
6. C₃×C₇×#63121	C₇	direct (N × H)
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. C₃×C₇²×C₁₃	AGL(1,𝔽₅)	semidirect
13. C₇×#63121	C₃×C₇	direct (N × H)
15. C₃×C₅×C₇×C₁₃	C₄×C₇	semidirect
20. C₃×#63121	C₇²	direct (N × H)
21. C₃×C₅×C₇²	#261	semidirect
22. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
26. C₅×C₇×C₁₃	C₃×C₄×C₇	semidirect
31. C₃×C₇×C₁₃	C₇×AGL(1,𝔽₅)	semidirect
32. #63121	C₃×C₇²	direct (N × H)
33. C₅×C₇²	C₃×#261	semidirect
36. C₃×C₅×C₁₃	C₄×C₇²	semidirect
39. C₃×C₇²	#63121	direct (N × H)
42. C₃×C₅×C₇	C₇×#261	semidirect
44. C₇×C₁₃	C₃×C₇×AGL(1,𝔽₅)	semidirect
48. C₅×C₁₃	C₃×C₄×C₇²	semidirect
51. C₇²	C₃×#63121	direct (N × H)
53. C₃×C₁₃	C₇²×AGL(1,𝔽₅)	semidirect
54. C₅×C₇	C₃×C₇×#261	semidirect
58. C₃×C₇	C₇×#63121	direct (N × H)
60. C₃×C₅	C₇²×#261	semidirect
62. C₁₃	C₃×C₇²×AGL(1,𝔽₅)	semidirect
65. C₇	C₃×C₇×#63121	direct (N × H)
67. C₅	C₃×C₇²×#261	semidirect
69. C₃	C₇²×#63121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇²×D₆₅	2
C₇²×#63121	3
C₃×C₇²×#261	5
C₃×C₇×#63121	7
C₃×C₇²×AGL(1,𝔽₅)	13

Subgroups — C3×C72×#63121 #50304353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×#63121 #50304353