Subgroups — C₂²×C₃×C₅×C₈×C₄₁ #72197331

Order 19680.

304 conjugacy classes of subgroups.

The subgroup lattice has 112 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₅×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₈×C₄₁	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×C₈×C₄₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₅×C₄₁	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄×C₅×C₄₁	C₄	1	3	yes	—
6	C₂×C₃×C₄×C₅×C₄₁	C₂²	1	3	yes	—
7	C₃×C₅×C₈×C₄₁	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₈×C₄₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₄×C₅×C₄₁	C₂×C₃	1	1	yes	characteristic
10	C₂×C₅×C₈×C₄₁	C₂×C₃	1	6	yes	direct product factor
11	C₂²×C₃×C₅×C₄₁	C₈	1	4	yes	direct product factor
12	C₂²×C₃×C₅×C₄₁	C₂×C₄	1	3	yes	—
13	C₃×C₄×C₅×C₄₁	C₂×C₄	1	3	yes	—
14	C₃×C₄×C₅×C₄₁	C₂³	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₄₁	C₂×C₅	1	1	yes	characteristic
16	C₂×C₃×C₈×C₄₁	C₂×C₅	1	6	yes	direct product factor
17	C₂³×C₅×C₄₁	C₃×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅×C₄₁	C₃×C₄	1	3	yes	—
19	C₂×C₄×C₅×C₄₁	C₂²×C₃	1	3	yes	—
20	C₅×C₈×C₄₁	C₂²×C₃	1	4	yes	direct product factor
21	C₂²×C₈×C₄₁	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₅×C₄₁	C₂×C₈	1	6	yes	direct product factor
23	C₂×C₃×C₅×C₄₁	C₂²×C₄	1	1	yes	characteristic
24	C₂³×C₃×C₄₁	C₄×C₅	1	1	yes	characteristic
25	C₂×C₃×C₄×C₄₁	C₄×C₅	1	3	yes	—
26	C₂×C₃×C₄×C₄₁	C₂²×C₅	1	3	yes	—
27	C₃×C₈×C₄₁	C₂²×C₅	1	4	yes	direct product factor
28	C₂²×C₅×C₄₁	C₃×C₈	1	4	yes	direct product factor
29	C₂²×C₅×C₄₁	C₂×C₃×C₄	1	3	yes	—
30	C₄×C₅×C₄₁	C₂×C₃×C₄	1	3	yes	—
31	C₄×C₅×C₄₁	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₄×C₄₁	C₂×C₃×C₅	1	1	yes	characteristic
33	C₂×C₈×C₄₁	C₂×C₃×C₅	1	6	yes	direct product factor
34	C₃×C₅×C₄₁	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
35	C₂²×C₃×C₄₁	C₅×C₈	1	4	yes	direct product factor
36	C₂²×C₃×C₄₁	C₂×C₄×C₅	1	3	yes	—
37	C₃×C₄×C₄₁	C₂×C₄×C₅	1	3	yes	—
38	C₃×C₄×C₄₁	C₂³×C₅	1	1	yes	characteristic
39	C₂²×C₃×C₅×C₈	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₅×C₄₁	C₂×C₃×C₈	1	6	yes	direct product factor
41	C₂×C₅×C₄₁	C₂²×C₃×C₄	1	1	yes	characteristic
42	C₂³×C₄₁	C₃×C₄×C₅	1	1	yes	characteristic
43	C₂×C₄×C₄₁	C₃×C₄×C₅	1	3	yes	—
44	C₂×C₄×C₄₁	C₂²×C₃×C₅	1	3	yes	—
45	C₈×C₄₁	C₂²×C₃×C₅	1	4	yes	direct product factor
46	C₂×C₃×C₄₁	C₂×C₅×C₈	1	6	yes	direct product factor
47	C₂×C₃×C₄₁	C₂²×C₄×C₅	1	1	yes	characteristic
48	C₂²×C₃×C₄×C₅	C₂×C₄₁	1	1	yes	characteristic
49	C₂×C₃×C₅×C₈	C₂×C₄₁	1	6	yes	direct product factor
