Subgroups — C₂×C₅₃×#64425 #44528439

Order 31164.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#3862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#64425	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅₃×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅₃×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅₃×C₇⋊D₇	C₂×C₃	1	2	yes	complemented
6	C₂×C₇²×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×AGL(1,𝔽₇)×C₅₃	—	7	2	no	maximal
8	C₇²×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	AGL(1,𝔽₇)×C₅₃	—	7	4	no	—
10	C₂×ΓL(1,𝔽₈)×C₅₃	—	7	2	no	—
11	C₂×D₇×C₅₃	—	7	2	no	—
12	C₂×D₇×C₅₃	—	21	2	no	—
13	ΓL(1,𝔽₈)×C₅₃	—	7	2	no	—
14	C₂×C₇×C₅₃	—	3	2	no	—
15	C₂×C₇×C₅₃	AGL(1,𝔽₇)	1	2	yes	complemented
16	D₇×C₅₃	—	7	4	no	—
17	D₇×C₅₃	—	21	4	no	—
18	C₂²×C₃×C₅₃	—	49	1	no	Hall
19	C₂×#64425	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₇×C₅₃	—	3	2	no	—
21	C₇×C₅₃	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
22	C₂×C₃×C₅₃	—	49	2	no	—
23	C₂×C₃×C₅₃	—	49	1	no	—
24	C₂×#3862	C₂×C₅₃	1	1	yes	characteristic, complemented
25	#64425	C₂×C₅₃	1	2	yes	direct product factor
26	C₂²×C₅₃	—	49	1	no	Hall
27	C₂×C₇⋊D₇	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
28	C₃×C₅₃	—	49	1	no	Hall
29	#3862	C₂²×C₅₃	1	1	yes	characteristic, complemented, Hall
30	C₂×C₅₃	—	49	2	no	—
31	C₂×C₅₃	#64425	1	1	yes	characteristic, direct product factor
32	C₂×C₇²	C₂×C₃×C₅₃	1	1	yes	characteristic, complemented
33	C₇⋊D₇	C₂×C₃×C₅₃	1	2	yes	complemented
34	C₂×AGL(1,𝔽₇)	—	7	2	no	—
35	C₅₃	C₂×#64425	1	1	yes	characteristic, direct product factor, Sylow
36	C₇²	C₂²×C₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
37	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
38	AGL(1,𝔽₇)	—	7	4	no	—
39	C₂×D₇	—	7	2	no	—
40	C₂×D₇	—	21	2	no	—
41	ΓL(1,𝔽₈)	—	7	2	no	—
42	C₂×C₇	—	3	2	no	—
43	C₂×C₇	AGL(1,𝔽₇)×C₅₃	1	2	yes	complemented
44	D₇	—	7	4	no	—
45	D₇	—	21	4	no	—
46	C₂²×C₃	—	49	1	no	Hall
47	C₇	—	3	2	no	—
48	C₇	C₂×AGL(1,𝔽₇)×C₅₃	1	2	yes	complemented
49	C₂×C₃	—	49	2	no	—
50	C₂×C₃	—	49	1	no	—
51	C₂²	—	49	1	no	Sylow
52	C₃	—	49	1	no	Sylow
53	C₂	—	49	2	no	—
54	C₂	C₅₃×#64425	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅₃	#64425
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₅₃	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#3862	C₂	semidirect
2. C₅₃×#64425	C₂	direct (N × H)
3. C₂×C₅₃×C₇⋊D₇	C₃	semidirect
4. C₅₃×#3862	C₂²	semidirect
5. C₅₃×C₇⋊D₇	C₂×C₃	semidirect
6. C₂×C₇²×C₅₃	C₂×C₃	semidirect
8. C₇²×C₅₃	C₂²×C₃	semidirect
15. C₂×C₇×C₅₃	AGL(1,𝔽₇)	semidirect
19. C₂×#64425	C₅₃	direct (N × H)
21. C₇×C₅₃	C₂×AGL(1,𝔽₇)	semidirect
24. C₂×#3862	C₂×C₅₃	semidirect
25. #64425	C₂×C₅₃	direct (N × H)
27. C₂×C₇⋊D₇	C₃×C₅₃	semidirect
29. #3862	C₂²×C₅₃	semidirect
31. C₂×C₅₃	#64425	direct (N × H)
32. C₂×C₇²	C₂×C₃×C₅₃	semidirect
33. C₇⋊D₇	C₂×C₃×C₅₃	semidirect
35. C₅₃	C₂×#64425	direct (N × H)
36. C₇²	C₂²×C₃×C₅₃	semidirect
43. C₂×C₇	AGL(1,𝔽₇)×C₅₃	semidirect
48. C₇	C₂×AGL(1,𝔽₇)×C₅₃	semidirect
54. C₂	C₅₃×#64425	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#3862	2
C₅₃×#64425	2
C₂×C₅₃×C₇⋊D₇	3
C₂×AGL(1,𝔽₇)×C₅₃	7
C₂×#64425	53

Subgroups — C2×C53×#64425 #44528439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#64425 #44528439