Subgroups — ΓL(1,𝔽₈)×D₁₃×C₁₉⋊C₃ #44527776

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×ΓL(1,𝔽₈)×C₁₉⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₁₃×#87705	C₃	1	1	yes	maximal, characteristic, complemented
3	D₁₃×#87704	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₉×ΓL(1,𝔽₈)×D₁₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₇×D₁₃×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#87705	C₂×C₃	1	1	yes	characteristic, complemented
7	C₁₃×#87704	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×C₁₃×C₁₉⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×D₁₃×C₁₉⋊C₃	—	7	1	no	maximal, Hall
11	C₇×C₁₉×D₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	—	13	1	no	maximal, Hall
13	C₃×C₁₃×C₁₉⋊C₃	—	7	1	no	Hall
14	C₇×C₁₃×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
15	C₃×ΓL(1,𝔽₈)×D₁₃	—	19	1	no	maximal, Hall
16	C₃×C₁₉×D₁₃	—	7	1	no	—
17	D₁₃×C₁₉⋊C₃	—	7	1	no	—
18	D₁₃×C₁₉⋊C₃	—	7	1	no	—
19	D₁₃×C₁₉⋊C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	ΓL(1,𝔽₈)×C₁₉⋊C₃	D₁₃	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₁₃×ΓL(1,𝔽₈)	—	19	1	no	Hall
22	C₂×#87705	—	13	1	no	—
23	C₂×#87704	—	13	1	no	—
24	C₂×C₁₉×ΓL(1,𝔽₈)	—	13	1	no	—
25	C₂×C₇×C₁₉⋊C₃	—	13	1	no	—
26	C₃×C₁₃×C₁₉	—	7	1	no	—
27	C₁₃×C₁₉⋊C₃	—	7	1	no	—
28	C₁₃×C₁₉⋊C₃	—	7	1	no	—
29	C₁₃×C₁₉⋊C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₃×C₇×D₁₃	—	19	1	no	—
31	ΓL(1,𝔽₈)×D₁₃	—	19	1	no	—
32	ΓL(1,𝔽₈)×D₁₃	—	19	1	no	—
33	ΓL(1,𝔽₈)×D₁₃	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
34	C₁₉×D₁₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	#87705	C₃×D₁₃	1	1	yes	characteristic, complemented
36	#87704	C₃×D₁₃	1	1	yes	characteristic, complemented
37	C₇×C₁₉⋊C₃	C₃×D₁₃	1	1	yes	characteristic, complemented
38	C₁₉×ΓL(1,𝔽₈)	C₃×D₁₃	1	1	yes	characteristic, complemented
39	C₂×C₃×C₁₉⋊C₃	—	91	1	no	Hall
40	C₃×C₇×C₁₃	—	19	1	no	—
41	C₁₃×ΓL(1,𝔽₈)	—	19	1	no	—
42	C₁₃×ΓL(1,𝔽₈)	—	19	1	no	—
43	C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉⋊C₃	1	1	yes	characteristic, complemented
44	C₂×C₇×C₁₉	—	13	1	no	Hall
45	C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
46	C₃²×D₁₃	—	133	1	no	Hall
47	C₇×D₁₃	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
48	C₃×C₁₉⋊C₃	—	7	1	no	Hall
49	C₇×C₁₉	C₃²×D₁₃	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×ΓL(1,𝔽₈)	—	247	1	no	Hall
51	C₃²×C₁₃	—	133	1	no	Hall
52	C₂×C₃×C₁₉	—	91	1	no	—
53	C₂×C₁₉⋊C₃	—	13	1	no	—
54	C₂×C₁₉⋊C₃	—	91	1	no	—
55	C₂×C₁₉⋊C₃	—	91	1	no	—
56	C₇×C₁₃	C₂×C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
57	C₃×D₁₃	—	7	1	no	—
58	C₃×D₁₃	—	19	1	no	—
59	C₃×D₁₃	—	133	1	no	—
