Subgroups — C₂²×C₃×C₇×C₈×C₄₃ #72205877

Order 28896.

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×C₇×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₈×C₄₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇×C₈×C₄₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₇×C₄₃	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄×C₇×C₄₃	C₄	1	3	yes	—
6	C₂×C₃×C₄×C₇×C₄₃	C₂²	1	3	yes	—
7	C₃×C₇×C₈×C₄₃	C₂²	1	4	yes	direct product factor
8	C₂²×C₄×C₇×C₄₃	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇×C₈×C₄₃	C₂×C₃	1	6	yes	direct product factor
10	C₂²×C₃×C₈×C₄₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₇×C₄₃	C₃×C₄	1	1	yes	characteristic
12	C₂×C₄×C₇×C₄₃	C₃×C₄	1	3	yes	—
13	C₂×C₄×C₇×C₄₃	C₂²×C₃	1	3	yes	—
14	C₇×C₈×C₄₃	C₂²×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₄×C₄₃	C₂×C₇	1	1	yes	characteristic
16	C₂×C₃×C₈×C₄₃	C₂×C₇	1	6	yes	direct product factor
17	C₂²×C₃×C₇×C₄₃	C₈	1	4	yes	direct product factor
18	C₂²×C₃×C₇×C₄₃	C₂×C₄	1	3	yes	—
19	C₃×C₄×C₇×C₄₃	C₂×C₄	1	3	yes	—
20	C₃×C₄×C₇×C₄₃	C₂³	1	1	yes	characteristic
21	C₂×C₃×C₇×C₄₃	C₂×C₈	1	6	yes	direct product factor
22	C₂×C₃×C₇×C₄₃	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₃	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
24	C₂²×C₇×C₄₃	C₃×C₈	1	4	yes	direct product factor
25	C₂²×C₇×C₄₃	C₂×C₃×C₄	1	3	yes	—
26	C₄×C₇×C₄₃	C₂×C₃×C₄	1	3	yes	—
27	C₄×C₇×C₄₃	C₂³×C₃	1	1	yes	characteristic
28	C₂³×C₃×C₄₃	C₄×C₇	1	1	yes	characteristic
29	C₂×C₃×C₄×C₄₃	C₄×C₇	1	3	yes	—
30	C₂×C₃×C₄×C₄₃	C₂²×C₇	1	3	yes	—
31	C₃×C₈×C₄₃	C₂²×C₇	1	4	yes	direct product factor
32	C₃×C₇×C₄₃	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
33	C₂²×C₄×C₄₃	C₂×C₃×C₇	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂×C₃×C₇	1	6	yes	direct product factor
35	C₂²×C₃×C₇×C₈	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₂×C₇×C₄₃	C₂×C₃×C₈	1	6	yes	direct product factor
37	C₂×C₇×C₄₃	C₂²×C₃×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄₃	C₇×C₈	1	4	yes	direct product factor
39	C₂²×C₃×C₄₃	C₂×C₄×C₇	1	3	yes	—
40	C₃×C₄×C₄₃	C₂×C₄×C₇	1	3	yes	—
41	C₃×C₄×C₄₃	C₂³×C₇	1	1	yes	characteristic
42	C₂³×C₄₃	C₃×C₄×C₇	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₃×C₄×C₇	1	3	yes	—
44	C₂×C₄×C₄₃	C₂²×C₃×C₇	1	3	yes	—
45	C₈×C₄₃	C₂²×C₃×C₇	1	4	yes	direct product factor
46	C₂²×C₃×C₄×C₇	C₂×C₄₃	1	1	yes	characteristic
47	C₂×C₃×C₇×C₈	C₂×C₄₃	1	6	yes	direct product factor
48	C₇×C₄₃	C₂²×C₃×C₈	1	1	yes	characteristic, direct product factor, Hall
49	C₂×C₃×C₄₃	C₂×C₇×C₈	1	6	yes	direct product factor
