Subgroups — ΓL(1,𝔽₈)×D₁₉×C₁₃⋊C₃ #44527770

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	D₁₉×#63271	C₃	1	1	yes	maximal, characteristic, complemented
3	D₁₉×#63270	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×ΓL(1,𝔽₈)×D₁₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₇×D₁₉×C₁₃⋊C₃	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×D₁₉×C₁₃⋊C₃	—	7	1	no	maximal, Hall
11	C₇×C₁₃×D₁₉	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	C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	—	19	1	no	maximal, Hall
16	C₃×C₁₃×D₁₉	—	7	1	no	—
17	D₁₉×C₁₃⋊C₃	—	7	1	no	—
18	D₁₉×C₁₃⋊C₃	—	7	1	no	—
19	D₁₉×C₁₃⋊C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₃×C₁₉×ΓL(1,𝔽₈)	—	13	1	no	Hall
21	ΓL(1,𝔽₈)×C₁₃⋊C₃	D₁₉	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₇×D₁₉	—	13	1	no	—
23	ΓL(1,𝔽₈)×D₁₉	—	13	1	no	—
24	ΓL(1,𝔽₈)×D₁₉	—	13	1	no	—
25	ΓL(1,𝔽₈)×D₁₉	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
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	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₂×#63271	—	19	1	no	—
31	C₂×#63270	—	19	1	no	—
32	C₂×C₁₃×ΓL(1,𝔽₈)	—	19	1	no	—
33	C₂×C₇×C₁₃⋊C₃	—	19	1	no	—
34	C₁₃×D₁₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, 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)	C₂×C₁₃⋊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	C₇×D₁₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
45	C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
46	C₂×C₃×C₁₃⋊C₃	—	133	1	no	Hall
47	C₂×C₇×C₁₃	—	19	1	no	Hall
48	C₃²×C₁₉	—	91	1	no	Hall
49	C₇×C₁₉	C₂×C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×ΓL(1,𝔽₈)	—	247	1	no	Hall
51	C₃×C₁₃⋊C₃	—	7	1	no	Hall
52	C₃×D₁₉	—	7	1	no	—
53	C₃×D₁₉	—	13	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	C₂×C₃×C₁₃	—	133	1	no	—
58	C₂×C₁₃⋊C₃	—	19	1	no	—
59	C₂×C₁₃⋊C₃	—	133	1	no	—
60	C₂×C₁₃⋊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	C₂×C₃×C₇	—	247	1	no	—
67	C₂×ΓL(1,𝔽₈)	—	19	1	no	—
68	C₂×ΓL(1,𝔽₈)	—	247	1	no	—
69	C₂×ΓL(1,𝔽₈)	—	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, direct product factor
74	D₁₉	ΓL(1,𝔽₈)×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
75	C₂×C₁₃	—	19	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)	D₁₉×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
80	C₁₉	C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃²	—	1729	1	no	Hall
82	C₂×C₇	—	19	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×D₁₉×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	133	1	no	—
87	C₂×C₃	—	1729	1	no	—
88	C₂×C₃	—	1729	1	no	—
89	C₂×C₃	—	247	1	no	—
90	C₃	—	7	1	no	—
91	C₃	—	13	1	no	—
92	C₃	—	91	1	no	—
93	C₃	—	91	1	no	—
94	C₂	—	19	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. D₁₉×#63271	C₃	semidirect
3. D₁₉×#63270	C₃	semidirect
4. C₁₃×ΓL(1,𝔽₈)×D₁₉	C₃	semidirect
5. C₇×D₁₉×C₁₃⋊C₃	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. C₇×C₁₃×D₁₉	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
19. D₁₉×C₁₃⋊C₃	ΓL(1,𝔽₈)	direct (N × H)
21. ΓL(1,𝔽₈)×C₁₃⋊C₃	D₁₉	direct (N × H)
25. ΓL(1,𝔽₈)×D₁₉	C₁₃⋊C₃	direct (N × H)
29. C₁₉×C₁₃⋊C₃	C₂×ΓL(1,𝔽₈)	semidirect
34. C₁₃×D₁₉	C₃×ΓL(1,𝔽₈)	semidirect
38. C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃⋊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
44. C₇×D₁₉	C₃×C₁₃⋊C₃	semidirect
45. C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
49. C₇×C₁₉	C₂×C₃×C₁₃⋊C₃	semidirect
56. C₇×C₁₃	C₃²×D₁₉	semidirect
73. C₁₃⋊C₃	ΓL(1,𝔽₈)×D₁₉	direct (N × H)
74. D₁₉	ΓL(1,𝔽₈)×C₁₃⋊C₃	direct (N × H)
79. ΓL(1,𝔽₈)	D₁₉×C₁₃⋊C₃	direct (N × H)
80. C₁₉	C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	semidirect
83. C₁₃	C₃×ΓL(1,𝔽₈)×D₁₉	semidirect
85. C₇	C₃×D₁₉×C₁₃⋊C₃	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
D₁₉×#63271	3
D₁₉×#63270	3
C₁₃×ΓL(1,𝔽₈)×D₁₉	3
C₇×D₁₉×C₁₃⋊C₃	3
C₃×D₁₉×C₁₃⋊C₃	7
C₃×ΓL(1,𝔽₈)×D₁₉	13
C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	19

Subgroups — ΓL(1,𝔽8)×D19×C13⋊C3 #44527770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₉×C₁₃⋊C₃ #44527770