Subgroups — C₅×AGL(1,𝔽₇)×C₃₁⋊C₃ #34086202

Order 19530.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×ΓL(1,𝔽₈)×C₃₁⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×#13049958	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅×#13049959	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅×C₃₁×AGL(1,𝔽₇)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₅×D₇×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
6	AGL(1,𝔽₇)×C₃₁⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×#10620252	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×#10620253	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×ΓL(1,𝔽₈)×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×C₇×C₃₁⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×C₅×C₃₁⋊C₃	—	7	1	no	maximal, Hall
12	C₅×D₇×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
13	ΓL(1,𝔽₈)×C₃₁⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
14	C₃×C₅×C₃₁⋊C₃	—	7	1	no	Hall
15	#13049959	C₃×C₅	1	1	yes	characteristic, complemented
16	#13049958	C₃×C₅	1	1	yes	characteristic, complemented
17	C₃₁×AGL(1,𝔽₇)	C₃×C₅	1	1	yes	characteristic, complemented
18	D₇×C₃₁⋊C₃	C₃×C₅	1	1	yes	characteristic, complemented
19	C₅×C₇×C₃₁	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×C₅×C₃₁	—	7	1	no	—
21	C₂×C₅×C₃₁⋊C₃	—	7	1	no	—
22	C₂×C₅×C₃₁⋊C₃	—	7	1	no	—
23	C₂×C₅×C₃₁⋊C₃	—	21	1	no	—
24	#10620253	C₂×C₃×C₅	1	1	yes	characteristic, complemented
25	#10620252	C₂×C₃×C₅	1	1	yes	characteristic, complemented
26	C₇×C₃₁⋊C₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
27	ΓL(1,𝔽₈)×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
28	C₃×C₅×AGL(1,𝔽₇)	—	31	1	no	maximal, Hall
29	C₂×C₃×C₃₁⋊C₃	—	7	1	no	Hall
30	C₃×C₅×C₃₁	—	7	1	no	—
31	C₅×C₃₁⋊C₃	—	7	1	no	—
32	C₅×C₃₁⋊C₃	—	7	1	no	—
33	C₅×C₃₁⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
34	D₇×C₃₁	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
35	C₃×C₅×ΓL(1,𝔽₈)	—	31	1	no	Hall
36	C₂×C₅×C₃₁	—	21	1	no	Hall
37	C₃×C₃₁⋊C₃	—	7	1	no	Hall
38	C₇×C₃₁	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×D₇	—	31	1	no	—
40	C₅×AGL(1,𝔽₇)	—	31	1	no	—
41	C₅×AGL(1,𝔽₇)	—	31	1	no	—
42	C₅×AGL(1,𝔽₇)	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
43	C₂×C₃×C₃₁	—	7	1	no	—
44	C₂×C₃₁⋊C₃	—	7	1	no	—
45	C₂×C₃₁⋊C₃	—	7	1	no	—
46	C₂×C₃₁⋊C₃	—	21	1	no	—
47	C₅×C₃₁	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
48	C₃×AGL(1,𝔽₇)	—	31	1	no	Hall
49	C₃×C₅×C₇	—	31	1	no	—
50	C₅×ΓL(1,𝔽₈)	—	31	1	no	—
51	C₅×ΓL(1,𝔽₈)	—	31	1	no	—
52	C₅×ΓL(1,𝔽₈)	C₂×C₃₁⋊C₃	1	1	yes	characteristic, complemented
53	C₃×C₃₁	—	7	1	no	—
54	C₃₁⋊C₃	—	7	1	no	—
55	C₃₁⋊C₃	—	7	1	no	—
56	C₃₁⋊C₃	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
57	C₂×C₃²×C₅	—	217	1	no	Hall
58	C₅×D₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
59	C₃×ΓL(1,𝔽₈)	—	31	1	no	Hall
60	C₂×C₃₁	—	21	1	no	Hall
61	C₃²×C₅	—	217	1	no	Hall
