Subgroups — C₂×C₃×C₅²×#3861 #36348058

Order 22050.

960 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×#3861	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₅²×#3861	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₅²×C₇²	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₅×#3861	C₅	1	6	yes	maximal, direct product factor
5	C₅²×#3861	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	8	no	maximal
8	C₂×C₅²×C₇²	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×#3861	C₂×C₅	1	6	yes	direct product factor
10	C₃×C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
11	C₂×C₃×C₅×C₇²	C₃×C₅	1	6	yes	complemented
12	C₂×C₅×#3861	C₃×C₅	1	18	yes	direct product factor
13	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅²×C₇	ΓL(1,𝔽₈)	1	8	yes	complemented
15	C₂×C₅²×ΓL(1,𝔽₈)	—	7	24	no	—
16	C₂×C₃×#3861	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₅×C₇²	C₂×C₃×C₅	1	6	yes	complemented
18	C₅×#3861	C₂×C₃×C₅	1	18	yes	direct product factor
19	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	7	48	no	—
20	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
21	C₅²×ΓL(1,𝔽₈)	—	7	24	no	—
22	C₂×C₅×C₇²	C₃²×C₅	1	6	yes	complemented
23	C₂×C₃²×C₅²	—	49	1	no	Hall
24	C₃×#3861	C₂×C₅²	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₅²×C₇	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
26	C₃×C₅×ΓL(1,𝔽₈)	—	7	48	no	—
27	C₂×C₃×C₇²	C₃×C₅²	1	1	yes	characteristic, complemented
28	C₂×#3861	C₃×C₅²	1	3	yes	direct product factor
29	C₅×C₇²	C₂×C₃²×C₅	1	6	yes	complemented
30	C₃²×C₅²	—	49	1	no	Hall
31	C₂×C₃×C₅×C₇	C₅×ΓL(1,𝔽₈)	1	48	yes	complemented
32	C₂×C₅×ΓL(1,𝔽₈)	—	7	144	no	—
33	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
34	C₂×C₃×C₅²	—	49	3	no	—
35	C₂×C₃×C₅²	#3861	1	1	yes	characteristic, direct product factor
36	C₃×C₇²	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
37	#3861	C₂×C₃×C₅²	1	3	yes	direct product factor
38	C₂×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
39	C₃×C₅×C₇	C₂×C₅×ΓL(1,𝔽₈)	1	48	yes	complemented
40	C₅×ΓL(1,𝔽₈)	—	7	144	no	—
41	C₂×C₇²	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃²×C₅	—	49	6	no	—
43	C₃×C₅²	—	49	3	no	—
44	C₃×C₅²	C₂×#3861	1	1	yes	characteristic, direct product factor
45	C₂×C₅×C₇	C₃×C₅×ΓL(1,𝔽₈)	1	48	yes	complemented
46	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
47	C₂×C₅²	C₃×#3861	1	1	yes	characteristic, direct product factor, Hall
48	C₇²	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
49	C₃²×C₅	—	49	6	no	—
50	C₂×C₃×C₇	C₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
51	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
52	C₅×C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	48	yes	complemented
53	C₂×C₃×C₅	—	49	18	no	—
54	C₂×C₃×C₅	C₅×#3861	1	6	yes	direct product factor
55	C₅²	C₂×C₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
56	C₃×C₇	C₂×C₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
57	ΓL(1,𝔽₈)	—	7	24	no	—
58	C₂×C₃²	—	49	1	no	Hall
59	C₃×C₅	—	49	18	no	—
60	C₃×C₅	C₂×C₅×#3861	1	6	yes	direct product factor
61	C₂×C₇	C₃×C₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
62	C₂×C₅	C₃×C₅×#3861	1	6	yes	direct product factor
63	C₃²	—	49	1	no	Sylow
64	C₇	C₂×C₃×C₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
65	C₂×C₃	—	49	3	no	—
66	C₂×C₃	C₅²×#3861	1	1	yes	characteristic, direct product factor
67	C₅	C₂×C₃×C₅×#3861	1	6	yes	direct product factor
68	C₃	—	49	3	no	—
69	C₃	C₂×C₅²×#3861	1	1	yes	characteristic, direct product factor
70	C₂	C₃×C₅²×#3861	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²	#3861
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×#3861	C₂	direct (N × H)
2. C₂×C₅²×#3861	C₃	direct (N × H)
3. C₂×C₃×C₅²×C₇²	C₃	semidirect
4. C₂×C₃×C₅×#3861	C₅	direct (N × H)
5. C₅²×#3861	C₂×C₃	direct (N × H)
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₂×C₅²×C₇²	C₃²	semidirect
9. C₃×C₅×#3861	C₂×C₅	direct (N × H)
11. C₂×C₃×C₅×C₇²	C₃×C₅	semidirect
12. C₂×C₅×#3861	C₃×C₅	direct (N × H)
13. C₅²×C₇²	C₂×C₃²	semidirect
14. C₂×C₃×C₅²×C₇	ΓL(1,𝔽₈)	semidirect
16. C₂×C₃×#3861	C₅²	direct (N × H)
17. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
18. C₅×#3861	C₂×C₃×C₅	direct (N × H)
20. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
22. C₂×C₅×C₇²	C₃²×C₅	semidirect
24. C₃×#3861	C₂×C₅²	direct (N × H)
25. C₂×C₅²×C₇	C₃×ΓL(1,𝔽₈)	semidirect
27. C₂×C₃×C₇²	C₃×C₅²	semidirect
28. C₂×#3861	C₃×C₅²	direct (N × H)
29. C₅×C₇²	C₂×C₃²×C₅	semidirect
31. C₂×C₃×C₅×C₇	C₅×ΓL(1,𝔽₈)	semidirect
33. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
35. C₂×C₃×C₅²	#3861	direct (N × H)
36. C₃×C₇²	C₂×C₃×C₅²	semidirect
37. #3861	C₂×C₃×C₅²	direct (N × H)
39. C₃×C₅×C₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
41. C₂×C₇²	C₃²×C₅²	semidirect
44. C₃×C₅²	C₂×#3861	direct (N × H)
45. C₂×C₅×C₇	C₃×C₅×ΓL(1,𝔽₈)	semidirect
47. C₂×C₅²	C₃×#3861	direct (N × H)
48. C₇²	C₂×C₃²×C₅²	semidirect
50. C₂×C₃×C₇	C₅²×ΓL(1,𝔽₈)	semidirect
52. C₅×C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	semidirect
54. C₂×C₃×C₅	C₅×#3861	direct (N × H)
55. C₅²	C₂×C₃×#3861	direct (N × H)
56. C₃×C₇	C₂×C₅²×ΓL(1,𝔽₈)	semidirect
60. C₃×C₅	C₂×C₅×#3861	direct (N × H)
61. C₂×C₇	C₃×C₅²×ΓL(1,𝔽₈)	semidirect
62. C₂×C₅	C₃×C₅×#3861	direct (N × H)
64. C₇	C₂×C₃×C₅²×ΓL(1,𝔽₈)	semidirect
66. C₂×C₃	C₅²×#3861	direct (N × H)
67. C₅	C₂×C₃×C₅×#3861	direct (N × H)
69. C₃	C₂×C₅²×#3861	direct (N × H)
70. C₂	C₃×C₅²×#3861	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×#3861	2
C₂×C₅²×#3861	3
C₂×C₃×C₅²×C₇²	3
C₂×C₃×C₅×#3861	5
C₂×C₃×C₅²×ΓL(1,𝔽₈)	7

Subgroups — C2×C3×C52×#3861 #36348058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅²×#3861 #36348058