Subgroups — C₃×C₇⋊(C₇⋊#63122) #50304360

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	C₇⋊(C₇⋊#63122)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₇²×C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇⋊(C₇⋊#261)	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	C₃×C₇⋊#63122	—	7	8	no	maximal
7	C₃×C₇²×D₁₃	—	5	1	no	—
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	—	13	1	no	maximal, Hall
10	C₃×C₇×D₆₅	D₇	1	8	yes	—
11	C₇⋊(C₇⋊#261)	—	5	1	no	Hall
12	C₃×C₇²×C₁₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₇⋊#63122	—	7	8	no	—
14	C₃×C₇²×D₅	—	13	1	no	—
15	C₃×C₅×C₇×C₁₃	Dic₇	1	8	yes	complemented
16	C₇²×D₁₃	—	5	1	no	—
17	C₃×C₇⋊#261	—	35	8	no	—
18	C₇⋊(C₇⋊AGL(1,𝔽₅))	—	13	1	no	Hall
19	C₇×D₆₅	C₃×D₇	1	8	yes	—
20	C₃×#63122	—	49	1	no	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×C₇⋊Dic₇	—	65	1	no	Hall
24	C₃×C₇×D₁₃	—	5	8	no	—
25	C₇²×D₅	—	13	1	no	—
26	C₅×C₇×C₁₃	C₃×Dic₇	1	8	yes	complemented
27	C₃×C₇⋊AGL(1,𝔽₅)	—	91	8	no	—
28	C₃×D₆₅	C₇⋊D₇	1	1	yes	characteristic
29	C₇⋊#261	—	35	8	no	—
30	C₂×C₃×C₇²	—	65	1	no	—
31	C₃×C₇×C₁₃	C₇⋊AGL(1,𝔽₅)	1	8	yes	complemented
32	#63122	—	49	1	no	Hall
33	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
34	C₃×C₇×D₅	—	13	8	no	—
35	C₇⋊Dic₇	—	65	1	no	Hall
36	C₃×C₅×C₁₃	C₇⋊Dic₇	1	1	yes	characteristic, complemented, Hall
37	C₇×D₁₃	—	5	8	no	—
38	C₃×#261	—	245	1	no	Hall
39	C₃×C₇²	#63122	1	1	yes	characteristic, complemented, Hall
40	C₇⋊AGL(1,𝔽₅)	—	91	8	no	—
41	D₆₅	C₃×C₇⋊D₇	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×Dic₇	—	455	8	no	—
46	C₃×D₁₃	—	5	1	no	—
47	C₇×D₅	—	13	8	no	—
48	C₅×C₁₃	C₃×C₇⋊Dic₇	1	1	yes	characteristic, complemented, Hall
49	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
50	#261	—	245	1	no	Hall
51	C₇²	C₃×#63122	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃×C₇	—	65	8	no	—
53	C₃×C₁₃	C₇⋊(C₇⋊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	Dic₇	—	455	8	no	—
57	D₁₃	—	5	1	no	—
58	C₃×C₇	C₇⋊#63122	1	8	yes	complemented
59	AGL(1,𝔽₅)	—	637	1	no	Hall
60	C₃×C₅	C₇⋊(C₇⋊#261)	1	1	yes	characteristic, complemented, Hall
61	C₂×C₇	—	65	8	no	—
62	C₁₃	C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	3185	1	no	Hall
64	D₅	—	13	1	no	—
65	C₇	C₃×C₇⋊#63122	1	8	yes	complemented
66	C₂×C₃	—	65	1	no	—
67	C₅	C₃×C₇⋊(C₇⋊#261)	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	3185	1	no	Sylow
69	C₃	C₇⋊(C₇⋊#63122)	1	1	yes	characteristic, direct product factor, Sylow
70	C₂	—	65	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₇⋊(C₇⋊#63122)
commutator, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇²×C₁₃	C₄
prime radical	C₃×C₇²×D₆₅	C₂
cosocle	C₇²×D₆₅	C₂×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₇⋊(C₇⋊#63122)	C₃	direct (N × H)
3. C₃×C₅×C₇²×C₁₃	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. C₃×C₇²×C₁₃	AGL(1,𝔽₅)	semidirect
15. C₃×C₅×C₇×C₁₃	Dic₇	semidirect
21. C₃×C₅×C₇²	#261	semidirect
22. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
26. C₅×C₇×C₁₃	C₃×Dic₇	semidirect
31. C₃×C₇×C₁₃	C₇⋊AGL(1,𝔽₅)	semidirect
33. C₅×C₇²	C₃×#261	semidirect
36. C₃×C₅×C₁₃	C₇⋊Dic₇	semidirect
39. C₃×C₇²	#63122	semidirect
42. C₃×C₅×C₇	C₇⋊#261	semidirect
44. C₇×C₁₃	C₃×C₇⋊AGL(1,𝔽₅)	semidirect
48. C₅×C₁₃	C₃×C₇⋊Dic₇	semidirect
51. C₇²	C₃×#63122	semidirect
53. C₃×C₁₃	C₇⋊(C₇⋊AGL(1,𝔽₅))	semidirect
54. C₅×C₇	C₃×C₇⋊#261	semidirect
58. C₃×C₇	C₇⋊#63122	semidirect
60. C₃×C₅	C₇⋊(C₇⋊#261)	semidirect
62. C₁₃	C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	semidirect
65. C₇	C₃×C₇⋊#63122	semidirect
67. C₅	C₃×C₇⋊(C₇⋊#261)	semidirect
69. C₃	C₇⋊(C₇⋊#63122)	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
C₇⋊(C₇⋊#63122)	3
C₃×C₇⋊(C₇⋊#261)	5
C₃×C₇⋊#63122	7
C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	13

Subgroups — C3×C7⋊(C7⋊#63122) #50304360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇⋊(C₇⋊#63122) #50304360