Subgroups — ΓL(1,𝔽₈)×#6898 #19103214

Order 5103.

692 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133584	C₃	1	16	yes	maximal, complemented
2	#13133583	C₃	1	2	yes	maximal, complemented
3	#13133582	C₃	1	8	yes	maximal, complemented
4	ΓL(1,𝔽₈)×C₉⋊C₉	C₃	1	8	yes	maximal, complemented
5	ΓL(1,𝔽₈)×C₃²⋊C₉	C₃	1	4	yes	maximal
6	C₉²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
7	C₇×#6898	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#6898	—	7	1	no	maximal, Sylow
9	C₃³×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
10	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	4	yes	—
11	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	8	yes	—
12	C₇×C₉⋊C₉	C₃²	1	8	yes	complemented
13	C₇×C₃²⋊C₉	C₃²	1	4	yes	—
14	C₇×C₉²	C₃²	1	1	yes	characteristic, complemented
15	C₃²⋊(C₇⋊C₉)	C₃²	1	8	yes	—
16	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	8	yes	—
17	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	16	yes	—
18	#10588036	C₃²	1	8	yes	complemented
19	#10588035	C₃²	1	8	yes	complemented
20	(C₇⋊C₉)⋊C₉	C₃²	1	16	yes	complemented
21	(C₇⋊C₉)⋊C₉	C₃²	1	32	yes	complemented
22	C₉×C₇⋊C₉	C₃²	1	8	yes	complemented
23	#6898	—	7	16	no	—
24	#6898	—	7	8	no	—
25	#6898	—	7	2	no	—
26	#6898	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
27	C₃×C₉⋊C₉	—	7	8	no	—
28	C₃×C₃²⋊C₉	—	7	4	no	—
29	C₃×C₉²	—	7	1	no	—
30	C₃³×C₇	C₃³	1	1	yes	characteristic
31	C₃²×ΓL(1,𝔽₈)	—	3	12	no	—
32	C₃²×ΓL(1,𝔽₈)	C₃³	1	2	yes	—
33	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
34	C₃×C₇×C₉	C₃³	1	4	yes	—
35	C₃×C₇×C₉	C₃³	1	8	yes	—
36	C₃×C₇⋊C₉	C₃³	1	8	yes	—
37	C₃×C₇⋊C₉	C₃³	1	16	yes	—
38	C₉×ΓL(1,𝔽₈)	—	3	12	no	—
39	C₉×ΓL(1,𝔽₈)	—	3	24	no	—
40	C₃⁴	—	7	1	no	—
41	C₃²×C₉	—	7	4	no	—
42	C₃²×C₉	—	7	8	no	—
43	C₉⋊C₉	—	7	32	no	—
44	C₉⋊C₉	—	7	16	no	—
45	C₉⋊C₉	—	7	8	no	—
46	C₉⋊C₉	—	7	8	no	—
47	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
48	C₃²⋊C₉	—	7	8	no	—
49	C₃²⋊C₉	—	7	16	no	—
50	C₃²⋊C₉	—	7	8	no	—
51	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	4	yes	—
52	C₉²	—	7	8	no	—
53	C₉²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₃²×C₇	—	3	4	no	—
55	C₃²×C₇	C₃⁴	1	1	yes	characteristic
56	C₃×ΓL(1,𝔽₈)	—	3	8	no	—
57	C₃×ΓL(1,𝔽₈)	—	9	9	no	—
58	C₃×ΓL(1,𝔽₈)	#737	1	12	yes	—
59	C₇×C₉	—	3	4	no	—
60	C₇×C₉	—	3	8	no	—
61	C₇⋊C₉	—	3	8	no	—
62	C₇⋊C₉	—	3	16	no	—
63	C₃³	—	7	1	no	—
64	C₃³	—	7	2	no	—
65	C₃³	—	21	12	no	—
66	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
67	C₃×C₉	—	7	8	no	—
68	C₃×C₉	—	7	16	no	—
69	C₃×C₉	—	21	12	no	—
70	C₃×C₉	—	21	24	no	—
71	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	4	yes	—
72	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	8	yes	—
73	C₃×C₇	—	9	1	no	—
74	C₃×C₇	C₃×#737	1	4	yes	—
75	ΓL(1,𝔽₈)	—	9	2	no	—
76	ΓL(1,𝔽₈)	#6898	1	9	yes	direct product factor
77	C₃²	—	3	4	no	—
78	C₃²	—	7	12	no	—
79	C₃²	—	21	8	no	—
80	C₃²	—	63	9	no	—
81	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
82	C₉	—	3	4	no	—
83	C₉	—	3	8	no	—
84	C₉	—	21	8	no	—
85	C₉	—	21	16	no	—
86	C₇	C₃×#6898	1	1	yes	characteristic, complemented, Sylow
87	C₃	—	7	9	no	—
88	C₃	—	9	1	no	—
89	C₃	—	63	2	no	—
90	C₃	ΓL(1,𝔽₈)×#737	1	4	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×#6898	C₃
hypercenter	#6898	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#6898
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6898

Complements

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

N	H ≅ G/N	type
1. #13133584	C₃	semidirect
2. #13133583	C₃	semidirect
3. #13133582	C₃	semidirect
4. ΓL(1,𝔽₈)×C₉⋊C₉	C₃	semidirect
6. C₉²×ΓL(1,𝔽₈)	C₃	semidirect
7. C₇×#6898	C₃	semidirect
12. C₇×C₉⋊C₉	C₃²	semidirect
14. C₇×C₉²	C₃²	semidirect
18. #10588036	C₃²	semidirect
19. #10588035	C₃²	semidirect
20. (C₇⋊C₉)⋊C₉	C₃²	semidirect
21. (C₇⋊C₉)⋊C₉	C₃²	semidirect
22. C₉×C₇⋊C₉	C₃²	semidirect
26. #6898	ΓL(1,𝔽₈)	direct (N × H)
47. C₉⋊C₉	C₃×ΓL(1,𝔽₈)	semidirect
53. C₉²	C₃×ΓL(1,𝔽₈)	semidirect
76. ΓL(1,𝔽₈)	#6898	direct (N × H)
86. C₇	C₃×#6898	semidirect

Subgroups — ΓL(1,𝔽8)×#6898 #19103214

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽₈)×#6898 #19103214