Subgroups — #21439613

Order 7644.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838459	C₂	1	2	yes	maximal, complemented
2	#16838449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×D₇²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	#14274262	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₃×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×C₁₃×D₇	C₂×C₃	1	2	yes	complemented
7	C₂×#10587721	—	7	2	no	maximal
8	C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	#64423⋊C₂	—	13	1	no	maximal, Hall
10	C₂×#63271	—	7	2	no	—
11	#10587721	—	7	2	no	—
12	#10587721	—	7	2	no	—
13	#10587721	—	14	3	no	—
14	C₂×C₁₃×D₇	—	7	2	no	—
15	#64424	—	13	1	no	—
16	#64423	—	13	2	no	—
17	#63271	—	7	2	no	—
18	#63271	—	14	3	no	—
19	D₇²	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
20	C₂×C₇×C₁₃	—	7	2	no	—
21	C₁₃×D₇	—	7	2	no	—
22	C₁₃×D₇	—	14	3	no	—
23	C₁₃×D₇	AGL(1,𝔽₇)	1	2	yes	complemented
24	C₂²×C₁₃⋊C₃	—	49	1	no	Hall
25	#3861	—	13	1	no	Hall
26	C₇⋊D₇	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
27	C₇×D₇	C₂×C₁₃⋊C₃	1	2	yes	complemented
28	C₇×C₁₃	—	2	3	no	—
29	C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
30	C₂×AGL(1,𝔽₇)	—	91	2	no	—
31	C₂×C₁₃⋊C₃	—	49	2	no	—
32	C₂×C₁₃⋊C₃	—	49	1	no	—
33	C₂²×C₁₃	—	49	1	no	Hall
34	C₇²	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
35	C₂×ΓL(1,𝔽₈)	—	91	2	no	—
36	AGL(1,𝔽₇)	—	91	2	no	—
37	AGL(1,𝔽₇)	—	91	2	no	—
38	AGL(1,𝔽₇)	—	182	3	no	—
39	C₁₃⋊C₃	—	49	1	no	Hall
40	C₂×D₇	—	7	2	no	—
41	C₂×C₁₃	—	7	2	no	—
42	C₂×C₁₃	—	49	1	no	—
43	ΓL(1,𝔽₈)	—	91	2	no	—
44	ΓL(1,𝔽₈)	—	182	3	no	—
45	C₂×C₇	—	7	2	no	—
46	D₇	—	7	2	no	—
47	D₇	—	14	3	no	—
48	D₇	#10587721	1	2	yes	complemented
49	C₁₃	#64423⋊C₂	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₃	—	637	1	no	Hall
51	C₇	—	2	3	no	—
52	C₇	C₂×#10587721	1	2	yes	complemented
53	C₂×C₃	—	637	2	no	—
54	C₂×C₃	—	637	1	no	—
55	C₂²	—	49	1	no	Sylow
56	C₃	—	637	1	no	Sylow
57	C₂	—	7	2	no	—
58	C₂	—	49	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₁₃	C₂²×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #16838459	C₂	semidirect
2. #16838449	C₂	semidirect
3. C₁₃×D₇²	C₃	semidirect
4. #14274262	C₂²	semidirect
5. C₁₃×C₇⋊D₇	C₂×C₃	semidirect
6. C₇×C₁₃×D₇	C₂×C₃	semidirect
8. C₇²×C₁₃	C₂²×C₃	semidirect
19. D₇²	C₁₃⋊C₃	semidirect
23. C₁₃×D₇	AGL(1,𝔽₇)	semidirect
26. C₇⋊D₇	C₂×C₁₃⋊C₃	semidirect
27. C₇×D₇	C₂×C₁₃⋊C₃	semidirect
29. C₇×C₁₃	C₂×AGL(1,𝔽₇)	semidirect
34. C₇²	C₂²×C₁₃⋊C₃	semidirect
48. D₇	#10587721	semidirect
49. C₁₃	#64423⋊C₂	semidirect
52. C₇	C₂×#10587721	semidirect