50	C₂×C₃×C₄₃	C₂²×C₄×C₇	1	1	yes	characteristic
51	C₂²×C₇×C₈	C₃×C₄₃	1	1	yes	characteristic, direct product factor, Hall
52	C₂²×C₄₃	C₃×C₇×C₈	1	4	yes	direct product factor
53	C₂²×C₄₃	C₂×C₃×C₄×C₇	1	3	yes	—
54	C₄×C₄₃	C₂×C₃×C₄×C₇	1	3	yes	—
55	C₄×C₄₃	C₂³×C₃×C₇	1	1	yes	characteristic
56	C₂³×C₃×C₇	C₄×C₄₃	1	1	yes	characteristic
57	C₂×C₃×C₄×C₇	C₄×C₄₃	1	3	yes	—
58	C₂×C₃×C₄×C₇	C₂²×C₄₃	1	3	yes	—
59	C₃×C₇×C₈	C₂²×C₄₃	1	4	yes	direct product factor
60	C₃×C₄₃	C₂²×C₇×C₈	1	1	yes	characteristic, direct product factor, Hall
61	C₂²×C₄×C₇	C₂×C₃×C₄₃	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×C₃×C₄₃	1	6	yes	direct product factor
63	C₂²×C₃×C₈	C₇×C₄₃	1	1	yes	characteristic, direct product factor, Hall
64	C₂×C₄₃	C₂×C₃×C₇×C₈	1	6	yes	direct product factor
65	C₂×C₄₃	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
66	C₂²×C₃×C₇	C₈×C₄₃	1	4	yes	direct product factor
67	C₂²×C₃×C₇	C₂×C₄×C₄₃	1	3	yes	—
68	C₃×C₄×C₇	C₂×C₄×C₄₃	1	3	yes	—
69	C₃×C₄×C₇	C₂³×C₄₃	1	1	yes	characteristic
70	C₂³×C₇	C₃×C₄×C₄₃	1	1	yes	characteristic
71	C₂×C₄×C₇	C₃×C₄×C₄₃	1	3	yes	—
72	C₂×C₄×C₇	C₂²×C₃×C₄₃	1	3	yes	—
73	C₇×C₈	C₂²×C₃×C₄₃	1	4	yes	direct product factor
74	C₂²×C₃×C₄	C₂×C₇×C₄₃	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×C₇×C₄₃	1	6	yes	direct product factor
76	C₄₃	C₂²×C₃×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₃×C₇	C₂×C₈×C₄₃	1	6	yes	direct product factor
78	C₂×C₃×C₇	C₂²×C₄×C₄₃	1	1	yes	characteristic
79	C₂²×C₈	C₃×C₇×C₄₃	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₇	C₃×C₈×C₄₃	1	4	yes	direct product factor
81	C₂²×C₇	C₂×C₃×C₄×C₄₃	1	3	yes	—
82	C₄×C₇	C₂×C₃×C₄×C₄₃	1	3	yes	—
83	C₄×C₇	C₂³×C₃×C₄₃	1	1	yes	characteristic
84	C₂³×C₃	C₄×C₇×C₄₃	1	1	yes	characteristic
85	C₂×C₃×C₄	C₄×C₇×C₄₃	1	3	yes	—
86	C₂×C₃×C₄	C₂²×C₇×C₄₃	1	3	yes	—
87	C₃×C₈	C₂²×C₇×C₄₃	1	4	yes	direct product factor
88	C₃×C₇	C₂²×C₈×C₄₃	1	1	yes	characteristic, direct product factor, Hall
89	C₂²×C₄	C₂×C₃×C₇×C₄₃	1	1	yes	characteristic
90	C₂×C₈	C₂×C₃×C₇×C₄₃	1	6	yes	direct product factor
91	C₂×C₇	C₂×C₃×C₈×C₄₃	1	6	yes	direct product factor
92	C₂×C₇	C₂²×C₃×C₄×C₄₃	1	1	yes	characteristic
93	C₂²×C₃	C₇×C₈×C₄₃	1	4	yes	direct product factor
94	C₂²×C₃	C₂×C₄×C₇×C₄₃	1	3	yes	—
95	C₃×C₄	C₂×C₄×C₇×C₄₃	1	3	yes	—
96	C₃×C₄	C₂³×C₇×C₄₃	1	1	yes	characteristic
97	C₂³	C₃×C₄×C₇×C₄₃	1	1	yes	characteristic
98	C₂×C₄	C₃×C₄×C₇×C₄₃	1	3	yes	—
99	C₂×C₄	C₂²×C₃×C₇×C₄₃	1	3	yes	—
100	C₈	C₂²×C₃×C₇×C₄₃	1	4	yes	direct product factor
101	C₇	C₂²×C₃×C₈×C₄₃	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×C₃	C₂×C₇×C₈×C₄₃	1	6	yes	direct product factor
