Subgroups — C₃×(C₇⋊D₇)⋊D₅₃ #44528449

Order 31164.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×D₃₇₁	C₂	1	2	yes	maximal, complemented
2	C₃×C₅₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₇⋊D₇)⋊D₅₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₇²×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅₃×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₃₇₁	C₂×C₃	1	2	yes	complemented
7	C₃×D₇×D₅₃	—	7	2	no	maximal
8	C₇²×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₃×D₃₇₁	D₇	1	2	yes	complemented
10	C₃×D₇×C₅₃	—	7	2	no	—
11	C₃×D₇×C₅₃	—	14	3	no	—
12	C₃×C₇×D₅₃	—	7	2	no	—
13	D₇×D₅₃	—	7	2	no	—
14	C₃×C₇×C₅₃	—	2	3	no	—
15	C₃×C₇×C₅₃	C₂×D₇	1	2	yes	complemented
16	D₃₇₁	C₃×D₇	1	2	yes	complemented
17	C₇×D₅₃	—	7	2	no	—
18	D₇×C₅₃	—	7	2	no	—
19	D₇×C₅₃	—	14	3	no	—
20	C₂×C₃×D₅₃	—	49	1	no	Hall
21	C₃×D₇²	—	53	1	no	maximal, Hall
22	C₇×C₅₃	—	2	3	no	—
23	C₇×C₅₃	C₂×C₃×D₇	1	2	yes	complemented
24	C₂×C₃×C₅₃	—	49	1	no	—
25	C₃×D₅₃	—	7	2	no	—
26	C₃×C₇⋊D₇	D₅₃	1	1	yes	characteristic, complemented
27	C₃×C₇×D₇	—	53	2	no	—
28	C₂×D₅₃	—	49	1	no	Hall
29	D₇²	—	53	1	no	Hall
30	C₃×C₅₃	D₇²	1	1	yes	characteristic, complemented, Hall
31	C₃×C₇²	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
32	C₂×C₅₃	—	49	1	no	—
33	D₅₃	—	7	2	no	—
34	C₇⋊D₇	C₃×D₅₃	1	1	yes	characteristic, complemented
35	C₇×D₇	—	53	2	no	—
36	C₂×C₃×D₇	—	371	2	no	—
37	C₅₃	C₃×D₇²	1	1	yes	characteristic, complemented, Sylow
38	C₇²	C₂×C₃×D₅₃	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃×C₇	—	371	2	no	—
40	C₃×D₇	—	7	2	no	—
41	C₃×D₇	—	14	3	no	—
42	C₃×D₇	—	53	2	no	—
43	C₂×D₇	—	371	2	no	—
44	C₃×C₇	—	2	3	no	—
45	C₃×C₇	D₇×D₅₃	1	2	yes	complemented
46	C₂×C₇	—	371	2	no	—
47	D₇	—	7	2	no	—
48	D₇	—	14	3	no	—
49	D₇	—	53	2	no	—
50	C₂²×C₃	—	2597	1	no	Hall
51	C₇	—	2	3	no	—
52	C₇	C₃×D₇×D₅₃	1	2	yes	complemented
53	C₂×C₃	—	49	1	no	—
54	C₂×C₃	—	371	2	no	—
55	C₂²	—	2597	1	no	Sylow
56	C₃	(C₇⋊D₇)⋊D₅₃	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	—	49	1	no	—
58	C₂	—	371	2	no	—

	subgroup	G/H
center, hypercenter	C₃	(C₇⋊D₇)⋊D₅₃
commutator, cosocle, hypocenter	C₇²×C₅₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₅₃	C₂²
LCS	G ▷ C₇²×C₅₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×D₃₇₁	C₂	semidirect
2. C₃×C₅₃×C₇⋊D₇	C₂	semidirect
3. (C₇⋊D₇)⋊D₅₃	C₃	direct (N × H)
4. C₃×C₇²×C₅₃	C₂²	semidirect
5. C₅₃×C₇⋊D₇	C₂×C₃	semidirect
6. C₇×D₃₇₁	C₂×C₃	semidirect
8. C₇²×C₅₃	C₂²×C₃	semidirect
9. C₃×D₃₇₁	D₇	semidirect
15. C₃×C₇×C₅₃	C₂×D₇	semidirect
16. D₃₇₁	C₃×D₇	semidirect
23. C₇×C₅₃	C₂×C₃×D₇	semidirect
26. C₃×C₇⋊D₇	D₅₃	semidirect
30. C₃×C₅₃	D₇²	semidirect
31. C₃×C₇²	C₂×D₅₃	semidirect
34. C₇⋊D₇	C₃×D₅₃	semidirect
37. C₅₃	C₃×D₇²	semidirect
38. C₇²	C₂×C₃×D₅₃	semidirect
45. C₃×C₇	D₇×D₅₃	semidirect
52. C₇	C₃×D₇×D₅₃	semidirect
56. C₃	(C₇⋊D₇)⋊D₅₃	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×C₇×D₃₇₁	2
C₃×C₅₃×C₇⋊D₇	2
(C₇⋊D₇)⋊D₅₃	3
C₃×D₇×D₅₃	7
C₃×D₇²	53

Subgroups — C3×(C7⋊D7)⋊D53 #44528449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₇⋊D₇)⋊D₅₃ #44528449