62	C₃×D₇	—	31	1	no	—
63	AGL(1,𝔽₇)	—	31	1	no	—
64	AGL(1,𝔽₇)	—	31	1	no	—
65	AGL(1,𝔽₇)	C₅×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
66	C₅×C₇	C₂×C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
67	C₃₁	C₃×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₃×C₅	—	7	1	no	—
69	C₂×C₃×C₅	—	651	1	no	—
70	C₂×C₃×C₅	—	217	1	no	—
71	C₂×C₃×C₅	—	217	1	no	—
72	C₃×C₇	—	31	1	no	—
73	ΓL(1,𝔽₈)	—	31	1	no	—
74	ΓL(1,𝔽₈)	—	31	1	no	—
75	ΓL(1,𝔽₈)	C₂×C₅×C₃₁⋊C₃	1	1	yes	characteristic, complemented
76	C₂×C₃²	—	217	1	no	Hall
77	C₃×C₅	—	7	1	no	—
78	C₃×C₅	—	31	1	no	—
79	C₃×C₅	—	217	1	no	—
80	C₃×C₅	—	217	1	no	—
81	D₇	C₃×C₅×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
82	C₂×C₅	—	21	1	no	Hall
83	C₃²	—	217	1	no	Sylow
84	C₇	C₂×C₃×C₅×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₃	—	7	1	no	—
86	C₂×C₃	—	651	1	no	—
87	C₂×C₃	—	217	1	no	—
88	C₂×C₃	—	217	1	no	—
89	C₅	AGL(1,𝔽₇)×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
90	C₃	—	7	1	no	—
91	C₃	—	31	1	no	—
92	C₃	—	217	1	no	—
93	C₃	—	217	1	no	—
94	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅	AGL(1,𝔽₇)×C₃₁⋊C₃
commutator, cosocle, hypocenter	C₇×C₃₁	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₃₁	C₂×C₃²
LCS	G ▷ C₇×C₃₁
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×ΓL(1,𝔽₈)×C₃₁⋊C₃	C₂	semidirect
2. C₅×#13049958	C₃	semidirect
3. C₅×#13049959	C₃	semidirect
4. C₅×C₃₁×AGL(1,𝔽₇)	C₃	semidirect
5. C₅×D₇×C₃₁⋊C₃	C₃	semidirect
6. AGL(1,𝔽₇)×C₃₁⋊C₃	C₅	direct (N × H)
7. C₅×#10620252	C₂×C₃	semidirect
8. C₅×#10620253	C₂×C₃	semidirect
9. C₅×ΓL(1,𝔽₈)×C₃₁	C₂×C₃	semidirect
10. C₅×C₇×C₃₁⋊C₃	C₂×C₃	semidirect
12. C₅×D₇×C₃₁	C₃²	semidirect
13. ΓL(1,𝔽₈)×C₃₁⋊C₃	C₂×C₅	semidirect
15. #13049959	C₃×C₅	semidirect
16. #13049958	C₃×C₅	semidirect
17. C₃₁×AGL(1,𝔽₇)	C₃×C₅	semidirect
18. D₇×C₃₁⋊C₃	C₃×C₅	semidirect
19. C₅×C₇×C₃₁	C₂×C₃²	semidirect
24. #10620253	C₂×C₃×C₅	semidirect
25. #10620252	C₂×C₃×C₅	semidirect
26. C₇×C₃₁⋊C₃	C₂×C₃×C₅	semidirect
27. ΓL(1,𝔽₈)×C₃₁	C₂×C₃×C₅	semidirect
33. C₅×C₃₁⋊C₃	AGL(1,𝔽₇)	direct (N × H)
34. D₇×C₃₁	C₃²×C₅	semidirect
38. C₇×C₃₁	C₂×C₃²×C₅	semidirect
42. C₅×AGL(1,𝔽₇)	C₃₁⋊C₃	direct (N × H)
47. C₅×C₃₁	C₃×AGL(1,𝔽₇)	semidirect
52. C₅×ΓL(1,𝔽₈)	C₂×C₃₁⋊C₃	semidirect
56. C₃₁⋊C₃	C₅×AGL(1,𝔽₇)	direct (N × H)
58. C₅×D₇	C₃×C₃₁⋊C₃	semidirect
65. AGL(1,𝔽₇)	C₅×C₃₁⋊C₃	direct (N × H)
66. C₅×C₇	C₂×C₃×C₃₁⋊C₃	semidirect
67. C₃₁	C₃×C₅×AGL(1,𝔽₇)	semidirect
75. ΓL(1,𝔽₈)	C₂×C₅×C₃₁⋊C₃	semidirect
81. D₇	C₃×C₅×C₃₁⋊C₃	semidirect
84. C₇	C₂×C₃×C₅×C₃₁⋊C₃	semidirect
89. C₅	AGL(1,𝔽₇)×C₃₁⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×ΓL(1,𝔽₈)×C₃₁⋊C₃	2
C₅×#13049958	3
C₅×#13049959	3
C₅×C₃₁×AGL(1,𝔽₇)	3
C₅×D₇×C₃₁⋊C₃	3
AGL(1,𝔽₇)×C₃₁⋊C₃	5
C₂×C₃×C₅×C₃₁⋊C₃	7
C₃×C₅×AGL(1,𝔽₇)	31

Subgroups — C5×AGL(1,𝔽7)×C31⋊C3 #34086202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×AGL(1,𝔽₇)×C₃₁⋊C₃ #34086202