Subgroups — C₂²×C₃×C₅×#87704 #37561627

Order 23940.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#87704	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅×#87704	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₅×C₇×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#87704	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃×#87704	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×#87704	C₂×C₃	1	9	yes	direct product factor
7	C₂×C₃×C₅×C₇×C₁₉	C₂×C₃	1	3	yes	complemented
8	C₂²×C₃×C₅×C₁₉⋊C₃	—	7	1	no	maximal, Hall
9	C₂²×C₅×C₇×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×#87704	C₂×C₅	1	3	yes	direct product factor
11	C₃×C₅×C₇×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₅×#87704	C₂²×C₃	1	3	yes	direct product factor
13	C₂×C₃×C₅×C₁₉⋊C₃	—	7	3	no	—
14	C₂²×C₃×C₇×C₁₉	C₃×C₅	1	1	yes	characteristic, complemented
15	C₂²×#87704	C₃×C₅	1	3	yes	direct product factor
16	C₂×C₅×C₇×C₁₉	C₂×C₃²	1	3	yes	complemented
17	C₂²×C₃×C₅×ΓL(1,𝔽₈)	—	19	1	no	maximal, Hall
18	C₃×#87704	C₂²×C₅	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₃×C₅×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	C₂²×C₅×C₁₉⋊C₃	—	7	3	no	—
21	C₃×C₅×C₁₉⋊C₃	—	7	1	no	Hall
22	C₂×C₃×C₇×C₁₉	C₂×C₃×C₅	1	3	yes	complemented
23	C₂×#87704	C₂×C₃×C₅	1	9	yes	direct product factor
24	C₂²×C₃×C₁₉⋊C₃	—	7	1	no	Hall
25	C₅×C₇×C₁₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	19	3	no	—
27	C₂×C₃×C₅×C₁₉	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
28	C₂×C₅×C₁₉⋊C₃	—	7	9	no	—
29	C₂²×C₇×C₁₉	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₃×C₅×C₇	C₁₉⋊C₃	1	1	yes	characteristic, complemented
31	C₂²×C₅×ΓL(1,𝔽₈)	—	19	3	no	—
32	C₃×C₇×C₁₉	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
33	#87704	C₂²×C₃×C₅	1	3	yes	direct product factor
34	C₂²×C₅×C₁₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₁₉⋊C₃	—	7	3	no	—
36	C₃×C₅×ΓL(1,𝔽₈)	—	19	1	no	Hall
37	C₃×C₅×C₁₉	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	C₅×C₁₉⋊C₃	—	7	3	no	—
39	C₂×C₇×C₁₉	C₂×C₃²×C₅	1	3	yes	complemented
40	C₂²×C₃×ΓL(1,𝔽₈)	—	19	1	no	Hall
41	C₂²×C₃×C₁₉	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
42	C₂²×C₁₉⋊C₃	—	7	3	no	—
43	C₂×C₃×C₅×C₇	C₂×C₁₉⋊C₃	1	3	yes	complemented
44	C₂×C₅×ΓL(1,𝔽₈)	—	19	9	no	—
45	C₂×C₅×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	3	yes	complemented
46	C₂²×C₃²×C₅	—	133	1	no	Hall
47	C₃×C₁₉⋊C₃	—	7	1	no	Hall
48	C₂²×C₅×C₇	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
