Subgroups — ΓL(1,𝔽₈)⋊#13093366 #44527672

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#10625202	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	#23780629	C₃	1	1	yes	maximal, characteristic, complemented
3	#23780625	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)⋊D₂₄₇	C₃	1	1	yes	maximal, characteristic, complemented
5	#23780619	C₃	1	1	yes	maximal, characteristic, complemented
6	#19104444	C₂×C₃	1	1	yes	characteristic, complemented
7	#19104440	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×#10625202	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×#13093366	—	7	1	no	maximal, Hall
11	D₁₇₂₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	—	13	1	no	maximal, Hall
13	C₃×#10625202	—	7	1	no	Hall
14	C₇×C₁₃×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
15	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	—	19	1	no	maximal, Hall
16	C₃×D₂₄₇	—	7	1	no	—
17	#13093366	—	7	1	no	—
18	#13093366	—	7	1	no	—
19	#13093366	—	7	1	no	—
20	ΓL(1,𝔽₈)×C₁₉⋊C₃	—	13	1	no	Hall
21	ΓL(1,𝔽₈)×C₁₃⋊C₃	—	19	1	no	Hall
22	#11716328	—	13	1	no	—
23	#11716327	—	13	1	no	—
24	ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	—
25	(C₁₉⋊C₃)⋊D₇	—	13	1	no	—
26	C₃×C₁₃×C₁₉	—	7	1	no	—
27	#10625202	—	7	1	no	—
28	#10625202	—	7	1	no	—
29	#10625202	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
30	#10587723	—	19	1	no	—
31	#10587722	—	19	1	no	—
32	ΓL(1,𝔽₈)⋊D₁₃	—	19	1	no	—
33	(C₁₃⋊C₃)⋊D₇	—	19	1	no	—
34	D₂₄₇	—	7	1	no	Hall
35	#87705	—	13	1	no	—
36	#87704	—	13	1	no	—
37	C₇×C₁₉⋊C₃	—	13	1	no	—
38	C₁₉×ΓL(1,𝔽₈)	D₁₃⋊C₃	1	1	yes	characteristic, complemented
39	C₃×D₁₉⋊C₃	—	91	1	no	Hall
40	#63271	—	19	1	no	—
41	#63270	—	19	1	no	—
42	C₇×C₁₃⋊C₃	—	19	1	no	—
43	C₁₃×ΓL(1,𝔽₈)	D₁₉⋊C₃	1	1	yes	characteristic, complemented
44	D₁₃₃	—	13	1	no	Hall
45	C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
46	C₃×D₁₃⋊C₃	—	133	1	no	Hall
47	D₉₁	—	19	1	no	Hall
48	C₃×C₁₉⋊C₃	—	91	1	no	Hall
49	C₇×C₁₉	C₃×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
50	C₃×AGL(1,𝔽₇)	—	247	1	no	Hall
51	C₃×C₁₃⋊C₃	—	133	1	no	Hall
52	C₃×D₁₉	—	91	1	no	—
53	D₁₉⋊C₃	—	91	1	no	—
54	D₁₉⋊C₃	—	91	1	no	—
55	D₁₉⋊C₃	—	91	1	no	—
56	C₇×C₁₃	C₃×D₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
57	C₃×D₁₃	—	133	1	no	—
58	D₁₃⋊C₃	—	133	1	no	—
59	D₁₃⋊C₃	—	133	1	no	—
60	D₁₃⋊C₃	—	133	1	no	—
61	C₃×ΓL(1,𝔽₈)	—	247	1	no	Hall
62	C₃×C₁₉	—	7	1	no	—
63	C₁₉⋊C₃	—	13	1	no	—
64	C₁₉⋊C₃	—	91	1	no	—
65	C₁₉⋊C₃	—	91	1	no	—
66	C₃×D₇	—	247	1	no	—
67	AGL(1,𝔽₇)	—	247	1	no	—
68	AGL(1,𝔽₇)	—	247	1	no	—
69	AGL(1,𝔽₇)	—	247	1	no	—
70	C₃×C₁₃	—	7	1	no	—
71	C₁₃⋊C₃	—	19	1	no	—
72	C₁₃⋊C₃	—	133	1	no	—
73	C₁₃⋊C₃	—	133	1	no	—
74	D₁₉	—	91	1	no	Hall
75	D₁₃	—	133	1	no	Hall
76	C₃×C₇	—	247	1	no	—
77	ΓL(1,𝔽₈)	—	247	1	no	—
78	ΓL(1,𝔽₈)	—	247	1	no	—
79	ΓL(1,𝔽₈)	#13093366	1	1	yes	characteristic, complemented
80	C₁₉	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃²	—	1729	1	no	Hall
82	D₇	—	247	1	no	Hall
83	C₁₃	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	1729	1	no	Sylow
85	C₇	C₃×#13093366	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	1729	1	no	—
87	C₂×C₃	—	1729	1	no	—
88	C₂×C₃	—	1729	1	no	—
89	C₂×C₃	—	1729	1	no	—
90	C₃	—	7	1	no	—
91	C₃	—	1729	1	no	—
92	C₃	—	1729	1	no	—
93	C₃	—	247	1	no	—
94	C₂	—	1729	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₁₉	C₂×C₃²
LCS	G ▷ C₇×C₁₃×C₁₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#10625202	C₂	semidirect
2. #23780629	C₃	semidirect
3. #23780625	C₃	semidirect
4. ΓL(1,𝔽₈)⋊D₂₄₇	C₃	semidirect
5. #23780619	C₃	semidirect
6. #19104444	C₂×C₃	semidirect
7. #19104440	C₂×C₃	semidirect
8. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
9. C₇×#10625202	C₂×C₃	semidirect
11. D₁₇₂₉	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
29. #10625202	AGL(1,𝔽₇)	semidirect
38. C₁₉×ΓL(1,𝔽₈)	D₁₃⋊C₃	semidirect
43. C₁₃×ΓL(1,𝔽₈)	D₁₉⋊C₃	semidirect
45. C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	semidirect
49. C₇×C₁₉	C₃×D₁₃⋊C₃	semidirect
56. C₇×C₁₃	C₃×D₁₉⋊C₃	semidirect
79. ΓL(1,𝔽₈)	#13093366	semidirect
80. C₁₉	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	semidirect
83. C₁₃	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	semidirect
85. C₇	C₃×#13093366	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)×#10625202	2
#23780629	3
#23780625	3
ΓL(1,𝔽₈)⋊D₂₄₇	3
#23780619	3
C₃×#13093366	7
ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	13
ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	19

Subgroups — ΓL(1,𝔽8)⋊#13093366 #44527672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊#13093366 #44527672