Subgroups — C₇×ΓL(1,𝔽₈)×D₅₇ #30660815

Order 16758.

88 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	C₇²×D₅₇	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	ΓL(1,𝔽₈)×D₅₇	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₇²×D₁₉	—	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	D₃×C₇×ΓL(1,𝔽₈)	—	19	1	no	maximal, Hall
14	C₇×D₅₇	—	3	2	no	—
15	C₇×D₅₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₇×D₅₇	C₃×C₇	1	1	yes	characteristic, complemented
17	C₃×C₇×D₁₉	—	21	1	no	—
18	ΓL(1,𝔽₈)×D₁₉	—	3	1	no	—
19	C₃×C₇×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented, Hall
20	C₃×C₇×C₁₉	—	3	2	no	—
21	C₃×C₇×C₁₉	—	7	1	no	—
22	C₃×C₇×C₁₉	—	14	1	no	—
23	C₃×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₃×C₇×C₁₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
25	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
26	C₁₉×ΓL(1,𝔽₈)	D₃×C₇	1	1	yes	characteristic, complemented
27	C₃×D₅₇	—	7	1	no	Hall
28	D₃×C₇²	—	19	1	no	—
29	C₂×C₇×ΓL(1,𝔽₈)	—	57	1	no	—
30	C₇×D₁₉	—	3	1	no	—
31	C₇×D₁₉	—	3	1	no	—
32	C₇×D₁₉	—	9	2	no	—
33	C₃²×C₁₉	—	7	1	no	Hall
34	C₃×C₇²	C₃×D₁₉	1	1	yes	characteristic, complemented
35	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
36	C₇×ΓL(1,𝔽₈)	D₅₇	1	1	yes	characteristic, direct product factor
37	C₇×C₁₉	—	3	2	no	—
38	C₇×C₁₉	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
39	C₇×C₁₉	C₃×D₃×C₇	1	1	yes	characteristic, complemented
40	C₃×D₃×C₇	—	133	1	no	—
41	D₃×ΓL(1,𝔽₈)	—	19	1	no	—
42	D₅₇	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
43	C₃×D₁₉	—	21	1	no	—
44	C₂×C₇²	—	57	1	no	Hall
45	C₃²×C₇	—	7	1	no	—
46	C₃×ΓL(1,𝔽₈)	C₇×D₁₉	1	1	yes	characteristic, complemented
47	C₃×C₁₉	—	7	1	no	—
48	C₃×C₁₉	—	14	1	no	—
49	C₃×C₁₉	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
50	C₇²	C₃×D₅₇	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃×C₇	—	399	1	no	—
52	D₃×C₇	—	19	1	no	—
53	D₃×C₇	—	19	1	no	—
54	D₃×C₇	—	57	2	no	—
55	C₂×ΓL(1,𝔽₈)	—	57	1	no	—
56	D₁₉	—	3	1	no	Hall
57	C₃×C₇	—	3	2	no	—
58	C₃×C₇	—	7	1	no	—
59	C₃×C₇	—	14	1	no	—
60	C₃×C₇	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented
61	C₃×C₇	C₃×C₇×D₁₉	1	1	yes	characteristic, complemented
62	ΓL(1,𝔽₈)	—	2	1	no	—
63	ΓL(1,𝔽₈)	C₇×D₅₇	1	1	yes	characteristic, direct product factor
64	C₁₉	D₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
65	C₃×D₃	—	133	1	no	Hall
66	C₂×C₇	—	57	1	no	—
67	C₂×C₇	—	57	1	no	—
68	C₂×C₇	—	171	2	no	—
69	C₃²	—	7	1	no	Sylow
70	C₇	—	3	2	no	—
71	C₇	ΓL(1,𝔽₈)×D₅₇	1	1	yes	characteristic, direct product factor
72	C₇	C₃×C₇×D₅₇	1	1	yes	characteristic, complemented
73	C₂×C₃	—	399	1	no	—
74	D₃	—	19	1	no	—
75	C₃	—	7	1	no	—
76	C₃	—	14	1	no	—
77	C₃	C₇×ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented
78	C₂	—	57	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)×D₅₇
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. C₇²×D₅₇	C₃	semidirect
5. C₇×C₁₉×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇²×C₁₉	C₂×C₃	semidirect
8. ΓL(1,𝔽₈)×D₅₇	C₇	direct (N × H)
11. C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₇	semidirect
12. C₇²×C₁₉	C₃×D₃	semidirect
15. C₇×D₅₇	ΓL(1,𝔽₈)	direct (N × H)
16. C₇×D₅₇	C₃×C₇	semidirect
19. C₃×C₇×ΓL(1,𝔽₈)	D₁₉	semidirect
23. C₃×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
24. C₃×C₇×C₁₉	C₂×C₃×C₇	semidirect
26. C₁₉×ΓL(1,𝔽₈)	D₃×C₇	semidirect
34. C₃×C₇²	C₃×D₁₉	semidirect
36. C₇×ΓL(1,𝔽₈)	D₅₇	direct (N × H)
38. C₇×C₁₉	D₃×ΓL(1,𝔽₈)	semidirect
39. C₇×C₁₉	C₃×D₃×C₇	semidirect
42. D₅₇	C₇×ΓL(1,𝔽₈)	direct (N × H)
46. C₃×ΓL(1,𝔽₈)	C₇×D₁₉	semidirect
49. C₃×C₁₉	C₂×C₇×ΓL(1,𝔽₈)	semidirect
50. C₇²	C₃×D₅₇	semidirect
60. C₃×C₇	ΓL(1,𝔽₈)×D₁₉	semidirect
61. C₃×C₇	C₃×C₇×D₁₉	semidirect
63. ΓL(1,𝔽₈)	C₇×D₅₇	direct (N × H)
64. C₁₉	D₃×C₇×ΓL(1,𝔽₈)	semidirect
71. C₇	ΓL(1,𝔽₈)×D₅₇	direct (N × H)
72. C₇	C₃×C₇×D₅₇	semidirect
77. 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
C₇²×D₅₇	3
C₇×ΓL(1,𝔽₈)×D₁₉	3
C₃×C₇×D₅₇	7
ΓL(1,𝔽₈)×D₅₇	7
D₃×C₇×ΓL(1,𝔽₈)	19

Subgroups — C7×ΓL(1,𝔽8)×D57 #30660815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)×D₅₇ #30660815