Subgroups — C₂×C₃×C₇²×D₆₅ #50304516

Order 38220.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×D₆₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅×C₇²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇²×D₆₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅×C₇²×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₇²×D₁₃	—	5	1	no	maximal, Hall
6	C₇²×D₆₅	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₇×D₆₅	C₇	1	8	yes	maximal, direct product factor
9	C₃×C₇²×D₁₃	—	5	2	no	—
10	C₂×C₃×C₇²×C₁₃	D₅	1	1	yes	characteristic, complemented
11	C₅×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₇²×D₅	—	13	1	no	maximal, Hall
13	C₃×C₇×D₆₅	C₂×C₇	1	16	yes	direct product factor
14	C₂×C₃×C₅×C₇×C₁₃	C₂×C₇	1	8	yes	complemented
15	C₂×C₇²×D₁₃	—	5	1	no	Hall
16	C₃×C₇²×C₁₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
17	C₂×C₇×D₆₅	C₃×C₇	1	8	yes	direct product factor
18	C₂×C₃×C₅×C₇²	D₁₃	1	1	yes	characteristic, complemented
19	C₃×C₇²×D₅	—	13	2	no	—
20	C₃×C₅×C₇×C₁₃	C₂²×C₇	1	8	yes	complemented
21	C₂×C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic, complemented
22	C₇²×D₁₃	—	5	2	no	—
23	C₂×C₃×C₇×D₁₃	—	5	8	no	—
24	C₂×C₇²×D₅	—	13	1	no	Hall
25	C₂×C₅×C₇×C₁₃	C₂×C₃×C₇	1	8	yes	complemented
26	C₇×D₆₅	C₂×C₃×C₇	1	16	yes	direct product factor
27	C₂×C₃×D₆₅	C₇²	1	1	yes	characteristic, direct product factor, Hall
28	C₃×C₅×C₇²	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
29	C₇²×C₁₃	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₃×C₇²	—	65	1	no	Hall
31	C₂×C₃×C₇×C₁₃	C₇×D₅	1	8	yes	complemented
32	C₃×C₇×D₁₃	—	5	16	no	—
33	C₂×C₅×C₇²	C₃×D₁₃	1	1	yes	characteristic, complemented
34	C₇²×D₅	—	13	2	no	—
35	C₅×C₇×C₁₃	C₂²×C₃×C₇	1	8	yes	complemented
36	C₂×C₃×C₇×D₅	—	13	8	no	—
37	C₂×C₃×C₅×C₁₃	C₂×C₇²	1	1	yes	characteristic, complemented
38	C₃×D₆₅	C₂×C₇²	1	2	yes	direct product factor
39	C₂×C₇×D₁₃	—	5	8	no	—
40	C₂×C₃×C₇²	—	65	2	no	—
41	C₂×C₃×C₇²	D₆₅	1	1	yes	characteristic, direct product factor
42	C₃×C₇×C₁₃	C₂×C₇×D₅	1	8	yes	complemented
43	C₂×D₆₅	C₃×C₇²	1	1	yes	characteristic, direct product factor, Hall
44	C₅×C₇²	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅×C₇	C₇×D₁₃	1	8	yes	complemented
46	C₃×C₇×D₅	—	13	16	no	—
47	C₂²×C₇²	—	65	1	no	Hall