50	C₅×C₄₁	C₂²×C₃×C₈	1	1	yes	characteristic, direct product factor, Hall
51	C₂²×C₄₁	C₃×C₅×C₈	1	4	yes	direct product factor
52	C₂²×C₄₁	C₂×C₃×C₄×C₅	1	3	yes	—
53	C₄×C₄₁	C₂×C₃×C₄×C₅	1	3	yes	—
54	C₄×C₄₁	C₂³×C₃×C₅	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₃×C₄₁	1	1	yes	characteristic, direct product factor, Hall
56	C₃×C₄₁	C₂²×C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
57	C₂³×C₃×C₅	C₄×C₄₁	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄×C₄₁	1	3	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄₁	1	3	yes	—
60	C₃×C₅×C₈	C₂²×C₄₁	1	4	yes	direct product factor
61	C₂²×C₃×C₈	C₅×C₄₁	1	1	yes	characteristic, direct product factor, Hall
62	C₂×C₄₁	C₂×C₃×C₅×C₈	1	6	yes	direct product factor
63	C₂×C₄₁	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×C₃×C₄₁	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂×C₃×C₄₁	1	6	yes	direct product factor
66	C₂²×C₃×C₅	C₈×C₄₁	1	4	yes	direct product factor
67	C₂²×C₃×C₅	C₂×C₄×C₄₁	1	3	yes	—
68	C₃×C₄×C₅	C₂×C₄×C₄₁	1	3	yes	—
69	C₃×C₄×C₅	C₂³×C₄₁	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×C₅×C₄₁	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×C₅×C₄₁	1	6	yes	direct product factor
72	C₄₁	C₂²×C₃×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₅	C₃×C₄×C₄₁	1	1	yes	characteristic
74	C₂×C₄×C₅	C₃×C₄×C₄₁	1	3	yes	—
75	C₂×C₄×C₅	C₂²×C₃×C₄₁	1	3	yes	—
76	C₅×C₈	C₂²×C₃×C₄₁	1	4	yes	direct product factor
77	C₂²×C₈	C₃×C₅×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×C₅	C₂×C₈×C₄₁	1	6	yes	direct product factor
79	C₂×C₃×C₅	C₂²×C₄×C₄₁	1	1	yes	characteristic
80	C₂³×C₃	C₄×C₅×C₄₁	1	1	yes	characteristic
81	C₂×C₃×C₄	C₄×C₅×C₄₁	1	3	yes	—
82	C₂×C₃×C₄	C₂²×C₅×C₄₁	1	3	yes	—
83	C₃×C₈	C₂²×C₅×C₄₁	1	4	yes	direct product factor
84	C₂²×C₅	C₃×C₈×C₄₁	1	4	yes	direct product factor
85	C₂²×C₅	C₂×C₃×C₄×C₄₁	1	3	yes	—
86	C₄×C₅	C₂×C₃×C₄×C₄₁	1	3	yes	—
87	C₄×C₅	C₂³×C₃×C₄₁	1	1	yes	characteristic
88	C₂²×C₄	C₂×C₃×C₅×C₄₁	1	1	yes	characteristic
89	C₂×C₈	C₂×C₃×C₅×C₄₁	1	6	yes	direct product factor
90	C₃×C₅	C₂²×C₈×C₄₁	1	1	yes	characteristic, direct product factor, Hall
91	C₂²×C₃	C₅×C₈×C₄₁	1	4	yes	direct product factor
92	C₂²×C₃	C₂×C₄×C₅×C₄₁	1	3	yes	—
93	C₃×C₄	C₂×C₄×C₅×C₄₁	1	3	yes	—
94	C₃×C₄	C₂³×C₅×C₄₁	1	1	yes	characteristic
95	C₂×C₅	C₂×C₃×C₈×C₄₁	1	6	yes	direct product factor
96	C₂×C₅	C₂²×C₃×C₄×C₄₁	1	1	yes	characteristic
97	C₂³	C₃×C₄×C₅×C₄₁	1	1	yes	characteristic
98	C₂×C₄	C₃×C₄×C₅×C₄₁	1	3	yes	—
99	C₂×C₄	C₂²×C₃×C₅×C₄₁	1	3	yes	—
100	C₈	C₂²×C₃×C₅×C₄₁	1	4	yes	direct product factor
101	C₂×C₃	C₂×C₅×C₈×C₄₁	1	6	yes	direct product factor
