Subgroups — C₅²×C₃⋊#64418 #36348020

Order 22050.

304 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	C₅²×C₇×D₂₁	—	3	1	no	maximal
3	C₅²×#64418	—	3	3	no	maximal
4	C₅×C₃⋊#64418	C₅	1	6	yes	maximal, direct product factor
5	C₅²×#3862	D₃	1	3	yes	complemented
6	C₃×C₅²×C₇²	D₃	1	1	yes	characteristic, complemented
7	C₅²×C₇×D₇	—	9	1	no	Hall
8	C₃×C₅×#3862	C₂×C₅	1	6	yes	complemented
9	C₃×C₅²×ΓL(1,𝔽₈)	—	14	1	no	—
10	C₅×C₇×D₂₁	—	3	6	no	—
11	C₅×#64418	—	3	18	no	—
12	C₅²×C₇²	C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
13	C₅²×D₂₁	—	3	1	no	—
14	C₅²×D₃×C₇	—	21	1	no	—
15	C₃⋊#64418	C₅²	1	1	yes	characteristic, direct product factor, Hall
16	C₃×C₅×C₇²	C₅×D₃	1	6	yes	complemented
17	C₅×#3862	C₅×D₃	1	18	yes	complemented
18	C₃×C₅²×C₇	—	2	1	no	—
19	C₃×C₅²×C₇	—	3	1	no	—
20	C₃×C₅²×C₇	—	3	1	no	—
21	C₅²×ΓL(1,𝔽₈)	—	14	3	no	—
22	C₅×C₇×D₇	—	9	6	no	—
23	C₅²×C₃⋊D₃	—	49	1	no	maximal, Hall
24	C₃×#3862	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
25	C₂×C₅²×C₇	—	63	1	no	—
26	C₅²×D₇	—	9	1	no	—
27	C₃×C₅×ΓL(1,𝔽₈)	—	14	6	no	—
28	C₇×D₂₁	—	3	1	no	—
29	#64418	—	3	3	no	—
30	C₅×C₇²	C₅×C₃⋊D₃	1	6	yes	complemented
31	C₃²×C₅²	—	49	1	no	Hall
32	C₅×D₂₁	—	3	6	no	—
33	C₅×D₃×C₇	—	21	6	no	—
34	C₅²×C₇	—	2	1	no	—
35	C₅²×C₇	—	3	1	no	—
36	C₅²×C₇	—	3	1	no	—
37	C₅²×D₃	—	21	1	no	—
38	C₅²×D₃	—	147	3	no	—
39	C₃×C₇²	C₅²×D₃	1	1	yes	characteristic, complemented
40	#3862	C₅²×D₃	1	3	yes	complemented
41	C₃×C₅×C₇	—	2	6	no	—
42	C₃×C₅×C₇	—	3	6	no	—
43	C₃×C₅×C₇	—	3	6	no	—
44	C₅×ΓL(1,𝔽₈)	—	14	18	no	—
45	C₇×D₇	—	9	1	no	Hall
46	C₅×C₃⋊D₃	—	49	6	no	—
47	C₃×C₅²	—	49	3	no	—
48	C₃×C₅²	#64418	1	1	yes	characteristic, complemented
49	C₂×C₅×C₇	—	63	6	no	—
50	C₅×D₇	—	9	6	no	—
51	C₃×ΓL(1,𝔽₈)	—	14	1	no	—
52	C₂×C₅²	—	63	1	no	Hall
53	C₇²	C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
54	C₃²×C₅	—	49	6	no	—
55	D₂₁	—	3	1	no	—
56	D₃×C₇	—	21	1	no	—
57	C₅×C₇	—	2	6	no	—
58	C₅×C₇	—	3	6	no	—
59	C₅×C₇	—	3	6	no	—
60	C₅×D₃	—	21	6	no	—
61	C₅×D₃	—	147	18	no	—
62	C₅²	C₃⋊#64418	1	1	yes	characteristic, direct product factor, Sylow
63	C₃×C₇	—	2	1	no	—
64	C₃×C₇	—	3	1	no	—
65	C₃×C₇	—	3	1	no	—
66	ΓL(1,𝔽₈)	—	14	3	no	—
67	C₃⋊D₃	—	49	1	no	Hall
68	C₃×C₅	—	49	18	no	—
69	C₃×C₅	C₅×#64418	1	6	yes	complemented
70	C₂×C₇	—	63	1	no	—
71	D₇	—	9	1	no	—
72	C₂×C₅	—	63	6	no	—
73	C₃²	—	49	1	no	Sylow
74	C₇	—	2	1	no	—
75	C₇	—	3	1	no	—
76	C₇	—	3	1	no	—
77	D₃	—	21	1	no	—
78	D₃	—	147	3	no	—
79	C₅	C₅×C₃⋊#64418	1	6	yes	direct product factor
80	C₃	—	49	3	no	—
81	C₃	C₅²×#64418	1	1	yes	characteristic, complemented
82	C₂	—	63	1	no	Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3862	C₂	semidirect
4. C₅×C₃⋊#64418	C₅	direct (N × H)
5. C₅²×#3862	D₃	semidirect
6. C₃×C₅²×C₇²	D₃	semidirect
8. C₃×C₅×#3862	C₂×C₅	semidirect
12. C₅²×C₇²	C₃⋊D₃	semidirect
15. C₃⋊#64418	C₅²	direct (N × H)
16. C₃×C₅×C₇²	C₅×D₃	semidirect
17. C₅×#3862	C₅×D₃	semidirect
24. C₃×#3862	C₂×C₅²	semidirect
30. C₅×C₇²	C₅×C₃⋊D₃	semidirect
39. C₃×C₇²	C₅²×D₃	semidirect
40. #3862	C₅²×D₃	semidirect
48. C₃×C₅²	#64418	semidirect
53. C₇²	C₅²×C₃⋊D₃	semidirect
62. C₅²	C₃⋊#64418	direct (N × H)
69. C₃×C₅	C₅×#64418	semidirect
79. C₅	C₅×C₃⋊#64418	direct (N × H)
81. C₃	C₅²×#64418	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
C₅²×C₇×D₂₁	3
C₅²×#64418	3
C₅×C₃⋊#64418	5
C₅²×C₃⋊D₃	49

Subgroups — C52×C3⋊#64418 #36348020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃⋊#64418 #36348020