60	C₃×D₁₃	—	133	1	no	—
61	C₃×ΓL(1,𝔽₈)	—	19	1	no	Hall
62	C₃×C₁₉	—	7	1	no	—
63	C₁₉⋊C₃	—	7	1	no	—
64	C₁₉⋊C₃	—	7	1	no	—
65	C₁₉⋊C₃	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, direct product factor
66	C₂×C₃×C₇	—	247	1	no	—
67	C₂×ΓL(1,𝔽₈)	—	13	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	C₃×C₁₃	—	19	1	no	—
72	C₃×C₁₃	—	133	1	no	—
73	C₃×C₁₃	—	133	1	no	—
74	C₂×C₁₉	—	13	1	no	Hall
75	D₁₃	ΓL(1,𝔽₈)×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Hall
76	C₃×C₇	—	19	1	no	—
77	ΓL(1,𝔽₈)	—	19	1	no	—
78	ΓL(1,𝔽₈)	—	19	1	no	—
79	ΓL(1,𝔽₈)	D₁₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
80	C₁₉	C₃×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃²	—	1729	1	no	Hall
82	C₂×C₇	—	13	1	no	Hall
83	C₁₃	C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	133	1	no	Sylow
85	C₇	C₃×D₁₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	91	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	—	19	1	no	—
92	C₃	—	133	1	no	—
93	C₃	—	133	1	no	—
94	C₂	—	13	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. C₁₃×ΓL(1,𝔽₈)×C₁₉⋊C₃	C₂	semidirect
2. D₁₃×#87705	C₃	semidirect
3. D₁₃×#87704	C₃	semidirect
4. C₁₉×ΓL(1,𝔽₈)×D₁₃	C₃	semidirect
5. C₇×D₁₃×C₁₉⋊C₃	C₃	semidirect
6. C₁₃×#87705	C₂×C₃	semidirect
7. C₁₃×#87704	C₂×C₃	semidirect
8. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
9. C₇×C₁₃×C₁₉⋊C₃	C₂×C₃	semidirect
11. C₇×C₁₉×D₁₃	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
19. D₁₃×C₁₉⋊C₃	ΓL(1,𝔽₈)	direct (N × H)
20. ΓL(1,𝔽₈)×C₁₉⋊C₃	D₁₃	direct (N × H)
29. C₁₃×C₁₉⋊C₃	C₂×ΓL(1,𝔽₈)	semidirect
33. ΓL(1,𝔽₈)×D₁₃	C₁₉⋊C₃	direct (N × H)
34. C₁₉×D₁₃	C₃×ΓL(1,𝔽₈)	semidirect
35. #87705	C₃×D₁₃	semidirect
36. #87704	C₃×D₁₃	semidirect
37. C₇×C₁₉⋊C₃	C₃×D₁₃	semidirect
38. C₁₉×ΓL(1,𝔽₈)	C₃×D₁₃	semidirect
43. C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉⋊C₃	semidirect
45. C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
47. C₇×D₁₃	C₃×C₁₉⋊C₃	semidirect
49. C₇×C₁₉	C₃²×D₁₃	semidirect
56. C₇×C₁₃	C₂×C₃×C₁₉⋊C₃	semidirect
65. C₁₉⋊C₃	ΓL(1,𝔽₈)×D₁₃	direct (N × H)
75. D₁₃	ΓL(1,𝔽₈)×C₁₉⋊C₃	direct (N × H)
79. ΓL(1,𝔽₈)	D₁₃×C₁₉⋊C₃	direct (N × H)
80. C₁₉	C₃×ΓL(1,𝔽₈)×D₁₃	semidirect
83. C₁₃	C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	semidirect
85. C₇	C₃×D₁₃×C₁₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×ΓL(1,𝔽₈)×C₁₉⋊C₃	2
D₁₃×#87705	3
D₁₃×#87704	3
C₁₉×ΓL(1,𝔽₈)×D₁₃	3
C₇×D₁₃×C₁₉⋊C₃	3
C₃×D₁₃×C₁₉⋊C₃	7
C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	13
C₃×ΓL(1,𝔽₈)×D₁₃	19

Subgroups — ΓL(1,𝔽8)×D13×C19⋊C3 #44527776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₃×C₁₉⋊C₃ #44527776