102	C₂×C₃	C₂²×C₄×C₅×C₄₁	1	1	yes	characteristic
103	C₅	C₂²×C₃×C₈×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²	C₃×C₅×C₈×C₄₁	1	4	yes	direct product factor
105	C₂²	C₂×C₃×C₄×C₅×C₄₁	1	3	yes	—
106	C₄	C₂×C₃×C₄×C₅×C₄₁	1	3	yes	—
107	C₄	C₂³×C₃×C₅×C₄₁	1	1	yes	characteristic
108	C₃	C₂²×C₅×C₈×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
109	C₂	C₂×C₃×C₅×C₈×C₄₁	1	6	yes	direct product factor
110	C₂	C₂²×C₃×C₄×C₅×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₃×C₅×C₄₁
socle, prime radical	C₂³×C₃×C₅×C₄₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₈×C₄₁	C₂	direct (N × H)
3. C₂²×C₅×C₈×C₄₁	C₃	direct (N × H)
7. C₃×C₅×C₈×C₄₁	C₂²	direct (N × H)
8. C₂²×C₃×C₈×C₄₁	C₅	direct (N × H)
10. C₂×C₅×C₈×C₄₁	C₂×C₃	direct (N × H)
11. C₂²×C₃×C₅×C₄₁	C₈	direct (N × H)
16. C₂×C₃×C₈×C₄₁	C₂×C₅	direct (N × H)
20. C₅×C₈×C₄₁	C₂²×C₃	direct (N × H)
21. C₂²×C₈×C₄₁	C₃×C₅	direct (N × H)
22. C₂×C₃×C₅×C₄₁	C₂×C₈	direct (N × H)
27. C₃×C₈×C₄₁	C₂²×C₅	direct (N × H)
28. C₂²×C₅×C₄₁	C₃×C₈	direct (N × H)
33. C₂×C₈×C₄₁	C₂×C₃×C₅	direct (N × H)
34. C₃×C₅×C₄₁	C₂²×C₈	direct (N × H)
35. C₂²×C₃×C₄₁	C₅×C₈	direct (N × H)
39. C₂²×C₃×C₅×C₈	C₄₁	direct (N × H)
40. C₂×C₅×C₄₁	C₂×C₃×C₈	direct (N × H)
45. C₈×C₄₁	C₂²×C₃×C₅	direct (N × H)
46. C₂×C₃×C₄₁	C₂×C₅×C₈	direct (N × H)
49. C₂×C₃×C₅×C₈	C₂×C₄₁	direct (N × H)
50. C₅×C₄₁	C₂²×C₃×C₈	direct (N × H)
51. C₂²×C₄₁	C₃×C₅×C₈	direct (N × H)
55. C₂²×C₅×C₈	C₃×C₄₁	direct (N × H)
56. C₃×C₄₁	C₂²×C₅×C₈	direct (N × H)
60. C₃×C₅×C₈	C₂²×C₄₁	direct (N × H)
61. C₂²×C₃×C₈	C₅×C₄₁	direct (N × H)
62. C₂×C₄₁	C₂×C₃×C₅×C₈	direct (N × H)
65. C₂×C₅×C₈	C₂×C₃×C₄₁	direct (N × H)
66. C₂²×C₃×C₅	C₈×C₄₁	direct (N × H)
71. C₂×C₃×C₈	C₂×C₅×C₄₁	direct (N × H)
72. C₄₁	C₂²×C₃×C₅×C₈	direct (N × H)
76. C₅×C₈	C₂²×C₃×C₄₁	direct (N × H)
77. C₂²×C₈	C₃×C₅×C₄₁	direct (N × H)
78. C₂×C₃×C₅	C₂×C₈×C₄₁	direct (N × H)
83. C₃×C₈	C₂²×C₅×C₄₁	direct (N × H)
84. C₂²×C₅	C₃×C₈×C₄₁	direct (N × H)
89. C₂×C₈	C₂×C₃×C₅×C₄₁	direct (N × H)
90. C₃×C₅	C₂²×C₈×C₄₁	direct (N × H)
91. C₂²×C₃	C₅×C₈×C₄₁	direct (N × H)
95. C₂×C₅	C₂×C₃×C₈×C₄₁	direct (N × H)
100. C₈	C₂²×C₃×C₅×C₄₁	direct (N × H)
101. C₂×C₃	C₂×C₅×C₈×C₄₁	direct (N × H)
103. C₅	C₂²×C₃×C₈×C₄₁	direct (N × H)
104. C₂²	C₃×C₅×C₈×C₄₁	direct (N × H)
108. C₃	C₂²×C₅×C₈×C₄₁	direct (N × H)
109. C₂	C₂×C₃×C₅×C₈×C₄₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₅×C₄₁	2
C₂×C₃×C₅×C₈×C₄₁	2
C₂²×C₅×C₈×C₄₁	3
C₂²×C₃×C₈×C₄₁	5
C₂²×C₃×C₅×C₈	41

Subgroups — C22×C3×C5×C8×C41 #72197331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×C₈×C₄₁ #72197331