Subgroups — C₃×C₅²×#64425 #36347967

Order 22050.

448 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×#3862	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅²×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅²×#64425	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅×#64425	C₅	1	6	yes	maximal, direct product factor
5	C₅²×#3862	C₂×C₃	1	3	yes	complemented
6	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×AGL(1,𝔽₇)	—	7	2	no	maximal
8	C₅²×C₇⋊D₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×#3862	C₂×C₅	1	6	yes	complemented
10	C₃×C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
11	C₃×C₅×C₇⋊D₇	C₃×C₅	1	6	yes	complemented
12	C₅×#64425	C₃×C₅	1	18	yes	direct product factor
13	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₃×C₅²×D₇	—	7	2	no	—
15	C₃×C₅²×D₇	—	21	2	no	—
16	C₅²×AGL(1,𝔽₇)	—	7	6	no	—
17	C₃×#64425	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₅×C₇²	C₂×C₃×C₅	1	6	yes	complemented
19	C₅×#3862	C₂×C₃×C₅	1	18	yes	complemented
20	C₃×C₅×AGL(1,𝔽₇)	—	7	12	no	—
21	C₃×C₅²×C₇	—	3	2	no	—
22	C₃×C₅²×C₇	AGL(1,𝔽₇)	1	2	yes	complemented
23	C₅²×ΓL(1,𝔽₈)	—	7	6	no	—
24	C₅×C₇⋊D₇	C₃²×C₅	1	6	yes	complemented
25	C₂×C₃²×C₅²	—	49	1	no	Hall
26	C₃×#3862	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
27	C₅²×D₇	—	7	2	no	—
28	C₅²×D₇	—	21	2	no	—
29	C₃×C₅×ΓL(1,𝔽₈)	—	7	12	no	—
30	C₃×C₇⋊D₇	C₃×C₅²	1	1	yes	characteristic, complemented
31	#64425	C₃×C₅²	1	3	yes	direct product factor
32	C₅×C₇²	C₂×C₃²×C₅	1	6	yes	complemented
33	C₃²×C₅²	—	49	1	no	Hall
34	C₃×C₅×D₇	—	7	12	no	—
35	C₃×C₅×D₇	—	21	12	no	—
36	C₅×AGL(1,𝔽₇)	—	7	36	no	—
37	C₅²×C₇	—	3	2	no	—
38	C₅²×C₇	C₃×AGL(1,𝔽₇)	1	2	yes	complemented
39	C₂×C₃×C₅²	—	49	3	no	—
40	C₂×C₃×C₅²	—	49	1	no	—
41	C₃×C₇²	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
42	#3862	C₂×C₃×C₅²	1	3	yes	complemented
43	C₃×AGL(1,𝔽₇)	—	7	2	no	—
44	C₃×C₅×C₇	—	3	12	no	—
45	C₃×C₅×C₇	C₅×AGL(1,𝔽₇)	1	12	yes	complemented
46	C₅×ΓL(1,𝔽₈)	—	7	36	no	—
47	C₇⋊D₇	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
48	C₂×C₃²×C₅	—	49	6	no	—
49	C₃×C₅²	—	49	3	no	—
50	C₃×C₅²	#64425	1	1	yes	characteristic, direct product factor
51	C₅×D₇	—	7	12	no	—
52	C₅×D₇	—	21	12	no	—
53	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
54	C₂×C₅²	—	49	1	no	Hall
55	C₇²	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
56	C₃²×C₅	—	49	6	no	—
57	C₃×D₇	—	7	2	no	—
58	C₃×D₇	—	21	2	no	—
59	AGL(1,𝔽₇)	—	7	6	no	—
60	C₅×C₇	—	3	12	no	—
61	C₅×C₇	C₃×C₅×AGL(1,𝔽₇)	1	12	yes	complemented
62	C₂×C₃×C₅	—	49	18	no	—
63	C₂×C₃×C₅	—	49	6	no	—
64	C₅²	C₃×#64425	1	1	yes	characteristic, direct product factor, Sylow
65	C₃×C₇	—	3	2	no	—
66	C₃×C₇	C₅²×AGL(1,𝔽₇)	1	2	yes	complemented
67	ΓL(1,𝔽₈)	—	7	6	no	—
68	C₂×C₃²	—	49	1	no	Hall
69	C₃×C₅	—	49	18	no	—
70	C₃×C₅	C₅×#64425	1	6	yes	direct product factor
71	D₇	—	7	2	no	—
72	D₇	—	21	2	no	—
73	C₂×C₅	—	49	6	no	—
74	C₃²	—	49	1	no	Sylow
75	C₇	—	3	2	no	—
76	C₇	C₃×C₅²×AGL(1,𝔽₇)	1	2	yes	complemented
77	C₂×C₃	—	49	3	no	—
78	C₂×C₃	—	49	1	no	—
79	C₅	C₃×C₅×#64425	1	6	yes	direct product factor
80	C₃	—	49	3	no	—
81	C₃	C₅²×#64425	1	1	yes	characteristic, direct product factor
82	C₂	—	49	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅²	#64425
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3862	C₂	semidirect
2. C₃×C₅²×C₇⋊D₇	C₃	semidirect
3. C₅²×#64425	C₃	direct (N × H)
4. C₃×C₅×#64425	C₅	direct (N × H)
5. C₅²×#3862	C₂×C₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₅²×C₇⋊D₇	C₃²	semidirect
9. C₃×C₅×#3862	C₂×C₅	semidirect
11. C₃×C₅×C₇⋊D₇	C₃×C₅	semidirect
12. C₅×#64425	C₃×C₅	direct (N × H)
13. C₅²×C₇²	C₂×C₃²	semidirect
17. C₃×#64425	C₅²	direct (N × H)
18. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
19. C₅×#3862	C₂×C₃×C₅	semidirect
22. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
24. C₅×C₇⋊D₇	C₃²×C₅	semidirect
26. C₃×#3862	C₂×C₅²	semidirect
30. C₃×C₇⋊D₇	C₃×C₅²	semidirect
31. #64425	C₃×C₅²	direct (N × H)
32. C₅×C₇²	C₂×C₃²×C₅	semidirect
38. C₅²×C₇	C₃×AGL(1,𝔽₇)	semidirect
41. C₃×C₇²	C₂×C₃×C₅²	semidirect
42. #3862	C₂×C₃×C₅²	semidirect
45. C₃×C₅×C₇	C₅×AGL(1,𝔽₇)	semidirect
47. C₇⋊D₇	C₃²×C₅²	semidirect
50. C₃×C₅²	#64425	direct (N × H)
55. C₇²	C₂×C₃²×C₅²	semidirect
61. C₅×C₇	C₃×C₅×AGL(1,𝔽₇)	semidirect
64. C₅²	C₃×#64425	direct (N × H)
66. C₃×C₇	C₅²×AGL(1,𝔽₇)	semidirect
70. C₃×C₅	C₅×#64425	direct (N × H)
76. C₇	C₃×C₅²×AGL(1,𝔽₇)	semidirect
79. C₅	C₃×C₅×#64425	direct (N × H)
81. C₃	C₅²×#64425	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×C₅²×#3862	2
C₃×C₅²×C₇⋊D₇	3
C₅²×#64425	3
C₃×C₅×#64425	5
C₃×C₅²×AGL(1,𝔽₇)	7

Subgroups — C3×C52×#64425 #36347967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×#64425 #36347967