Subgroups — C₂×C₃×C₅²×#3862 #36348059

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²	#3862
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₇²	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3862	C₂	direct (N × H)
2. C₂×C₅²×#3862	C₃	direct (N × H)
3. C₂×C₃×C₅²×C₇²	C₃	semidirect
4. C₂×C₃×C₅×#3862	C₅	direct (N × H)
5. C₅²×#3862	C₂×C₃	direct (N × H)
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₂×C₅²×C₇²	C₃²	semidirect
9. C₃×C₅×#3862	C₂×C₅	direct (N × H)
11. C₂×C₃×C₅×C₇²	C₃×C₅	semidirect
12. C₂×C₅×#3862	C₃×C₅	direct (N × H)
13. C₅²×C₇²	C₂×C₃²	semidirect
15. C₂×C₃×C₅²×C₇	ΓL(1,𝔽₈)	semidirect
17. C₂×C₃×#3862	C₅²	direct (N × H)
18. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
19. C₅×#3862	C₂×C₃×C₅	direct (N × H)
22. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
24. C₂×C₅×C₇²	C₃²×C₅	semidirect
26. C₃×#3862	C₂×C₅²	direct (N × H)
28. C₂×C₅²×C₇	C₃×ΓL(1,𝔽₈)	semidirect
30. C₂×C₃×C₇²	C₃×C₅²	semidirect
31. C₂×#3862	C₃×C₅²	direct (N × H)
32. C₅×C₇²	C₂×C₃²×C₅	semidirect
35. C₂×C₃×C₅×C₇	C₅×ΓL(1,𝔽₈)	semidirect
38. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
40. C₂×C₃×C₅²	#3862	direct (N × H)
41. C₃×C₇²	C₂×C₃×C₅²	semidirect
42. #3862	C₂×C₃×C₅²	direct (N × H)
45. C₃×C₅×C₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
47. C₂×C₇²	C₃²×C₅²	semidirect
50. C₃×C₅²	C₂×#3862	direct (N × H)
52. C₂×C₅×C₇	C₃×C₅×ΓL(1,𝔽₈)	semidirect
54. C₂×C₅²	C₃×#3862	direct (N × H)
55. C₇²	C₂×C₃²×C₅²	semidirect
58. C₂×C₃×C₇	C₅²×ΓL(1,𝔽₈)	semidirect
61. C₅×C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	semidirect
63. C₂×C₃×C₅	C₅×#3862	direct (N × H)
64. C₅²	C₂×C₃×#3862	direct (N × H)
66. C₃×C₇	C₂×C₅²×ΓL(1,𝔽₈)	semidirect
70. C₃×C₅	C₂×C₅×#3862	direct (N × H)
72. C₂×C₇	C₃×C₅²×ΓL(1,𝔽₈)	semidirect
73. C₂×C₅	C₃×C₅×#3862	direct (N × H)
76. C₇	C₂×C₃×C₅²×ΓL(1,𝔽₈)	semidirect
78. C₂×C₃	C₅²×#3862	direct (N × H)
79. C₅	C₂×C₃×C₅×#3862	direct (N × H)
81. C₃	C₂×C₅²×#3862	direct (N × H)
82. C₂	C₃×C₅²×#3862	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
C₂×C₅²×#3862	3
C₂×C₃×C₅²×C₇²	3
C₂×C₃×C₅×#3862	5
C₂×C₃×C₅²×ΓL(1,𝔽₈)	7

Subgroups — C2×C3×C52×#3862 #36348059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅²×#3862 #36348059