103	C₂×C₃	C₂²×C₄×C₇×C₄₃	1	1	yes	characteristic
104	C₂²	C₃×C₇×C₈×C₄₃	1	4	yes	direct product factor
105	C₂²	C₂×C₃×C₄×C₇×C₄₃	1	3	yes	—
106	C₄	C₂×C₃×C₄×C₇×C₄₃	1	3	yes	—
107	C₄	C₂³×C₃×C₇×C₄₃	1	1	yes	characteristic
108	C₃	C₂²×C₇×C₈×C₄₃	1	1	yes	characteristic, direct product factor, Sylow
109	C₂	C₂×C₃×C₇×C₈×C₄₃	1	6	yes	direct product factor
110	C₂	C₂²×C₃×C₄×C₇×C₄₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₃×C₇×C₄₃
socle, prime radical	C₂³×C₃×C₇×C₄₃	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×C₇×C₈×C₄₃	C₂	direct (N × H)
3. C₂²×C₇×C₈×C₄₃	C₃	direct (N × H)
7. C₃×C₇×C₈×C₄₃	C₂²	direct (N × H)
9. C₂×C₇×C₈×C₄₃	C₂×C₃	direct (N × H)
10. C₂²×C₃×C₈×C₄₃	C₇	direct (N × H)
14. C₇×C₈×C₄₃	C₂²×C₃	direct (N × H)
16. C₂×C₃×C₈×C₄₃	C₂×C₇	direct (N × H)
17. C₂²×C₃×C₇×C₄₃	C₈	direct (N × H)
21. C₂×C₃×C₇×C₄₃	C₂×C₈	direct (N × H)
23. C₂²×C₈×C₄₃	C₃×C₇	direct (N × H)
24. C₂²×C₇×C₄₃	C₃×C₈	direct (N × H)
31. C₃×C₈×C₄₃	C₂²×C₇	direct (N × H)
32. C₃×C₇×C₄₃	C₂²×C₈	direct (N × H)
34. C₂×C₈×C₄₃	C₂×C₃×C₇	direct (N × H)
35. C₂²×C₃×C₇×C₈	C₄₃	direct (N × H)
36. C₂×C₇×C₄₃	C₂×C₃×C₈	direct (N × H)
38. C₂²×C₃×C₄₃	C₇×C₈	direct (N × H)
45. C₈×C₄₃	C₂²×C₃×C₇	direct (N × H)
47. C₂×C₃×C₇×C₈	C₂×C₄₃	direct (N × H)
48. C₇×C₄₃	C₂²×C₃×C₈	direct (N × H)
49. C₂×C₃×C₄₃	C₂×C₇×C₈	direct (N × H)
51. C₂²×C₇×C₈	C₃×C₄₃	direct (N × H)
52. C₂²×C₄₃	C₃×C₇×C₈	direct (N × H)
59. C₃×C₇×C₈	C₂²×C₄₃	direct (N × H)
60. C₃×C₄₃	C₂²×C₇×C₈	direct (N × H)
62. C₂×C₇×C₈	C₂×C₃×C₄₃	direct (N × H)
63. C₂²×C₃×C₈	C₇×C₄₃	direct (N × H)
64. C₂×C₄₃	C₂×C₃×C₇×C₈	direct (N × H)
66. C₂²×C₃×C₇	C₈×C₄₃	direct (N × H)
73. C₇×C₈	C₂²×C₃×C₄₃	direct (N × H)
75. C₂×C₃×C₈	C₂×C₇×C₄₃	direct (N × H)
76. C₄₃	C₂²×C₃×C₇×C₈	direct (N × H)
77. C₂×C₃×C₇	C₂×C₈×C₄₃	direct (N × H)
79. C₂²×C₈	C₃×C₇×C₄₃	direct (N × H)
80. C₂²×C₇	C₃×C₈×C₄₃	direct (N × H)
87. C₃×C₈	C₂²×C₇×C₄₃	direct (N × H)
88. C₃×C₇	C₂²×C₈×C₄₃	direct (N × H)
90. C₂×C₈	C₂×C₃×C₇×C₄₃	direct (N × H)
91. C₂×C₇	C₂×C₃×C₈×C₄₃	direct (N × H)
93. C₂²×C₃	C₇×C₈×C₄₃	direct (N × H)
100. C₈	C₂²×C₃×C₇×C₄₃	direct (N × H)
101. C₇	C₂²×C₃×C₈×C₄₃	direct (N × H)
102. C₂×C₃	C₂×C₇×C₈×C₄₃	direct (N × H)
104. C₂²	C₃×C₇×C₈×C₄₃	direct (N × H)
108. C₃	C₂²×C₇×C₈×C₄₃	direct (N × H)
109. C₂	C₂×C₃×C₇×C₈×C₄₃	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×C₇×C₄₃	2
C₂×C₃×C₇×C₈×C₄₃	2
C₂²×C₇×C₈×C₄₃	3
C₂²×C₃×C₈×C₄₃	7
C₂²×C₃×C₇×C₈	43

Subgroups — C22×C3×C7×C8×C43 #72205877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₇×C₈×C₄₃ #72205877