Subgroups — #3861×C₃⋊(C₅⋊D₅) #36348047

Order 22050.

720 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	C₇²×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊D₅×#3861	—	3	1	no	maximal
4	D₁₅×#3861	—	5	6	no	maximal
5	C₅²×#3861	—	2	1	no	—
6	C₅²×#3861	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	—	7	8	no	maximal
9	C₇²×C₅⋊D₅	—	3	1	no	Hall
10	C₃×C₅×#3861	D₅	1	6	yes	complemented
11	C₃×C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
12	C₇²×D₁₅	—	5	6	no	—
13	D₅×#3861	—	15	6	no	—
14	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	1	8	yes	complemented
16	ΓL(1,𝔽₈)×C₅⋊D₅	—	21	8	no	—
17	D₃×#3861	—	25	1	no	Hall
18	C₃×C₅×C₇²	C₃×D₅	1	6	yes	complemented
19	C₅×#3861	—	2	6	no	—
20	C₅×#3861	D₁₅	1	6	yes	complemented
21	ΓL(1,𝔽₈)×D₁₅	—	35	48	no	—
22	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
23	C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
24	C₅²×ΓL(1,𝔽₈)	—	14	8	no	—
25	C₇²×D₅	—	15	6	no	—
26	C₃×C₃⋊(C₅⋊D₅)	—	49	1	no	Hall
27	C₃×#3861	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
28	C₇×C₅⋊D₅	—	3	8	no	—
29	C₃×C₅×ΓL(1,𝔽₈)	—	7	48	no	—
30	D₃×C₇²	—	25	1	no	—
31	C₂×#3861	—	75	1	no	—
32	C₅×C₇²	C₃×D₁₅	1	6	yes	complemented
33	C₃²×C₅²	—	49	1	no	Hall
34	C₇×D₁₅	—	5	48	no	—
35	D₅×ΓL(1,𝔽₈)	—	105	48	no	—
36	C₅²×C₇	D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
37	C₃⋊(C₅⋊D₅)	#3861	1	1	yes	characteristic, direct product factor
38	C₃×C₅⋊D₅	—	147	1	no	—
39	C₃×C₇²	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
40	#3861	—	2	1	no	—
41	#3861	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
42	D₃×ΓL(1,𝔽₈)	—	175	8	no	—
43	C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	1	48	yes	complemented
44	C₅×ΓL(1,𝔽₈)	—	7	48	no	—
45	C₅×ΓL(1,𝔽₈)	—	14	48	no	—
46	C₂×C₇²	—	75	1	no	Hall
47	C₃×D₁₅	—	245	6	no	—
48	C₃×C₅²	—	49	1	no	—
49	C₃×C₅²	—	98	1	no	—
50	C₃×C₅²	C₂×#3861	1	1	yes	characteristic, complemented
51	C₇×D₅	—	15	48	no	—
52	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
53	C₅⋊D₅	—	3	1	no	Hall
54	C₇²	C₃×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
55	C₃²×C₅	—	49	6	no	—
56	D₃×C₇	—	25	8	no	—
57	C₂×ΓL(1,𝔽₈)	—	525	8	no	—
58	C₅×C₇	ΓL(1,𝔽₈)×D₁₅	1	48	yes	complemented
59	D₁₅	—	5	6	no	—
60	C₃×D₅	—	735	6	no	—
61	C₅²	D₃×#3861	1	1	yes	characteristic, complemented, Sylow
62	C₃×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	1	8	yes	complemented
63	ΓL(1,𝔽₈)	—	7	8	no	—
64	ΓL(1,𝔽₈)	—	14	8	no	—
65	C₃×D₃	—	1225	1	no	Hall
66	C₃×C₅	—	49	6	no	—
67	C₃×C₅	—	98	6	no	—
68	C₃×C₅	D₅×#3861	1	6	yes	complemented
69	C₂×C₇	—	75	8	no	—
70	D₅	—	15	6	no	—
71	C₃²	—	49	1	no	Sylow
72	C₇	ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	1	8	yes	complemented
73	C₂×C₃	—	3675	1	no	—
74	D₃	—	25	1	no	—
75	C₅	D₁₅×#3861	1	6	yes	complemented
76	C₃	—	49	1	no	—
77	C₃	—	98	1	no	—
78	C₃	C₅⋊D₅×#3861	1	1	yes	characteristic, complemented
79	C₂	—	75	1	no	Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3861	C₂	semidirect
2. C₇²×C₃⋊(C₅⋊D₅)	C₃	semidirect
6. C₅²×#3861	D₃	semidirect
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
10. C₃×C₅×#3861	D₅	semidirect
14. C₅²×C₇²	C₃×D₃	semidirect
15. C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	semidirect
18. C₃×C₅×C₇²	C₃×D₅	semidirect
20. C₅×#3861	D₁₅	semidirect
22. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
27. C₃×#3861	C₅⋊D₅	semidirect
32. C₅×C₇²	C₃×D₁₅	semidirect
36. C₅²×C₇	D₃×ΓL(1,𝔽₈)	semidirect
37. C₃⋊(C₅⋊D₅)	#3861	direct (N × H)
39. C₃×C₇²	C₃×C₅⋊D₅	semidirect
41. #3861	C₃⋊(C₅⋊D₅)	direct (N × H)
43. C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	semidirect
50. C₃×C₅²	C₂×#3861	semidirect
54. C₇²	C₃×C₃⋊(C₅⋊D₅)	semidirect
58. C₅×C₇	ΓL(1,𝔽₈)×D₁₅	semidirect
61. C₅²	D₃×#3861	semidirect
62. C₃×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	semidirect
68. C₃×C₅	D₅×#3861	semidirect
72. C₇	ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	semidirect
75. C₅	D₁₅×#3861	semidirect
78. C₃	C₅⋊D₅×#3861	semidirect

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
C₇²×C₃⋊(C₅⋊D₅)	3
C₅⋊D₅×#3861	3
D₁₅×#3861	5
ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	7

Subgroups — #3861×C3⋊(C5⋊D5) #36348047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3861×C₃⋊(C₅⋊D₅) #36348047