Subgroups — C₃×C₅²×#64418 #36347960

Order 22050.

256 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	C₃×C₅²×C₇×D₇	—	3	1	no	maximal
3	C₅²×#64418	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×C₅×#64418	C₅	1	6	yes	maximal, direct product factor
5	C₅²×#3862	—	2	1	no	—
6	C₅²×#3862	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₇²	D₃	1	1	yes	characteristic, complemented
8	C₅²×C₇×D₇	—	3	1	no	Hall
9	C₃×C₅×#3862	C₂×C₅	1	6	yes	complemented
10	C₃×C₅²×ΓL(1,𝔽₈)	—	14	1	no	—
11	C₃×C₅×C₇×D₇	—	3	6	no	—
12	C₅×#64418	C₃×C₅	1	6	yes	direct product factor
13	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅²×C₇	—	21	1	no	—
15	C₃×C₅²×D₇	—	3	1	no	—
16	C₃×#64418	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₅×C₇²	C₅×D₃	1	6	yes	complemented
18	C₅×#3862	—	2	6	no	—
19	C₅×#3862	C₂×C₃×C₅	1	6	yes	complemented
20	C₃×C₅²×C₇	—	2	1	no	—
21	C₃×C₅²×C₇	—	3	1	no	—
22	C₃×C₅²×C₇	—	3	1	no	—
23	C₅²×ΓL(1,𝔽₈)	—	14	1	no	—
24	C₅²×ΓL(1,𝔽₈)	—	14	2	no	—
25	C₅×C₇×D₇	—	3	6	no	—
26	C₃×C₅²×D₃	—	49	1	no	maximal, Hall
27	C₃×#3862	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅²×C₇	—	21	1	no	—
29	C₅²×D₇	—	3	1	no	—
30	C₃×C₅×ΓL(1,𝔽₈)	—	14	6	no	—
31	C₃×C₇×D₇	—	3	1	no	—
32	#64418	C₃×C₅²	1	1	yes	characteristic, direct product factor
33	C₅×C₇²	C₃×C₅×D₃	1	6	yes	complemented
34	C₃²×C₅²	—	49	1	no	Hall
35	C₂×C₃×C₅×C₇	—	21	6	no	—
36	C₃×C₅×D₇	—	3	6	no	—
37	C₅²×C₇	—	2	1	no	—
38	C₅²×C₇	—	3	1	no	—
39	C₅²×C₇	—	3	1	no	—
40	C₂×C₃×C₅²	—	21	1	no	—
41	C₅²×D₃	—	49	1	no	—
42	C₃×C₇²	C₅²×D₃	1	1	yes	characteristic, complemented
43	#3862	—	2	1	no	—
44	#3862	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
45	C₃×C₅×C₇	—	2	6	no	—
46	C₃×C₅×C₇	—	3	6	no	—
47	C₃×C₅×C₇	—	3	6	no	—
48	C₅×ΓL(1,𝔽₈)	—	14	6	no	—
49	C₅×ΓL(1,𝔽₈)	—	14	12	no	—
50	C₇×D₇	—	3	1	no	Hall
51	C₃×C₅×D₃	—	49	6	no	—
52	C₃×C₅²	—	49	1	no	—
53	C₃×C₅²	—	98	1	no	—
54	C₃×C₅²	#64418	1	1	yes	characteristic, direct product factor
55	C₂×C₅×C₇	—	21	6	no	—
56	C₅×D₇	—	3	6	no	—
57	C₃×ΓL(1,𝔽₈)	—	14	1	no	—
58	C₂×C₅²	—	21	1	no	Hall
59	C₇²	C₃×C₅²×D₃	1	1	yes	characteristic, complemented, Sylow
60	C₃²×C₅	—	49	6	no	—
61	C₂×C₃×C₇	—	21	1	no	—
62	C₃×D₇	—	3	1	no	—
63	C₅×C₇	—	2	6	no	—
64	C₅×C₇	—	3	6	no	—
65	C₅×C₇	—	3	6	no	—
66	C₂×C₃×C₅	—	21	6	no	—
67	C₅×D₃	—	49	6	no	—
68	C₅²	C₃×#64418	1	1	yes	characteristic, direct product factor, Sylow
69	C₃×C₇	—	2	1	no	—
70	C₃×C₇	—	3	1	no	—
71	C₃×C₇	—	3	1	no	—
72	ΓL(1,𝔽₈)	—	14	1	no	—
73	ΓL(1,𝔽₈)	—	14	2	no	—
74	C₃×D₃	—	49	1	no	Hall
75	C₃×C₅	—	49	6	no	—
76	C₃×C₅	—	98	6	no	—
77	C₃×C₅	C₅×#64418	1	6	yes	direct product factor
78	C₂×C₇	—	21	1	no	—
79	D₇	—	3	1	no	—
80	C₂×C₅	—	21	6	no	—
81	C₃²	—	49	1	no	Sylow
82	C₇	—	2	1	no	—
83	C₇	—	3	1	no	—
84	C₇	—	3	1	no	—
85	C₂×C₃	—	21	1	no	—
86	D₃	—	49	1	no	—
87	C₅	C₃×C₅×#64418	1	6	yes	direct product factor
88	C₃	—	49	1	no	—
89	C₃	—	98	1	no	—
90	C₃	C₅²×#64418	1	1	yes	characteristic, direct product factor
91	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅²	#64418
commutator, cosocle, hypocenter	#3862	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	D₃
LCS	G ▷ #3862
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3862	C₂	semidirect
3. C₅²×#64418	C₃	direct (N × H)
4. C₃×C₅×#64418	C₅	direct (N × H)
6. C₅²×#3862	C₂×C₃	semidirect
7. C₃×C₅²×C₇²	D₃	semidirect
9. C₃×C₅×#3862	C₂×C₅	semidirect
12. C₅×#64418	C₃×C₅	direct (N × H)
13. C₅²×C₇²	C₃×D₃	semidirect
16. C₃×#64418	C₅²	direct (N × H)
17. C₃×C₅×C₇²	C₅×D₃	semidirect
19. C₅×#3862	C₂×C₃×C₅	semidirect
27. C₃×#3862	C₂×C₅²	semidirect
32. #64418	C₃×C₅²	direct (N × H)
33. C₅×C₇²	C₃×C₅×D₃	semidirect
42. C₃×C₇²	C₅²×D₃	semidirect
44. #3862	C₂×C₃×C₅²	semidirect
54. C₃×C₅²	#64418	direct (N × H)
59. C₇²	C₃×C₅²×D₃	semidirect
68. C₅²	C₃×#64418	direct (N × H)
77. C₃×C₅	C₅×#64418	direct (N × H)
87. C₅	C₃×C₅×#64418	direct (N × H)
90. C₃	C₅²×#64418	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
C₃×C₅²×C₇×D₇	3
C₅²×#64418	3
C₃×C₅×#64418	5
C₃×C₅²×D₃	49

Subgroups — C3×C52×#64418 #36347960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×#64418 #36347960