48	C₃×C₅×C₁₃	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
49	C₂×C₇×C₁₃	C₃×C₇×D₅	1	8	yes	complemented
50	C₇×D₁₃	—	5	16	no	—
51	C₂×C₃×D₁₃	—	5	1	no	Hall
52	C₃×C₇²	C₂×D₆₅	1	1	yes	characteristic, direct product factor, Hall
53	C₂×C₇×D₅	—	13	8	no	—
54	C₂×C₅×C₁₃	C₂×C₃×C₇²	1	1	yes	characteristic, complemented
55	D₆₅	C₂×C₃×C₇²	1	2	yes	direct product factor
56	C₃×C₅×C₇	C₂×C₇×D₁₃	1	8	yes	complemented
57	C₂×C₇²	—	65	2	no	—
58	C₂×C₇²	C₃×D₆₅	1	1	yes	characteristic, direct product factor
59	C₇×C₁₃	C₂×C₃×C₇×D₅	1	8	yes	complemented
60	C₂²×C₃×C₇	—	65	8	no	—
61	C₂×C₃×C₁₃	C₇²×D₅	1	1	yes	characteristic, complemented
62	C₃×D₁₃	—	5	2	no	—
63	C₂×C₅×C₇	C₃×C₇×D₁₃	1	8	yes	complemented
64	C₇×D₅	—	13	16	no	—
65	C₅×C₁₃	C₂²×C₃×C₇²	1	1	yes	characteristic, complemented, Hall
66	C₂×C₃×D₅	—	13	1	no	Hall
67	C₂×D₁₃	—	5	1	no	Hall
68	C₇²	C₂×C₃×D₆₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×C₇	—	65	16	no	—
70	C₂×C₃×C₇	C₇×D₆₅	1	8	yes	direct product factor
71	C₃×C₁₃	C₂×C₇²×D₅	1	1	yes	characteristic, complemented, Hall
72	C₅×C₇	C₂×C₃×C₇×D₁₃	1	8	yes	complemented
73	C₂×C₃×C₅	C₇²×D₁₃	1	1	yes	characteristic, complemented
74	C₃×D₅	—	13	2	no	—
75	C₂²×C₇	—	65	8	no	—
76	C₂×C₁₃	C₃×C₇²×D₅	1	1	yes	characteristic, complemented
77	D₁₃	—	5	2	no	—
78	C₃×C₇	C₂×C₇×D₆₅	1	8	yes	direct product factor
79	C₂×D₅	—	13	1	no	Hall
80	C₃×C₅	C₂×C₇²×D₁₃	1	1	yes	characteristic, complemented, Hall
81	C₂×C₇	—	65	16	no	—
82	C₂×C₇	C₃×C₇×D₆₅	1	8	yes	direct product factor
83	C₁₃	C₂×C₃×C₇²×D₅	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₃	—	65	1	no	Hall
85	C₂×C₅	C₃×C₇²×D₁₃	1	1	yes	characteristic, complemented
86	D₅	—	13	2	no	—
87	C₇	C₂×C₃×C₇×D₆₅	1	8	yes	direct product factor
88	C₂×C₃	—	65	2	no	—
89	C₂×C₃	C₇²×D₆₅	1	1	yes	characteristic, direct product factor
90	C₅	C₂×C₃×C₇²×D₁₃	1	1	yes	characteristic, complemented, Sylow
91	C₂²	—	65	1	no	Sylow
92	C₃	C₂×C₇²×D₆₅	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	—	65	2	no	—
94	C₂	C₃×C₇²×D₆₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇²	D₆₅
commutator, cosocle, hypocenter	C₅×C₁₃	C₂²×C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇²×C₁₃	C₂
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₂×C₃×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃×C₇²×D₆₅	C₂	direct (N × H)
