Subgroups — #3862×C₃⋊(C₅⋊D₅) #36348051

Order 22050.

352 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	C₇²×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊D₅×#3862	—	3	1	no	maximal
4	D₁₅×#3862	—	5	6	no	maximal
5	C₅²×#3862	—	2	1	no	—
6	C₅²×#3862	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	2	no	maximal
9	C₇²×C₅⋊D₅	—	3	1	no	Hall
10	C₃×C₅×#3862	D₅	1	6	yes	complemented
11	C₃×C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
12	C₇²×D₁₅	—	5	6	no	—
13	D₅×#3862	—	15	6	no	—
14	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₇×C₃⋊(C₅⋊D₅)	—	3	2	no	—
16	C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	1	2	yes	complemented
17	ΓL(1,𝔽₈)×C₅⋊D₅	—	21	2	no	—
18	D₃×#3862	—	25	1	no	Hall
19	C₃×C₅×C₇²	C₃×D₅	1	6	yes	complemented
20	C₅×#3862	—	2	6	no	—
21	C₅×#3862	D₁₅	1	6	yes	complemented
22	ΓL(1,𝔽₈)×D₁₅	—	35	12	no	—
23	C₃×C₅²×C₇	—	3	2	no	—
24	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
25	C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
26	C₅²×ΓL(1,𝔽₈)	—	14	2	no	—
27	C₇²×D₅	—	15	6	no	—
28	C₃×C₃⋊(C₅⋊D₅)	—	49	1	no	Hall
29	C₃×#3862	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
30	C₇×C₅⋊D₅	—	3	2	no	—
31	C₇×C₅⋊D₅	—	9	2	no	—
32	C₃×C₅×ΓL(1,𝔽₈)	—	7	12	no	—
33	D₃×C₇²	—	25	1	no	—
34	C₂×#3862	—	75	1	no	—
35	C₅×C₇²	C₃×D₁₅	1	6	yes	complemented
36	C₃²×C₅²	—	49	1	no	Hall
37	C₇×D₁₅	—	5	12	no	—
38	C₇×D₁₅	—	15	12	no	—
39	D₅×ΓL(1,𝔽₈)	—	105	12	no	—
40	C₅²×C₇	—	3	2	no	—
41	C₅²×C₇	D₃×ΓL(1,𝔽₈)	1	2	yes	complemented
42	C₃⋊(C₅⋊D₅)	#3862	1	1	yes	characteristic, direct product factor
43	C₃×C₅⋊D₅	—	147	1	no	—
44	C₃×C₇²	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
45	#3862	—	2	1	no	—
46	#3862	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
47	D₃×ΓL(1,𝔽₈)	—	175	2	no	—
48	C₃×C₅×C₇	—	3	12	no	—
49	C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	1	12	yes	complemented
50	C₅×ΓL(1,𝔽₈)	—	7	12	no	—
51	C₅×ΓL(1,𝔽₈)	—	14	12	no	—
52	C₂×C₇²	—	75	1	no	Hall
53	C₃×D₁₅	—	245	6	no	—
54	C₃×C₅²	—	49	1	no	—
55	C₃×C₅²	—	98	1	no	—
56	C₃×C₅²	C₂×#3862	1	1	yes	characteristic, complemented
57	C₇×D₅	—	15	12	no	—
58	C₇×D₅	—	45	12	no	—
59	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
60	C₅⋊D₅	—	3	1	no	Hall
61	C₇²	C₃×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
62	C₃²×C₅	—	49	6	no	—
63	D₃×C₇	—	25	2	no	—
64	D₃×C₇	—	75	2	no	—
65	C₂×ΓL(1,𝔽₈)	—	525	2	no	—
66	C₅×C₇	—	3	12	no	—
67	C₅×C₇	ΓL(1,𝔽₈)×D₁₅	1	12	yes	complemented
68	D₁₅	—	5	6	no	—
69	C₃×D₅	—	735	6	no	—
70	C₅²	D₃×#3862	1	1	yes	characteristic, complemented, Sylow
71	C₃×C₇	—	3	2	no	—
72	C₃×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	1	2	yes	complemented
73	ΓL(1,𝔽₈)	—	7	2	no	—
74	ΓL(1,𝔽₈)	—	14	2	no	—
75	C₃×D₃	—	1225	1	no	Hall
76	C₃×C₅	—	49	6	no	—
77	C₃×C₅	—	98	6	no	—
78	C₃×C₅	D₅×#3862	1	6	yes	complemented
79	C₂×C₇	—	75	2	no	—
80	C₂×C₇	—	225	2	no	—
81	D₅	—	15	6	no	—
82	C₃²	—	49	1	no	Sylow
83	C₇	—	3	2	no	—
84	C₇	ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	1	2	yes	complemented
85	C₂×C₃	—	3675	1	no	—
86	D₃	—	25	1	no	—
87	C₅	D₁₅×#3862	1	6	yes	complemented
88	C₃	—	49	1	no	—
89	C₃	—	98	1	no	—
90	C₃	C₅⋊D₅×#3862	1	1	yes	characteristic, complemented
91	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×#3862	C₂	semidirect
2. C₇²×C₃⋊(C₅⋊D₅)	C₃	semidirect
6. C₅²×#3862	D₃	semidirect
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
10. C₃×C₅×#3862	D₅	semidirect
14. C₅²×C₇²	C₃×D₃	semidirect
16. C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	semidirect
19. C₃×C₅×C₇²	C₃×D₅	semidirect
21. C₅×#3862	D₁₅	semidirect
24. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
29. C₃×#3862	C₅⋊D₅	semidirect
35. C₅×C₇²	C₃×D₁₅	semidirect
41. C₅²×C₇	D₃×ΓL(1,𝔽₈)	semidirect
42. C₃⋊(C₅⋊D₅)	#3862	direct (N × H)
44. C₃×C₇²	C₃×C₅⋊D₅	semidirect
46. #3862	C₃⋊(C₅⋊D₅)	direct (N × H)
49. C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	semidirect
56. C₃×C₅²	C₂×#3862	semidirect
61. C₇²	C₃×C₃⋊(C₅⋊D₅)	semidirect
67. C₅×C₇	ΓL(1,𝔽₈)×D₁₅	semidirect
70. C₅²	D₃×#3862	semidirect
72. C₃×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	semidirect
78. C₃×C₅	D₅×#3862	semidirect
84. C₇	ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	semidirect
87. C₅	D₁₅×#3862	semidirect
90. C₃	C₅⋊D₅×#3862	semidirect

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
C₇²×C₃⋊(C₅⋊D₅)	3
C₅⋊D₅×#3862	3
D₁₅×#3862	5
ΓL(1,𝔽₈)×C₃⋊(C₅⋊D₅)	7

Subgroups — #3862×C3⋊(C5⋊D5) #36348051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3862×C₃⋊(C₅⋊D₅) #36348051