Subgroups — C₇×C₃⋊(ΓL(1,𝔽₈)⋊D₁₉) #30660809

Order 16758.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₃₉₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×ΓL(1,𝔽₈)⋊D₁₉	—	3	1	no	maximal
4	C₇×C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₇×C₁₉×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₇×D₅₇	—	7	1	no	maximal
8	C₃⋊(ΓL(1,𝔽₈)⋊D₁₉)	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₇×D₁₃₃	—	3	1	no	Hall
10	C₃²×C₇×C₁₉	—	7	1	no	—
11	C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₇	1	1	yes	characteristic, complemented
12	C₇²×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₇×C₃⋊AGL(1,𝔽₇)	—	19	1	no	maximal, Hall
14	D₃₉₉	C₃×C₇	1	1	yes	characteristic, complemented
15	C₇×D₅₇	—	7	1	no	—
16	C₃×C₇×D₁₉	—	21	1	no	—
17	ΓL(1,𝔽₈)⋊D₁₉	—	3	1	no	—
18	C₃×C₇×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented, Hall
19	C₃×C₇×C₁₉	—	6	1	no	—
20	C₃×C₇×C₁₉	—	14	1	no	—
21	C₃×C₇×C₁₉	—	14	1	no	—
22	C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
23	C₃×C₇×C₁₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
24	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
25	C₁₉×ΓL(1,𝔽₈)	D₃×C₇	1	1	yes	characteristic, complemented
26	C₃×D₅₇	—	7	1	no	Hall
27	C₇×D₂₁	—	19	1	no	—
28	C₇×AGL(1,𝔽₇)	—	57	1	no	—
29	D₁₃₃	—	3	1	no	—
30	C₇×D₁₉	—	21	1	no	—
31	C₃²×C₁₉	—	14	1	no	Hall
32	C₃×C₇²	C₃×D₁₉	1	1	yes	characteristic, complemented
33	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
34	C₇×ΓL(1,𝔽₈)	D₅₇	1	1	yes	characteristic, complemented
35	C₇×C₁₉	—	6	1	no	—
36	C₇×C₁₉	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
37	C₇×C₁₉	C₃×D₃×C₇	1	1	yes	characteristic, complemented
38	C₃×D₃×C₇	—	133	1	no	—
39	C₃⋊AGL(1,𝔽₇)	—	19	1	no	—
40	D₅₇	—	7	1	no	—
41	C₃×D₁₉	—	21	1	no	—
42	C₇×D₇	—	57	1	no	Hall
43	C₃²×C₇	—	14	1	no	—
44	C₃×ΓL(1,𝔽₈)	C₇×D₁₉	1	1	yes	characteristic, complemented
45	C₃×C₁₉	—	7	1	no	—
46	C₃×C₁₉	—	14	1	no	—
47	C₃×C₁₉	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
48	C₇²	C₃×D₅₇	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃×C₇	—	399	1	no	—
50	D₂₁	—	19	1	no	—
51	D₃×C₇	—	133	1	no	—
52	AGL(1,𝔽₇)	—	57	1	no	—
53	D₁₉	—	21	1	no	Hall
54	C₃×C₇	—	6	1	no	—
55	C₃×C₇	—	7	1	no	—
56	C₃×C₇	—	14	1	no	—
57	C₃×C₇	ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented
58	C₃×C₇	C₃×C₇×D₁₉	1	1	yes	characteristic, complemented
59	ΓL(1,𝔽₈)	—	2	1	no	—
60	ΓL(1,𝔽₈)	C₇×D₅₇	1	1	yes	characteristic, complemented
61	C₁₉	C₇×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
62	C₃×D₃	—	133	1	no	Hall
63	C₂×C₇	—	399	1	no	—
64	D₇	—	57	1	no	—
65	C₃²	—	14	1	no	Sylow
66	C₇	—	6	1	no	—
67	C₇	C₃⋊(ΓL(1,𝔽₈)⋊D₁₉)	1	1	yes	characteristic, direct product factor
68	C₇	C₃×C₇×D₅₇	1	1	yes	characteristic, complemented
69	C₂×C₃	—	399	1	no	—
70	D₃	—	133	1	no	—
71	C₃	—	14	1	no	—
72	C₃	—	14	1	no	—
73	C₃	C₇×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented
74	C₂	—	399	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	C₃⋊(ΓL(1,𝔽₈)⋊D₁₉)
commutator, cosocle, hypocenter	C₃×C₇×C₁₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₉	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₉
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. C₇×D₃₉₉	C₃	semidirect
5. C₇×C₁₉×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇²×C₁₉	C₂×C₃	semidirect
8. C₃⋊(ΓL(1,𝔽₈)⋊D₁₉)	C₇	direct (N × H)
11. C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₇	semidirect
12. C₇²×C₁₉	C₃×D₃	semidirect
14. D₃₉₉	C₃×C₇	semidirect
18. C₃×C₇×ΓL(1,𝔽₈)	D₁₉	semidirect
22. C₃×C₇×C₁₉	AGL(1,𝔽₇)	semidirect
23. C₃×C₇×C₁₉	C₂×C₃×C₇	semidirect
25. C₁₉×ΓL(1,𝔽₈)	D₃×C₇	semidirect
32. C₃×C₇²	C₃×D₁₉	semidirect
34. C₇×ΓL(1,𝔽₈)	D₅₇	semidirect
36. C₇×C₁₉	C₃⋊AGL(1,𝔽₇)	semidirect
37. C₇×C₁₉	C₃×D₃×C₇	semidirect
44. C₃×ΓL(1,𝔽₈)	C₇×D₁₉	semidirect
47. C₃×C₁₉	C₇×AGL(1,𝔽₇)	semidirect
48. C₇²	C₃×D₅₇	semidirect
57. C₃×C₇	ΓL(1,𝔽₈)⋊D₁₉	semidirect
58. C₃×C₇	C₃×C₇×D₁₉	semidirect
60. ΓL(1,𝔽₈)	C₇×D₅₇	semidirect
61. C₁₉	C₇×C₃⋊AGL(1,𝔽₇)	semidirect
67. C₇	C₃⋊(ΓL(1,𝔽₈)⋊D₁₉)	direct (N × H)
68. C₇	C₃×C₇×D₅₇	semidirect
73. C₃	C₇×ΓL(1,𝔽₈)⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×C₁₉×ΓL(1,𝔽₈)	2
C₇×D₃₉₉	3
C₇×ΓL(1,𝔽₈)⋊D₁₉	3
C₃×C₇×D₅₇	7
C₃⋊(ΓL(1,𝔽₈)⋊D₁₉)	7
C₇×C₃⋊AGL(1,𝔽₇)	19

Subgroups — C7×C3⋊(ΓL(1,𝔽8)⋊D19) #30660809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊(ΓL(1,𝔽₈)⋊D₁₉) #30660809