2. C₂×C₃×C₅×C₇²×C₁₃	C₂	semidirect
3. C₂×C₇²×D₆₅	C₃	direct (N × H)
4. C₃×C₅×C₇²×C₁₃	C₂²	semidirect
6. C₇²×D₆₅	C₂×C₃	direct (N × H)
7. C₂×C₅×C₇²×C₁₃	C₂×C₃	semidirect
8. C₂×C₃×C₇×D₆₅	C₇	direct (N × H)
10. C₂×C₃×C₇²×C₁₃	D₅	semidirect
11. C₅×C₇²×C₁₃	C₂²×C₃	semidirect
13. C₃×C₇×D₆₅	C₂×C₇	direct (N × H)
14. C₂×C₃×C₅×C₇×C₁₃	C₂×C₇	semidirect
16. C₃×C₇²×C₁₃	C₂×D₅	semidirect
17. C₂×C₇×D₆₅	C₃×C₇	direct (N × H)
18. C₂×C₃×C₅×C₇²	D₁₃	semidirect
20. C₃×C₅×C₇×C₁₃	C₂²×C₇	semidirect
21. C₂×C₇²×C₁₃	C₃×D₅	semidirect
25. C₂×C₅×C₇×C₁₃	C₂×C₃×C₇	semidirect
26. C₇×D₆₅	C₂×C₃×C₇	direct (N × H)
27. C₂×C₃×D₆₅	C₇²	direct (N × H)
28. C₃×C₅×C₇²	C₂×D₁₃	semidirect
29. C₇²×C₁₃	C₂×C₃×D₅	semidirect
31. C₂×C₃×C₇×C₁₃	C₇×D₅	semidirect
33. C₂×C₅×C₇²	C₃×D₁₃	semidirect
35. C₅×C₇×C₁₃	C₂²×C₃×C₇	semidirect
37. C₂×C₃×C₅×C₁₃	C₂×C₇²	semidirect
38. C₃×D₆₅	C₂×C₇²	direct (N × H)
41. C₂×C₃×C₇²	D₆₅	direct (N × H)
42. C₃×C₇×C₁₃	C₂×C₇×D₅	semidirect
43. C₂×D₆₅	C₃×C₇²	direct (N × H)
44. C₅×C₇²	C₂×C₃×D₁₃	semidirect
45. C₂×C₃×C₅×C₇	C₇×D₁₃	semidirect
48. C₃×C₅×C₁₃	C₂²×C₇²	semidirect
49. C₂×C₇×C₁₃	C₃×C₇×D₅	semidirect
52. C₃×C₇²	C₂×D₆₅	direct (N × H)
54. C₂×C₅×C₁₃	C₂×C₃×C₇²	semidirect
55. D₆₅	C₂×C₃×C₇²	direct (N × H)
56. C₃×C₅×C₇	C₂×C₇×D₁₃	semidirect
58. C₂×C₇²	C₃×D₆₅	direct (N × H)
59. C₇×C₁₃	C₂×C₃×C₇×D₅	semidirect
61. C₂×C₃×C₁₃	C₇²×D₅	semidirect
63. C₂×C₅×C₇	C₃×C₇×D₁₃	semidirect
65. C₅×C₁₃	C₂²×C₃×C₇²	semidirect
68. C₇²	C₂×C₃×D₆₅	direct (N × H)
70. C₂×C₃×C₇	C₇×D₆₅	direct (N × H)
71. C₃×C₁₃	C₂×C₇²×D₅	semidirect
72. C₅×C₇	C₂×C₃×C₇×D₁₃	semidirect
73. C₂×C₃×C₅	C₇²×D₁₃	semidirect
76. C₂×C₁₃	C₃×C₇²×D₅	semidirect
78. C₃×C₇	C₂×C₇×D₆₅	direct (N × H)
80. C₃×C₅	C₂×C₇²×D₁₃	semidirect
82. C₂×C₇	C₃×C₇×D₆₅	direct (N × H)
83. C₁₃	C₂×C₃×C₇²×D₅	semidirect
85. C₂×C₅	C₃×C₇²×D₁₃	semidirect
87. C₇	C₂×C₃×C₇×D₆₅	direct (N × H)
89. C₂×C₃	C₇²×D₆₅	direct (N × H)
90. C₅	C₂×C₃×C₇²×D₁₃	semidirect
92. C₃	C₂×C₇²×D₆₅	direct (N × H)
94. C₂	C₃×C₇²×D₆₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇²×D₆₅	2
C₂×C₃×C₅×C₇²×C₁₃	2
C₂×C₇²×D₆₅	3
C₂×C₃×C₇²×D₁₃	5
C₂×C₃×C₇×D₆₅	7
C₂×C₃×C₇²×D₅	13

Subgroups — C2×C3×C72×D65 #50304516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇²×D₆₅ #50304516