Subgroups — C₂²×C₅×C₁₉×C₆₄ #72201608

Order 24320.

284 conjugacy classes of subgroups.

The subgroup lattice has 116 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₁₉×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₁₉×C₆₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×C₁₆×C₁₉	C₄	1	1	yes	characteristic
4	C₂×C₅×C₁₉×C₃₂	C₄	1	3	yes	—
5	C₂×C₅×C₁₉×C₃₂	C₂²	1	3	yes	—
6	C₅×C₁₉×C₆₄	C₂²	1	4	yes	direct product factor
7	C₂²×C₅×C₈×C₁₉	C₈	1	1	yes	characteristic
8	C₂×C₅×C₁₆×C₁₉	C₈	1	3	yes	—
9	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	3	yes	—
10	C₅×C₁₉×C₃₂	C₂×C₄	1	3	yes	—
11	C₅×C₁₉×C₃₂	C₂³	1	1	yes	characteristic
12	C₂²×C₁₉×C₆₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₁₉×C₃₂	C₂×C₅	1	1	yes	characteristic
14	C₂×C₁₉×C₆₄	C₂×C₅	1	6	yes	direct product factor
15	C₂²×C₄×C₅×C₁₉	C₁₆	1	1	yes	characteristic
16	C₂×C₅×C₈×C₁₉	C₁₆	1	3	yes	—
17	C₂×C₅×C₈×C₁₉	C₂×C₈	1	3	yes	—
18	C₅×C₁₆×C₁₉	C₂×C₈	1	3	yes	—
19	C₅×C₁₆×C₁₉	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₆₄	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₁₆×C₁₉	C₄×C₅	1	1	yes	characteristic
22	C₂×C₁₉×C₃₂	C₄×C₅	1	3	yes	—
23	C₂×C₁₉×C₃₂	C₂²×C₅	1	3	yes	—
24	C₁₉×C₆₄	C₂²×C₅	1	4	yes	direct product factor
25	C₂³×C₅×C₁₉	C₃₂	1	1	yes	characteristic
26	C₂×C₄×C₅×C₁₉	C₃₂	1	3	yes	—
27	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	3	yes	—
28	C₅×C₈×C₁₉	C₂×C₁₆	1	3	yes	—
29	C₅×C₈×C₁₉	C₂²×C₈	1	1	yes	characteristic
30	C₂²×C₅×C₃₂	C₂×C₁₉	1	1	yes	characteristic
31	C₂×C₅×C₆₄	C₂×C₁₉	1	6	yes	direct product factor
32	C₂²×C₈×C₁₉	C₅×C₈	1	1	yes	characteristic
33	C₂×C₁₆×C₁₉	C₅×C₈	1	3	yes	—
34	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	3	yes	—
35	C₁₉×C₃₂	C₂×C₄×C₅	1	3	yes	—
36	C₁₉×C₃₂	C₂³×C₅	1	1	yes	characteristic
37	C₂²×C₅×C₁₉	C₆₄	1	4	yes	direct product factor
38	C₂²×C₅×C₁₉	C₂×C₃₂	1	3	yes	—
39	C₄×C₅×C₁₉	C₂×C₃₂	1	3	yes	—
40	C₄×C₅×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₅×C₁₆	C₄×C₁₉	1	1	yes	characteristic
42	C₂×C₅×C₃₂	C₄×C₁₉	1	3	yes	—
43	C₂×C₅×C₃₂	C₂²×C₁₉	1	3	yes	—
44	C₅×C₆₄	C₂²×C₁₉	1	4	yes	direct product factor
45	C₂²×C₄×C₁₉	C₅×C₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₉	C₅×C₁₆	1	3	yes	—
47	C₂×C₈×C₁₉	C₂×C₅×C₈	1	3	yes	—
48	C₁₆×C₁₉	C₂×C₅×C₈	1	3	yes	—
49	C₁₆×C₁₉	C₂²×C₄×C₅	1	1	yes	characteristic
50	C₂²×C₆₄	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₅×C₁₉	C₂×C₆₄	1	6	yes	direct product factor
52	C₂×C₅×C₁₉	C₂²×C₃₂	1	1	yes	characteristic
53	C₂²×C₅×C₈	C₈×C₁₉	1	1	yes	characteristic
54	C₂×C₅×C₁₆	C₈×C₁₉	1	3	yes	—
55	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	3	yes	—
56	C₅×C₃₂	C₂×C₄×C₁₉	1	3	yes	—
57	C₅×C₃₂	C₂³×C₁₉	1	1	yes	characteristic
58	C₂³×C₁₉	C₅×C₃₂	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₅×C₃₂	1	3	yes	—
60	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	3	yes	—
61	C₈×C₁₉	C₂×C₅×C₁₆	1	3	yes	—
62	C₈×C₁₉	C₂²×C₅×C₈	1	1	yes	characteristic
63	C₂²×C₃₂	C₂×C₅×C₁₉	1	1	yes	characteristic
64	C₂×C₆₄	C₂×C₅×C₁₉	1	6	yes	direct product factor
65	C₅×C₁₉	C₂²×C₆₄	1	1	yes	characteristic, direct product factor, Hall
66	C₂²×C₄×C₅	C₁₆×C₁₉	1	1	yes	characteristic
67	C₂×C₅×C₈	C₁₆×C₁₉	1	3	yes	—
