Subgroups — C₃×C₅⋊D₅×#3861 #36347984

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, complemented, Hall
2	C₃×C₇²×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊D₅×#3861	C₃	1	3	yes	maximal, direct product factor
4	C₃×D₅×#3861	—	5	6	no	maximal
5	C₅²×#3861	C₂×C₃	1	3	yes	complemented
6	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×ΓL(1,𝔽₈)×C₅⋊D₅	—	7	8	no	maximal
8	C₇²×C₅⋊D₅	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×#3861	D₅	1	6	yes	complemented
10	C₃×C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
11	C₃×C₇²×D₅	—	5	6	no	—
12	D₅×#3861	—	5	18	no	—
13	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	1	8	yes	complemented
15	ΓL(1,𝔽₈)×C₅⋊D₅	—	7	24	no	—
16	C₂×C₃×#3861	—	25	1	no	Hall
17	C₃×C₅×C₇²	C₃×D₅	1	6	yes	complemented
18	C₅×#3861	C₃×D₅	1	18	yes	complemented
19	C₃×D₅×ΓL(1,𝔽₈)	—	35	48	no	—
20	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
21	C₅²×ΓL(1,𝔽₈)	—	7	24	no	—
22	C₇²×D₅	—	5	6	no	—
23	C₃²×C₅⋊D₅	—	49	1	no	Hall
24	C₃×#3861	C₅⋊D₅	1	1	yes	characteristic, direct product factor, Hall
25	C₇×C₅⋊D₅	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
26	C₃×C₅×ΓL(1,𝔽₈)	—	7	48	no	—
27	C₂×C₃×C₇²	—	25	1	no	—
28	C₂×#3861	—	25	3	no	—
29	C₅×C₇²	C₃²×D₅	1	6	yes	complemented
30	C₃²×C₅²	—	49	1	no	Hall
31	C₃×C₇×D₅	—	5	48	no	—
32	D₅×ΓL(1,𝔽₈)	—	35	144	no	—
33	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
34	C₃×C₅⋊D₅	—	49	3	no	—
35	C₃×C₅⋊D₅	#3861	1	1	yes	characteristic, direct product factor
36	C₃×C₇²	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
37	#3861	C₃×C₅⋊D₅	1	3	yes	direct product factor
38	C₂×C₃×ΓL(1,𝔽₈)	—	175	8	no	—
39	C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	1	48	yes	complemented
40	C₅×ΓL(1,𝔽₈)	—	7	144	no	—
41	C₂×C₇²	—	25	1	no	Hall
42	C₃²×D₅	—	245	6	no	—
43	C₃×C₅²	—	49	3	no	—
44	C₃×C₅²	C₂×#3861	1	1	yes	characteristic, complemented
45	C₇×D₅	—	5	48	no	—
46	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
47	C₅⋊D₅	C₃×#3861	1	1	yes	characteristic, direct product factor, Hall
48	C₇²	C₃²×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
49	C₃²×C₅	—	49	6	no	—
50	C₂×C₃×C₇	—	25	8	no	—
51	C₂×ΓL(1,𝔽₈)	—	175	24	no	—
52	C₅×C₇	C₃×D₅×ΓL(1,𝔽₈)	1	48	yes	complemented
53	C₃×D₅	—	5	6	no	—
54	C₃×D₅	—	245	18	no	—
55	C₅²	C₂×C₃×#3861	1	1	yes	characteristic, complemented, Sylow
56	C₃×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	1	8	yes	complemented
57	ΓL(1,𝔽₈)	—	7	24	no	—
58	C₂×C₃²	—	1225	1	no	Hall
59	C₃×C₅	—	49	18	no	—
60	C₃×C₅	D₅×#3861	1	6	yes	complemented
61	C₂×C₇	—	25	8	no	—
62	D₅	—	5	6	no	—
63	C₃²	—	49	1	no	Sylow
64	C₇	C₃×ΓL(1,𝔽₈)×C₅⋊D₅	1	8	yes	complemented
65	C₂×C₃	—	25	1	no	—
66	C₂×C₃	—	1225	3	no	—
67	C₅	C₃×D₅×#3861	1	6	yes	complemented
68	C₃	—	49	3	no	—
69	C₃	C₅⋊D₅×#3861	1	1	yes	characteristic, direct product factor
70	C₂	—	25	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	C₅⋊D₅×#3861
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3861	C₂	semidirect
2. C₃×C₇²×C₅⋊D₅	C₃	semidirect
3. C₅⋊D₅×#3861	C₃	direct (N × H)
5. C₅²×#3861	C₂×C₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₇²×C₅⋊D₅	C₃²	semidirect
9. C₃×C₅×#3861	D₅	semidirect
13. C₅²×C₇²	C₂×C₃²	semidirect
14. C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	semidirect
17. C₃×C₅×C₇²	C₃×D₅	semidirect
18. C₅×#3861	C₃×D₅	semidirect
20. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
24. C₃×#3861	C₅⋊D₅	direct (N × H)
25. C₇×C₅⋊D₅	C₃×ΓL(1,𝔽₈)	semidirect
29. C₅×C₇²	C₃²×D₅	semidirect
33. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
35. C₃×C₅⋊D₅	#3861	direct (N × H)
36. C₃×C₇²	C₃×C₅⋊D₅	semidirect
37. #3861	C₃×C₅⋊D₅	direct (N × H)
39. C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	semidirect
44. C₃×C₅²	C₂×#3861	semidirect
47. C₅⋊D₅	C₃×#3861	direct (N × H)
48. C₇²	C₃²×C₅⋊D₅	semidirect
52. C₅×C₇	C₃×D₅×ΓL(1,𝔽₈)	semidirect
55. C₅²	C₂×C₃×#3861	semidirect
56. C₃×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	semidirect
60. C₃×C₅	D₅×#3861	semidirect
64. C₇	C₃×ΓL(1,𝔽₈)×C₅⋊D₅	semidirect
67. C₅	C₃×D₅×#3861	semidirect
69. C₃	C₅⋊D₅×#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
C₃×C₇²×C₅⋊D₅	3
C₅⋊D₅×#3861	3
C₃×D₅×#3861	5
C₃×ΓL(1,𝔽₈)×C₅⋊D₅	7

Subgroups — C3×C5⋊D5×#3861 #36347984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅⋊D₅×#3861 #36347984