49	C₇×C₁₉	C₂²×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×ΓL(1,𝔽₈)	—	19	3	no	—
51	C₂×C₃×C₁₉	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	complemented
52	C₂×C₁₉⋊C₃	—	7	9	no	—
53	C₃×C₅×C₇	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, complemented
54	C₅×ΓL(1,𝔽₈)	—	19	3	no	—
55	C₅×C₁₉	C₂²×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
56	C₂×C₃²×C₅	—	133	3	no	—
57	C₂²×C₃×C₇	C₅×C₁₉⋊C₃	1	1	yes	characteristic, complemented
58	C₂²×ΓL(1,𝔽₈)	—	19	3	no	—
59	C₂²×C₁₉	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
60	C₂×C₅×C₇	C₂×C₃×C₁₉⋊C₃	1	3	yes	complemented
61	C₃×ΓL(1,𝔽₈)	—	19	1	no	Hall
62	C₂²×C₃×C₅	—	133	3	no	—
63	C₂²×C₃×C₅	#87704	1	1	yes	characteristic, direct product factor
64	C₃×C₁₉	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
65	C₁₉⋊C₃	—	7	3	no	—
66	C₃²×C₅	—	133	1	no	Hall
67	C₂×C₃×C₇	C₂×C₅×C₁₉⋊C₃	1	3	yes	complemented
68	C₂×ΓL(1,𝔽₈)	—	19	9	no	—
69	C₂×C₁₉	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	3	yes	complemented
70	C₂²×C₃²	—	133	1	no	Hall
71	C₅×C₇	C₂²×C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
72	C₂×C₃×C₅	—	133	9	no	—
73	C₂×C₃×C₅	C₂×#87704	1	3	yes	direct product factor
74	C₂²×C₇	C₃×C₅×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
75	C₃×C₇	C₂²×C₅×C₁₉⋊C₃	1	1	yes	characteristic, complemented
76	ΓL(1,𝔽₈)	—	19	3	no	—
77	C₂²×C₅	C₃×#87704	1	1	yes	characteristic, direct product factor, Hall
78	C₁₉	C₂²×C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃²	—	133	3	no	—
80	C₃×C₅	—	133	3	no	—
81	C₃×C₅	C₂²×#87704	1	1	yes	characteristic, direct product factor
82	C₂×C₇	C₂×C₃×C₅×C₁₉⋊C₃	1	3	yes	complemented
83	C₂²×C₃	—	133	3	no	—
84	C₂²×C₃	C₅×#87704	1	1	yes	characteristic, direct product factor
85	C₂×C₅	C₂×C₃×#87704	1	3	yes	direct product factor
86	C₃²	—	133	1	no	Sylow
87	C₇	C₂²×C₃×C₅×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	133	9	no	—
89	C₂×C₃	C₂×C₅×#87704	1	3	yes	direct product factor
90	C₅	C₂²×C₃×#87704	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²	C₃×C₅×#87704	1	1	yes	characteristic, direct product factor, Sylow
92	C₃	—	133	3	no	—
93	C₃	C₂²×C₅×#87704	1	1	yes	characteristic, direct product factor
94	C₂	C₂×C₃×C₅×#87704	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅	#87704
commutator, cosocle, hypocenter	C₇×C₁₉	C₂²×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅×C₇×C₁₉	C₃
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂²×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#87704	C₂	direct (N × H)