68	C₂×C₅×C₈	C₂×C₈×C₁₉	1	3	yes	—
69	C₅×C₁₆	C₂×C₈×C₁₉	1	3	yes	—
70	C₅×C₁₆	C₂²×C₄×C₁₉	1	1	yes	characteristic
71	C₂²×C₁₉	C₅×C₆₄	1	4	yes	direct product factor
72	C₂²×C₁₉	C₂×C₅×C₃₂	1	3	yes	—
73	C₄×C₁₉	C₂×C₅×C₃₂	1	3	yes	—
74	C₄×C₁₉	C₂²×C₅×C₁₆	1	1	yes	characteristic
75	C₂²×C₁₆	C₄×C₅×C₁₉	1	1	yes	characteristic
76	C₂×C₃₂	C₄×C₅×C₁₉	1	3	yes	—
77	C₂×C₃₂	C₂²×C₅×C₁₉	1	3	yes	—
78	C₆₄	C₂²×C₅×C₁₉	1	4	yes	direct product factor
79	C₂³×C₅	C₁₉×C₃₂	1	1	yes	characteristic
80	C₂×C₄×C₅	C₁₉×C₃₂	1	3	yes	—
81	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	3	yes	—
82	C₅×C₈	C₂×C₁₆×C₁₉	1	3	yes	—
83	C₅×C₈	C₂²×C₈×C₁₉	1	1	yes	characteristic
84	C₂×C₁₉	C₂×C₅×C₆₄	1	6	yes	direct product factor
85	C₂×C₁₉	C₂²×C₅×C₃₂	1	1	yes	characteristic
86	C₂²×C₈	C₅×C₈×C₁₉	1	1	yes	characteristic
87	C₂×C₁₆	C₅×C₈×C₁₉	1	3	yes	—
88	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	3	yes	—
89	C₃₂	C₂×C₄×C₅×C₁₉	1	3	yes	—
90	C₃₂	C₂³×C₅×C₁₉	1	1	yes	characteristic
91	C₂²×C₅	C₁₉×C₆₄	1	4	yes	direct product factor
92	C₂²×C₅	C₂×C₁₉×C₃₂	1	3	yes	—
93	C₄×C₅	C₂×C₁₉×C₃₂	1	3	yes	—
94	C₄×C₅	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
95	C₁₉	C₂²×C₅×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×C₄	C₅×C₁₆×C₁₉	1	1	yes	characteristic
97	C₂×C₈	C₅×C₁₆×C₁₉	1	3	yes	—
98	C₂×C₈	C₂×C₅×C₈×C₁₉	1	3	yes	—
99	C₁₆	C₂×C₅×C₈×C₁₉	1	3	yes	—
100	C₁₆	C₂²×C₄×C₅×C₁₉	1	1	yes	characteristic
101	C₂×C₅	C₂×C₁₉×C₆₄	1	6	yes	direct product factor
102	C₂×C₅	C₂²×C₁₉×C₃₂	1	1	yes	characteristic
103	C₂³	C₅×C₁₉×C₃₂	1	1	yes	characteristic
104	C₂×C₄	C₅×C₁₉×C₃₂	1	3	yes	—
105	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	3	yes	—
106	C₈	C₂×C₅×C₁₆×C₁₉	1	3	yes	—
107	C₈	C₂²×C₅×C₈×C₁₉	1	1	yes	characteristic
108	C₅	C₂²×C₁₉×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²	C₅×C₁₉×C₆₄	1	4	yes	direct product factor
110	C₂²	C₂×C₅×C₁₉×C₃₂	1	3	yes	—
111	C₄	C₂×C₅×C₁₉×C₃₂	1	3	yes	—
112	C₄	C₂²×C₅×C₁₆×C₁₉	1	1	yes	characteristic
113	C₂	C₂×C₅×C₁₉×C₆₄	1	6	yes	direct product factor
114	C₂	C₂²×C₅×C₁₉×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂³×C₅×C₁₉
socle, prime radical	C₂³×C₅×C₁₉	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₁₉×C₆₄	C₂	direct (N × H)
6. C₅×C₁₉×C₆₄	C₂²	direct (N × H)
12. C₂²×C₁₉×C₆₄	C₅	direct (N × H)
14. C₂×C₁₉×C₆₄	C₂×C₅	direct (N × H)
20. C₂²×C₅×C₆₄	C₁₉	direct (N × H)
24. C₁₉×C₆₄	C₂²×C₅	direct (N × H)
31. C₂×C₅×C₆₄	C₂×C₁₉	direct (N × H)
37. C₂²×C₅×C₁₉	C₆₄	direct (N × H)
44. C₅×C₆₄	C₂²×C₁₉	direct (N × H)
50. C₂²×C₆₄	C₅×C₁₉	direct (N × H)
51. C₂×C₅×C₁₉	C₂×C₆₄	direct (N × H)
64. C₂×C₆₄	C₂×C₅×C₁₉	direct (N × H)
65. C₅×C₁₉	C₂²×C₆₄	direct (N × H)
71. C₂²×C₁₉	C₅×C₆₄	direct (N × H)
78. C₆₄	C₂²×C₅×C₁₉	direct (N × H)
84. C₂×C₁₉	C₂×C₅×C₆₄	direct (N × H)
91. C₂²×C₅	C₁₉×C₆₄	direct (N × H)
95. C₁₉	C₂²×C₅×C₆₄	direct (N × H)
101. C₂×C₅	C₂×C₁₉×C₆₄	direct (N × H)
108. C₅	C₂²×C₁₉×C₆₄	direct (N × H)
109. C₂²	C₅×C₁₉×C₆₄	direct (N × H)
113. C₂	C₂×C₅×C₁₉×C₆₄	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×C₅×C₁₉×C₃₂	2
C₂×C₅×C₁₉×C₆₄	2
C₂²×C₁₉×C₆₄	5
C₂²×C₅×C₆₄	19

Subgroups — C22×C5×C19×C64 #72201608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₁₉×C₆₄ #72201608