Subgroups — D₃×C₅₃×C₇⋊D₇ #44528472

Order 31164.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₇²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₃⋊(C₇⋊D₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×C₇⋊D₇	—	3	1	no	maximal, Hall
5	C₃×C₇²×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅₃×C₇⋊D₇	—	3	1	no	—
7	C₅₃×C₇⋊D₇	D₃	1	1	yes	characteristic, direct product factor
8	C₂×C₇²×C₅₃	—	3	1	no	—
9	D₃×D₇×C₅₃	—	7	8	no	maximal
10	C₇²×C₅₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	D₂₁×C₅₃	—	7	8	no	—
12	D₃×C₇×C₅₃	D₇	1	8	yes	complemented
13	C₃×D₇×C₅₃	—	7	8	no	—
14	C₂×D₇×C₅₃	—	21	8	no	—
15	C₃×C₇×C₅₃	C₂×D₇	1	8	yes	complemented
16	C₂×C₇×C₅₃	—	3	8	no	—
17	D₇×C₅₃	—	7	8	no	—
18	D₇×C₅₃	—	21	8	no	—
19	C₂×D₃×C₅₃	—	49	1	no	Hall
20	D₃×C₇⋊D₇	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₇×C₅₃	D₃×D₇	1	8	yes	complemented
22	C₂×C₃×C₅₃	—	49	1	no	—
23	D₃×C₅₃	—	49	1	no	—
24	D₃×C₅₃	C₇⋊D₇	1	1	yes	characteristic, direct product factor
25	C₃⋊(C₇⋊D₇)	C₂×C₅₃	1	1	yes	characteristic, complemented
26	D₃×C₇²	C₂×C₅₃	1	1	yes	characteristic, complemented
27	C₃×C₇⋊D₇	C₂×C₅₃	1	1	yes	characteristic, complemented
28	C₂²×C₅₃	—	147	1	no	Hall
29	C₂×C₇⋊D₇	—	3	1	no	Hall
30	C₃×C₅₃	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
31	C₃×C₇²	C₂²×C₅₃	1	1	yes	characteristic, complemented, Hall
32	C₂×C₅₃	—	3	1	no	—
33	C₂×C₅₃	—	49	1	no	—
34	C₂×C₅₃	—	147	1	no	—
35	C₂×C₇²	—	3	1	no	—
36	C₇⋊D₇	—	3	1	no	—
37	C₇⋊D₇	D₃×C₅₃	1	1	yes	characteristic, direct product factor
38	D₃×D₇	—	7	8	no	—
39	C₅₃	D₃×C₇⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
40	C₇²	C₂×D₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
41	D₂₁	—	7	8	no	—
42	C₃×D₇	—	7	8	no	—
43	D₃×C₇	D₇×C₅₃	1	8	yes	complemented
44	C₂×D₇	—	21	8	no	—
45	C₃×C₇	C₂×D₇×C₅₃	1	8	yes	complemented
46	C₂×C₇	—	3	8	no	—
47	D₇	—	7	8	no	—
48	D₇	—	21	8	no	—
49	C₂×D₃	—	49	1	no	Hall
50	C₇	D₃×D₇×C₅₃	1	8	yes	complemented
51	C₂×C₃	—	49	1	no	—
52	D₃	—	49	1	no	—
53	D₃	C₅₃×C₇⋊D₇	1	1	yes	characteristic, direct product factor
54	C₂²	—	147	1	no	Sylow
55	C₃	C₂×C₅₃×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	3	1	no	—
57	C₂	—	49	1	no	—
58	C₂	—	147	1	no	—

	subgroup	G/H
center, hypercenter	C₅₃	D₃×C₇⋊D₇
commutator, cosocle, hypocenter	C₃×C₇²	C₂²×C₅₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₅₃	C₂²
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₅₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×C₇⋊D₇	C₂	semidirect
2. D₃×C₇²×C₅₃	C₂	semidirect
3. C₅₃×C₃⋊(C₇⋊D₇)	C₂	semidirect
5. C₃×C₇²×C₅₃	C₂²	semidirect
7. C₅₃×C₇⋊D₇	D₃	direct (N × H)
10. C₇²×C₅₃	C₂×D₃	semidirect
12. D₃×C₇×C₅₃	D₇	semidirect
15. C₃×C₇×C₅₃	C₂×D₇	semidirect
20. D₃×C₇⋊D₇	C₅₃	direct (N × H)
21. C₇×C₅₃	D₃×D₇	semidirect
24. D₃×C₅₃	C₇⋊D₇	direct (N × H)
25. C₃⋊(C₇⋊D₇)	C₂×C₅₃	semidirect
26. D₃×C₇²	C₂×C₅₃	semidirect
27. C₃×C₇⋊D₇	C₂×C₅₃	semidirect
30. C₃×C₅₃	C₂×C₇⋊D₇	semidirect
31. C₃×C₇²	C₂²×C₅₃	semidirect
37. C₇⋊D₇	D₃×C₅₃	direct (N × H)
39. C₅₃	D₃×C₇⋊D₇	direct (N × H)
40. C₇²	C₂×D₃×C₅₃	semidirect
43. D₃×C₇	D₇×C₅₃	semidirect
45. C₃×C₇	C₂×D₇×C₅₃	semidirect
50. C₇	D₃×D₇×C₅₃	semidirect
53. D₃	C₅₃×C₇⋊D₇	direct (N × H)
55. C₃	C₂×C₅₃×C₇⋊D₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅₃×C₇⋊D₇	2
D₃×C₇²×C₅₃	2
C₅₃×C₃⋊(C₇⋊D₇)	2
C₂×C₅₃×C₇⋊D₇	3
D₃×D₇×C₅₃	7
D₃×C₇⋊D₇	53

Subgroups — D3×C53×C7⋊D7 #44528472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₅₃×C₇⋊D₇ #44528472