Subgroups — (C₃⋊AGL(1,𝔽₇))⋊D₇ #13183641

Order 1764.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₃⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₁⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
5	D₇×AGL(1,𝔽₇)	—	3	1	no	maximal
6	C₃×C₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₇×D₂₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₂₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)⋊D₇	—	2	1	no	—
11	ΓL(1,𝔽₈)⋊D₇	D₃	1	1	yes	characteristic, complemented
12	D₇×ΓL(1,𝔽₈)	—	3	1	no	—
13	C₇×AGL(1,𝔽₇)	—	3	1	no	—
14	C₃×D₃×D₇	—	7	1	no	maximal
15	D₃×AGL(1,𝔽₇)	—	7	1	no	maximal
16	D₇²	—	3	1	no	Hall
17	C₃×C₇²	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
19	C₇×ΓL(1,𝔽₈)	C₂×D₃	1	1	yes	characteristic, complemented
20	C₃×D₂₁	—	7	1	no	—
21	C₃×D₃×C₇	—	7	1	no	—
22	C₃²×D₇	—	7	1	no	—
23	C₃⋊AGL(1,𝔽₇)	D₇	1	1	yes	characteristic, complemented
24	D₃×ΓL(1,𝔽₈)	—	7	1	no	—
25	C₃×AGL(1,𝔽₇)	—	7	1	no	—
26	C₇⋊D₇	C₃×D₃	1	1	yes	characteristic, complemented
27	C₇×D₇	—	3	1	no	—
28	C₇×D₇	—	3	1	no	—
29	C₂×C₃×D₇	—	21	1	no	—
30	D₃×D₇	—	7	1	no	—
31	D₃×D₇	—	7	1	no	—
32	C₂×AGL(1,𝔽₇)	—	21	1	no	—
33	C₃²×C₇	—	7	1	no	—
34	C₃×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic, complemented
35	C₇²	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃×C₇	—	21	1	no	—
37	D₂₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
38	D₂₁	C₃×D₇	1	1	yes	characteristic, complemented
39	C₃×D₇	—	7	1	no	—
40	C₃×D₇	—	7	1	no	—
41	C₃×D₇	—	7	1	no	—
42	C₃×D₇	—	14	1	no	—
43	C₃×D₇	—	21	1	no	—
44	C₃×D₇	—	42	1	no	—
45	D₃×C₇	—	7	1	no	—
46	D₃×C₇	—	7	1	no	—
47	C₂×ΓL(1,𝔽₈)	—	21	1	no	—
48	AGL(1,𝔽₇)	—	3	1	no	—
49	AGL(1,𝔽₇)	—	7	1	no	—
50	AGL(1,𝔽₇)	—	14	1	no	—
51	C₂×C₃×D₃	—	49	1	no	Hall
52	C₂×D₇	—	21	1	no	—
53	C₂×D₇	—	21	1	no	—
54	C₃×C₇	—	6	1	no	—
55	C₃×C₇	—	7	1	no	—
56	C₃×C₇	—	14	1	no	—
57	C₃×C₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
58	C₃×C₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
59	ΓL(1,𝔽₈)	—	2	1	no	—
60	ΓL(1,𝔽₈)	D₃×D₇	1	1	yes	characteristic, complemented
61	C₂×C₃²	—	49	1	no	—
62	C₃×D₃	—	7	1	no	—
63	C₃×D₃	—	49	1	no	—
64	C₂×C₇	—	21	1	no	—
65	C₂×C₇	—	21	1	no	—
66	D₇	—	3	1	no	—
67	D₇	—	3	1	no	—
68	D₇	—	7	1	no	—
69	D₇	—	7	1	no	—
70	D₇	—	42	1	no	—
71	C₂²×C₃	—	147	1	no	—
72	C₂×D₃	—	49	1	no	—
73	C₃²	—	7	1	no	Sylow
74	C₇	—	6	1	no	—
75	C₇	D₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
76	C₇	C₃×D₃×D₇	1	1	yes	characteristic, complemented
77	C₂×C₃	—	21	1	no	—
78	C₂×C₃	—	49	1	no	—
79	C₂×C₃	—	49	1	no	—
80	C₂×C₃	—	98	1	no	—
81	C₂×C₃	—	147	1	no	—
82	D₃	—	7	1	no	—
83	D₃	—	7	1	no	—
84	C₂²	—	147	1	no	Sylow
85	C₃	—	7	1	no	—
86	C₃	—	14	1	no	—
87	C₃	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
88	C₂	—	21	1	no	—
89	C₂	—	21	1	no	—
90	C₂	—	49	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₂₁	C₂	semidirect
2. C₇×C₃⋊AGL(1,𝔽₇)	C₂	semidirect
3. C₃×ΓL(1,𝔽₈)⋊D₇	C₂	semidirect
4. D₂₁⋊D₇	C₃	semidirect
6. C₃×C₇×ΓL(1,𝔽₈)	C₂²	semidirect
7. C₇×D₂₁	C₂×C₃	semidirect
8. C₇×D₂₁	C₂×C₃	semidirect
9. C₃×C₇⋊D₇	C₂×C₃	semidirect
11. ΓL(1,𝔽₈)⋊D₇	D₃	semidirect
17. C₃×C₇²	C₂²×C₃	semidirect
19. C₇×ΓL(1,𝔽₈)	C₂×D₃	semidirect
23. C₃⋊AGL(1,𝔽₇)	D₇	semidirect
26. C₇⋊D₇	C₃×D₃	semidirect
34. C₃×ΓL(1,𝔽₈)	C₂×D₇	semidirect
35. C₇²	C₂×C₃×D₃	semidirect
37. D₂₁	AGL(1,𝔽₇)	semidirect
38. D₂₁	C₃×D₇	semidirect
57. C₃×C₇	C₂×AGL(1,𝔽₇)	semidirect
58. C₃×C₇	C₂×C₃×D₇	semidirect
60. ΓL(1,𝔽₈)	D₃×D₇	semidirect
75. C₇	D₃×AGL(1,𝔽₇)	semidirect
76. C₇	C₃×D₃×D₇	semidirect
87. C₃	D₇×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₂₁	2
C₇×C₃⋊AGL(1,𝔽₇)	2
C₃×ΓL(1,𝔽₈)⋊D₇	2
D₂₁⋊D₇	3
D₇×AGL(1,𝔽₇)	3
C₃×D₃×D₇	7
D₃×AGL(1,𝔽₇)	7

Subgroups — (C3⋊AGL(1,𝔽7))⋊D7 #13183641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃⋊AGL(1,𝔽₇))⋊D₇ #13183641