Subgroups — ΓL(1,𝔽₈)⋊((C₁₃⋊C₃)⋊D₁₉) #44527686

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×ΓL(1,𝔽₈)×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	#23780627	C₃	1	1	yes	maximal, characteristic, complemented
3	#23780624	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)⋊D₂₄₇	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₁₃⋊C₃)⋊D₁₃₃	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#63271	C₂×C₃	1	1	yes	characteristic, complemented
7	C₁₉×#63270	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×C₁₉×C₁₃⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×(C₁₃⋊C₃)⋊D₁₉	—	7	1	no	maximal, Hall
11	D₁₇₂₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	maximal, Hall
13	C₃×C₁₉×C₁₃⋊C₃	—	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	(C₁₃⋊C₃)⋊D₁₉	—	7	1	no	—
18	(C₁₃⋊C₃)⋊D₁₉	—	7	1	no	—
19	(C₁₃⋊C₃)⋊D₁₉	—	7	1	no	—
20	C₃×C₁₉×ΓL(1,𝔽₈)	—	13	1	no	Hall
21	ΓL(1,𝔽₈)×C₁₃⋊C₃	D₁₉	1	1	yes	characteristic, complemented, Hall
22	C₃×D₁₃₃	—	13	1	no	—
23	ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	—
24	ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	—
25	ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	—
26	C₃×C₁₃×C₁₉	—	7	1	no	—
27	C₁₉×C₁₃⋊C₃	—	7	1	no	—
28	C₁₉×C₁₃⋊C₃	—	7	1	no	—
29	C₁₉×C₁₃⋊C₃	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	C₃×C₇×C₁₉	—	13	1	no	—
36	C₁₉×ΓL(1,𝔽₈)	—	13	1	no	—
37	C₁₉×ΓL(1,𝔽₈)	—	13	1	no	—
38	C₁₉×ΓL(1,𝔽₈)	D₁₃⋊C₃	1	1	yes	characteristic, complemented
39	C₃²×D₁₉	—	91	1	no	Hall
40	#63271	C₃×D₁₉	1	1	yes	characteristic, complemented
41	#63270	C₃×D₁₉	1	1	yes	characteristic, complemented
42	C₇×C₁₃⋊C₃	C₃×D₁₉	1	1	yes	characteristic, complemented
43	C₁₃×ΓL(1,𝔽₈)	C₃×D₁₉	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	C₃²×C₁₉	—	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	—	7	1	no	Hall
52	C₃×D₁₉	—	91	1	no	—
53	C₃×D₁₉	—	91	1	no	—
54	C₃×D₁₉	—	91	1	no	—
55	C₃×D₁₉	—	91	1	no	—
56	C₇×C₁₃	C₃²×D₁₉	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)	—	13	1	no	Hall
62	C₃×C₁₉	—	7	1	no	—
63	C₃×C₁₉	—	13	1	no	—
64	C₃×C₁₉	—	91	1	no	—
65	C₃×C₁₉	—	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	—	7	1	no	—
72	C₁₃⋊C₃	—	7	1	no	—
73	C₁₃⋊C₃	ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented
74	D₁₉	—	91	1	no	Hall
75	D₁₃	—	133	1	no	Hall
76	C₃×C₇	—	13	1	no	—
77	ΓL(1,𝔽₈)	—	13	1	no	—
78	ΓL(1,𝔽₈)	—	13	1	no	—
79	ΓL(1,𝔽₈)	(C₁₃⋊C₃)⋊D₁₉	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	C₃×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	91	1	no	Sylow
85	C₇	C₃×(C₁₃⋊C₃)⋊D₁₉	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	—	13	1	no	—
92	C₃	—	91	1	no	—
93	C₃	—	91	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. C₁₉×ΓL(1,𝔽₈)×C₁₃⋊C₃	C₂	semidirect
2. #23780627	C₃	semidirect
3. #23780624	C₃	semidirect
4. ΓL(1,𝔽₈)⋊D₂₄₇	C₃	semidirect
5. (C₁₃⋊C₃)⋊D₁₃₃	C₃	semidirect
6. C₁₉×#63271	C₂×C₃	semidirect
7. C₁₉×#63270	C₂×C₃	semidirect
8. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
9. C₇×C₁₉×C₁₃⋊C₃	C₂×C₃	semidirect
11. D₁₇₂₉	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
21. ΓL(1,𝔽₈)×C₁₃⋊C₃	D₁₉	semidirect
29. C₁₉×C₁₃⋊C₃	AGL(1,𝔽₇)	semidirect
38. C₁₉×ΓL(1,𝔽₈)	D₁₃⋊C₃	semidirect
40. #63271	C₃×D₁₉	semidirect
41. #63270	C₃×D₁₉	semidirect
42. C₇×C₁₃⋊C₃	C₃×D₁₉	semidirect
43. C₁₃×ΓL(1,𝔽₈)	C₃×D₁₉	semidirect
45. C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	semidirect
49. C₇×C₁₉	C₃×D₁₃⋊C₃	semidirect
56. C₇×C₁₃	C₃²×D₁₉	semidirect
73. C₁₃⋊C₃	ΓL(1,𝔽₈)⋊D₁₉	semidirect
79. ΓL(1,𝔽₈)	(C₁₃⋊C₃)⋊D₁₉	semidirect
80. C₁₉	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	semidirect
83. C₁₃	C₃×ΓL(1,𝔽₈)⋊D₁₉	semidirect
85. C₇	C₃×(C₁₃⋊C₃)⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×ΓL(1,𝔽₈)×C₁₃⋊C₃	2
#23780627	3
#23780624	3
ΓL(1,𝔽₈)⋊D₂₄₇	3
(C₁₃⋊C₃)⋊D₁₃₃	3
C₃×(C₁₃⋊C₃)⋊D₁₉	7
C₃×ΓL(1,𝔽₈)⋊D₁₉	13
ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	19

Subgroups — ΓL(1,𝔽8)⋊((C13⋊C3)⋊D19) #44527686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊((C₁₃⋊C₃)⋊D₁₉) #44527686