Subgroups — 3₋¹⁺²×AGL(1,𝔽₇) #11765713

Order 1134.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃²×AGL(1,𝔽₇)	C₃	1	1	yes	maximal, characteristic
3	D₇×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
4	#4430⋊C₂	C₃	1	2	yes	maximal, complemented
5	#67356	C₃	1	3	yes	maximal, complemented
6	#67355	C₃	1	3	yes	maximal, complemented
7	C₉×AGL(1,𝔽₇)	C₃	1	3	yes	maximal, complemented
8	C₃²×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic
9	C₇×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic, complemented
10	#4430	C₂×C₃	1	2	yes	complemented
11	#4428	C₂×C₃	1	3	yes	complemented
12	#4427	C₂×C₃	1	3	yes	complemented
13	C₉×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
14	C₂×C₃×3₋¹⁺²	—	7	1	no	maximal, Hall
15	C₃²×D₇	C₃²	1	1	yes	characteristic
16	C₃×AGL(1,𝔽₇)	—	3	3	no	—
17	C₃×AGL(1,𝔽₇)	C₃²	1	1	yes	characteristic
18	C₃×AGL(1,𝔽₇)	C₃²	1	1	yes	characteristic
19	C₃×AGL(1,𝔽₇)	C₃²	1	1	yes	characteristic
20	C₉×D₇	C₃²	1	3	yes	complemented
21	D₇⋊C₉	C₃²	1	6	yes	complemented
22	C₃×3₋¹⁺²	—	7	1	no	Sylow
23	C₃²×C₇	C₂×C₃²	1	1	yes	characteristic
24	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
25	C₃×ΓL(1,𝔽₈)	C₂×C₃²	1	1	yes	characteristic
26	C₃×ΓL(1,𝔽₈)	C₂×C₃²	1	1	yes	characteristic
27	C₃×ΓL(1,𝔽₈)	C₂×C₃²	1	1	yes	characteristic
28	C₇×C₉	C₂×C₃²	1	3	yes	complemented
29	C₇⋊C₉	C₂×C₃²	1	6	yes	complemented
30	C₂×C₃³	—	7	1	no	—
31	C₂×3₋¹⁺²	—	7	3	no	—
32	C₂×3₋¹⁺²	—	7	3	no	—
33	C₂×3₋¹⁺²	—	7	2	no	—
34	C₂×3₋¹⁺²	—	7	1	no	—
35	C₂×C₃×C₉	—	7	3	no	—
36	C₃×D₇	—	3	1	no	—
37	C₃×D₇	C₃³	1	1	yes	characteristic
38	AGL(1,𝔽₇)	—	3	1	no	—
39	AGL(1,𝔽₇)	—	3	1	no	—
40	AGL(1,𝔽₇)	3₋¹⁺²	1	3	yes	direct product factor
41	C₃³	—	7	1	no	—
42	3₋¹⁺²	—	7	2	no	—
43	3₋¹⁺²	—	7	3	no	—
44	3₋¹⁺²	—	7	3	no	—
45	3₋¹⁺²	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
46	C₃×C₉	—	7	3	no	—
47	C₃×C₇	—	3	1	no	—
48	C₃×C₇	C₂×C₃³	1	1	yes	characteristic
49	ΓL(1,𝔽₈)	—	3	1	no	—
50	ΓL(1,𝔽₈)	—	3	1	no	—
51	ΓL(1,𝔽₈)	C₂×3₋¹⁺²	1	3	yes	complemented
52	C₂×C₃²	—	7	1	no	—
53	C₂×C₃²	—	7	1	no	—
54	C₂×C₃²	—	7	1	no	—
55	C₂×C₃²	—	7	1	no	—
56	C₂×C₃²	—	21	3	no	—
57	C₂×C₉	—	7	6	no	—
58	C₂×C₉	—	7	3	no	—
59	D₇	C₃×3₋¹⁺²	1	1	yes	characteristic, complemented, Hall
60	C₃²	—	7	1	no	—
61	C₃²	—	7	1	no	—
62	C₃²	—	7	1	no	—
63	C₃²	—	21	3	no	—
64	C₃²	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
65	C₉	—	7	6	no	—
66	C₉	C₃×AGL(1,𝔽₇)	1	3	yes	complemented
67	C₇	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₃	—	7	3	no	—
69	C₂×C₃	—	7	1	no	—
70	C₂×C₃	—	21	1	no	—
71	C₂×C₃	—	21	1	no	—
72	C₂×C₃	—	21	1	no	—
73	C₃	—	3	1	no	—
74	C₃	—	7	3	no	—
75	C₃	—	21	1	no	—
76	C₃	—	21	1	no	—
77	C₃	C₃²×AGL(1,𝔽₇)	1	1	yes	characteristic
78	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, Frattini	C₃	C₃²×AGL(1,𝔽₇)
commutator, socle, cosocle	C₃×C₇	C₂×C₃³
Fitting	C₇×3₋¹⁺²	C₂×C₃
prime radical	C₃²×AGL(1,𝔽₇)	C₃
hypercenter	3₋¹⁺²	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×3₋¹⁺²	C₂	semidirect
3. D₇×3₋¹⁺²	C₃	semidirect
4. #4430⋊C₂	C₃	semidirect
5. #67356	C₃	semidirect
6. #67355	C₃	semidirect
7. C₉×AGL(1,𝔽₇)	C₃	semidirect
9. C₇×3₋¹⁺²	C₂×C₃	semidirect
10. #4430	C₂×C₃	semidirect
11. #4428	C₂×C₃	semidirect
12. #4427	C₂×C₃	semidirect
13. C₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
20. C₉×D₇	C₃²	semidirect
21. D₇⋊C₉	C₃²	semidirect
28. C₇×C₉	C₂×C₃²	semidirect
29. C₇⋊C₉	C₂×C₃²	semidirect
40. AGL(1,𝔽₇)	3₋¹⁺²	direct (N × H)
45. 3₋¹⁺²	AGL(1,𝔽₇)	direct (N × H)
51. ΓL(1,𝔽₈)	C₂×3₋¹⁺²	semidirect
59. D₇	C₃×3₋¹⁺²	semidirect
66. C₉	C₃×AGL(1,𝔽₇)	semidirect
67. C₇	C₂×C₃×3₋¹⁺²	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)×3₋¹⁺²	2
C₃²×AGL(1,𝔽₇)	3
D₇×3₋¹⁺²	3
#4430⋊C₂	3
#67356	3
#67355	3
C₉×AGL(1,𝔽₇)	3
C₂×C₃×3₋¹⁺²	7

Subgroups — 3−1+2×AGL(1,𝔽7) #11765713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×AGL(1,𝔽₇) #11765713