2. C₂²×C₅×#87704	C₃	direct (N × H)
3. C₂²×C₃×C₅×C₇×C₁₉	C₃	semidirect
4. C₃×C₅×#87704	C₂²	direct (N × H)
5. C₂²×C₃×#87704	C₅	direct (N × H)
6. C₂×C₅×#87704	C₂×C₃	direct (N × H)
7. C₂×C₃×C₅×C₇×C₁₉	C₂×C₃	semidirect
9. C₂²×C₅×C₇×C₁₉	C₃²	semidirect
10. C₂×C₃×#87704	C₂×C₅	direct (N × H)
11. C₃×C₅×C₇×C₁₉	C₂²×C₃	semidirect
12. C₅×#87704	C₂²×C₃	direct (N × H)
14. C₂²×C₃×C₇×C₁₉	C₃×C₅	semidirect
15. C₂²×#87704	C₃×C₅	direct (N × H)
16. C₂×C₅×C₇×C₁₉	C₂×C₃²	semidirect
18. C₃×#87704	C₂²×C₅	direct (N × H)
19. C₂²×C₃×C₅×C₁₉	ΓL(1,𝔽₈)	semidirect
22. C₂×C₃×C₇×C₁₉	C₂×C₃×C₅	semidirect
23. C₂×#87704	C₂×C₃×C₅	direct (N × H)
25. C₅×C₇×C₁₉	C₂²×C₃²	semidirect
27. C₂×C₃×C₅×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
29. C₂²×C₇×C₁₉	C₃²×C₅	semidirect
30. C₂²×C₃×C₅×C₇	C₁₉⋊C₃	semidirect
32. C₃×C₇×C₁₉	C₂²×C₃×C₅	semidirect
33. #87704	C₂²×C₃×C₅	direct (N × H)
34. C₂²×C₅×C₁₉	C₃×ΓL(1,𝔽₈)	semidirect
37. C₃×C₅×C₁₉	C₂²×ΓL(1,𝔽₈)	semidirect
39. C₂×C₇×C₁₉	C₂×C₃²×C₅	semidirect
41. C₂²×C₃×C₁₉	C₅×ΓL(1,𝔽₈)	semidirect
43. C₂×C₃×C₅×C₇	C₂×C₁₉⋊C₃	semidirect
45. C₂×C₅×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
48. C₂²×C₅×C₇	C₃×C₁₉⋊C₃	semidirect
49. C₇×C₁₉	C₂²×C₃²×C₅	semidirect
51. C₂×C₃×C₁₉	C₂×C₅×ΓL(1,𝔽₈)	semidirect
53. C₃×C₅×C₇	C₂²×C₁₉⋊C₃	semidirect
55. C₅×C₁₉	C₂²×C₃×ΓL(1,𝔽₈)	semidirect
57. C₂²×C₃×C₇	C₅×C₁₉⋊C₃	semidirect
59. C₂²×C₁₉	C₃×C₅×ΓL(1,𝔽₈)	semidirect
60. C₂×C₅×C₇	C₂×C₃×C₁₉⋊C₃	semidirect
63. C₂²×C₃×C₅	#87704	direct (N × H)
64. C₃×C₁₉	C₂²×C₅×ΓL(1,𝔽₈)	semidirect
67. C₂×C₃×C₇	C₂×C₅×C₁₉⋊C₃	semidirect
69. C₂×C₁₉	C₂×C₃×C₅×ΓL(1,𝔽₈)	semidirect
71. C₅×C₇	C₂²×C₃×C₁₉⋊C₃	semidirect
73. C₂×C₃×C₅	C₂×#87704	direct (N × H)
74. C₂²×C₇	C₃×C₅×C₁₉⋊C₃	semidirect
75. C₃×C₇	C₂²×C₅×C₁₉⋊C₃	semidirect
77. C₂²×C₅	C₃×#87704	direct (N × H)
78. C₁₉	C₂²×C₃×C₅×ΓL(1,𝔽₈)	semidirect
81. C₃×C₅	C₂²×#87704	direct (N × H)
82. C₂×C₇	C₂×C₃×C₅×C₁₉⋊C₃	semidirect
84. C₂²×C₃	C₅×#87704	direct (N × H)
85. C₂×C₅	C₂×C₃×#87704	direct (N × H)
87. C₇	C₂²×C₃×C₅×C₁₉⋊C₃	semidirect
89. C₂×C₃	C₂×C₅×#87704	direct (N × H)
90. C₅	C₂²×C₃×#87704	direct (N × H)
91. C₂²	C₃×C₅×#87704	direct (N × H)
93. C₃	C₂²×C₅×#87704	direct (N × H)
94. C₂	C₂×C₃×C₅×#87704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#87704	2
C₂²×C₅×#87704	3
C₂²×C₃×C₅×C₇×C₁₉	3
C₂²×C₃×#87704	5
C₂²×C₃×C₅×C₁₉⋊C₃	7
C₂²×C₃×C₅×ΓL(1,𝔽₈)	19

Subgroups — C22×C3×C5×#87704 #37561627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×#87704 #37561627