Subgroups — #21439608

Order 7644.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838458	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838456	C₂	1	1	yes	maximal, characteristic, complemented
3	#16838446	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₉₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	#14274263	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇⋊D₉₁	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₉₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₃×D₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×#10587722	—	7	1	no	maximal
10	#10587716⋊C₂	—	7	1	no	maximal
11	C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	#64423⋊C₂	—	13	1	no	maximal, Hall
13	C₂×#63270	—	7	1	no	—
14	#10587722	—	7	1	no	—
15	#10587722	—	7	1	no	—
16	#10587722	—	7	1	no	—
17	#10587722	—	14	3	no	—
18	#10587720	—	7	1	no	—
19	#10587716	—	7	1	no	—
20	C₂×D₉₁	—	7	1	no	—
21	D₇×D₁₃	—	7	1	no	—
22	#64424	—	13	1	no	—
23	#64423	—	13	1	no	—
24	#64423	—	13	1	no	—
25	#63270	—	7	1	no	—
26	#63270	—	7	1	no	—
27	#63270	—	14	3	no	—
28	D₇²	—	13	1	no	Hall
29	C₂×C₇×C₁₃	—	7	1	no	—
30	D₉₁	—	7	1	no	—
31	D₉₁	—	7	1	no	—
32	D₉₁	—	14	3	no	—
33	D₉₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	C₇×D₁₃	—	7	1	no	—
35	C₁₃×D₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
36	C₂×D₁₃⋊C₃	—	49	1	no	Hall
37	#3861	—	13	1	no	Hall
38	C₇⋊D₇	—	13	1	no	—
39	C₇×D₇	—	13	1	no	—
40	C₇×D₇	D₁₃⋊C₃	1	1	yes	characteristic, complemented
41	C₇×C₁₃	—	2	3	no	—
42	C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
43	C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
44	C₂×AGL(1,𝔽₇)	—	91	1	no	—
45	C₂×AGL(1,𝔽₇)	—	91	1	no	—
46	C₂×C₁₃⋊C₃	—	49	1	no	—
47	D₁₃⋊C₃	—	49	1	no	—
48	D₁₃⋊C₃	—	49	1	no	—
49	C₂×D₁₃	—	49	1	no	Hall
50	C₇²	C₂×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
51	C₂×ΓL(1,𝔽₈)	—	91	1	no	—
52	C₂×ΓL(1,𝔽₈)	—	91	1	no	—
53	AGL(1,𝔽₇)	—	91	1	no	—
54	AGL(1,𝔽₇)	—	91	1	no	—
55	AGL(1,𝔽₇)	—	91	1	no	—
56	AGL(1,𝔽₇)	—	91	1	no	—
57	AGL(1,𝔽₇)	—	182	3	no	—
58	C₁₃⋊C₃	—	49	1	no	Hall
59	C₂×D₇	—	91	1	no	—
60	C₂×D₇	—	91	1	no	—
61	C₂×C₁₃	—	7	1	no	—
62	D₁₃	—	7	1	no	—
63	D₁₃	—	49	1	no	—
64	ΓL(1,𝔽₈)	—	91	1	no	—
65	ΓL(1,𝔽₈)	—	91	1	no	—
66	ΓL(1,𝔽₈)	—	182	3	no	—
67	C₂×C₇	—	7	1	no	—
68	C₂×C₇	—	91	1	no	—
69	D₇	—	13	1	no	—
70	D₇	—	91	1	no	—
71	D₇	—	91	1	no	—
72	D₇	—	182	3	no	—
73	D₇	#10587722	1	1	yes	characteristic, complemented
74	C₁₃	#64423⋊C₂	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₃	—	637	1	no	Hall
76	C₇	—	2	3	no	—
77	C₇	#10587716⋊C₂	1	1	yes	characteristic, complemented
78	C₇	C₂×#10587722	1	1	yes	characteristic, complemented
79	C₂×C₃	—	637	1	no	—
80	C₂×C₃	—	637	1	no	—
81	C₂×C₃	—	637	1	no	—
82	C₂²	—	637	1	no	Sylow
83	C₃	—	637	1	no	Sylow
84	C₂	—	7	1	no	—
85	C₂	—	91	1	no	—
86	C₂	—	637	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. #16838458	C₂	semidirect
2. #16838456	C₂	semidirect
3. #16838446	C₂	semidirect
4. D₇×D₉₁	C₃	semidirect
5. #14274263	C₂²	semidirect
6. C₇⋊D₉₁	C₂×C₃	semidirect
7. C₇×D₉₁	C₂×C₃	semidirect
8. C₇×C₁₃×D₇	C₂×C₃	semidirect
11. C₇²×C₁₃	C₂²×C₃	semidirect
33. D₉₁	AGL(1,𝔽₇)	semidirect
35. C₁₃×D₇	AGL(1,𝔽₇)	semidirect
40. C₇×D₇	D₁₃⋊C₃	semidirect
42. C₇×C₁₃	C₂×AGL(1,𝔽₇)	semidirect
43. C₇×C₁₃	C₂×AGL(1,𝔽₇)	semidirect
50. C₇²	C₂×D₁₃⋊C₃	semidirect
73. D₇	#10587722	semidirect
74. C₁₃	#64423⋊C₂	semidirect
77. C₇	#10587716⋊C₂	semidirect
78. C₇	C₂×#10587722	semidirect