Subgroups — C₂×ΓL(1,𝔽₈)×#10625202 #44527809

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, direct product factor, Hall
2	C₂×#19104444	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×#19104440	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#10625202	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	C₂×C₃×#10625202	—	7	1	no	maximal, Hall
11	C₂×C₇×C₁₃×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×ΓL(1,𝔽₈)×C₁₉⋊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	C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	—	19	1	no	maximal, Hall
16	C₂×C₃×C₁₃×C₁₉	—	7	1	no	—
17	C₂×#10625202	—	7	1	no	—
18	C₂×#10625202	—	7	1	no	—
19	C₂×#10625202	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	ΓL(1,𝔽₈)×C₁₉⋊C₃	—	13	1	no	Hall
21	ΓL(1,𝔽₈)×C₁₃⋊C₃	—	19	1	no	Hall
22	C₂×#87705	—	13	1	no	—
23	C₂×#87704	—	13	1	no	—
24	C₂×C₁₉×ΓL(1,𝔽₈)	C₁₃⋊C₃	1	1	yes	characteristic, complemented
25	C₂×C₇×C₁₉⋊C₃	—	13	1	no	—
26	C₃×C₁₃×C₁₉	—	7	1	no	—
27	#10625202	—	7	1	no	—
28	#10625202	—	7	1	no	—
29	#10625202	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
30	C₂×#63271	—	19	1	no	—
31	C₂×#63270	—	19	1	no	—
32	C₂×C₁₃×ΓL(1,𝔽₈)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
33	C₂×C₇×C₁₃⋊C₃	—	19	1	no	—
34	C₂×C₁₃×C₁₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	#87705	—	13	1	no	—
36	#87704	—	13	1	no	—
37	C₇×C₁₉⋊C₃	—	13	1	no	—
38	C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
39	C₂×C₃×C₁₉⋊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)	C₂×C₁₉⋊C₃	1	1	yes	characteristic, complemented
44	C₂×C₇×C₁₉	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	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
48	C₃×C₁₉⋊C₃	—	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	—	133	1	no	Hall
52	C₂×C₃×C₁₉	—	7	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	C₂×C₃×C₁₃	—	7	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)	—	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	C₂×C₃×C₇	—	247	1	no	—
67	C₂×ΓL(1,𝔽₈)	—	247	1	no	—
68	C₂×ΓL(1,𝔽₈)	—	247	1	no	—
69	C₂×ΓL(1,𝔽₈)	#10625202	1	1	yes	characteristic, direct product factor
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	C₂×C₁₉	ΓL(1,𝔽₈)×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
75	C₂×C₁₃	ΓL(1,𝔽₈)×C₁₉⋊C₃	1	1	yes	characteristic, complemented, 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)	C₂×#10625202	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	C₃×#10625202	1	1	yes	characteristic, complemented, Hall
83	C₁₃	C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	1729	1	no	Sylow
85	C₇	C₂×C₃×#10625202	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	7	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	—	1729	1	no	—
92	C₃	—	1729	1	no	—
93	C₃	—	247	1	no	—
94	C₂	ΓL(1,𝔽₈)×#10625202	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×#10625202
commutator, cosocle, hypocenter	C₇×C₁₃×C₁₉	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₃×C₁₉	C₃²
LCS	G ▷ C₇×C₁₃×C₁₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#10625202	C₂	direct (N × H)
2. C₂×#19104444	C₃	semidirect
3. C₂×#19104440	C₃	semidirect
4. C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₃	semidirect
5. C₂×C₇×#10625202	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. C₂×C₇×C₁₃×C₁₉	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
19. C₂×#10625202	ΓL(1,𝔽₈)	direct (N × H)
24. C₂×C₁₉×ΓL(1,𝔽₈)	C₁₃⋊C₃	semidirect
29. #10625202	C₂×ΓL(1,𝔽₈)	direct (N × H)
32. C₂×C₁₃×ΓL(1,𝔽₈)	C₁₉⋊C₃	semidirect
34. C₂×C₁₃×C₁₉	C₃×ΓL(1,𝔽₈)	semidirect
38. C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃⋊C₃	semidirect
43. C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉⋊C₃	semidirect
44. C₂×C₇×C₁₉	C₃×C₁₃⋊C₃	semidirect
45. C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
47. C₂×C₇×C₁₃	C₃×C₁₉⋊C₃	semidirect
49. C₇×C₁₉	C₂×C₃×C₁₃⋊C₃	semidirect
56. C₇×C₁₃	C₂×C₃×C₁₉⋊C₃	semidirect
69. C₂×ΓL(1,𝔽₈)	#10625202	direct (N × H)
74. C₂×C₁₉	ΓL(1,𝔽₈)×C₁₃⋊C₃	semidirect
75. C₂×C₁₃	ΓL(1,𝔽₈)×C₁₉⋊C₃	semidirect
79. ΓL(1,𝔽₈)	C₂×#10625202	direct (N × H)
80. C₁₉	C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	semidirect
82. C₂×C₇	C₃×#10625202	semidirect
83. C₁₃	C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	semidirect
85. C₇	C₂×C₃×#10625202	semidirect
94. C₂	ΓL(1,𝔽₈)×#10625202	direct (N × H)

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
C₂×#19104444	3
C₂×#19104440	3
C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	3
C₂×C₇×#10625202	3
C₂×C₃×#10625202	7
C₂×ΓL(1,𝔽₈)×C₁₉⋊C₃	13
C₂×ΓL(1,𝔽₈)×C₁₃⋊C₃	19

Subgroups — C2×ΓL(1,𝔽8)×#10625202 #44527809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×#10625202 #44527809