Subgroups — C₄×C₅×C₈²×C₁₉ #72201617

Order 24320.

1500 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₈×C₁₉	C₂	1	3	yes	maximal
2	C₂×C₅×C₈²×C₁₉	C₂	1	4	yes	maximal
3	C₄³×C₅×C₁₉	C₂²	1	1	yes	characteristic
4	C₂×C₄×C₅×C₈×C₁₉	C₄	1	12	yes	—
5	C₂×C₄×C₅×C₈×C₁₉	C₂²	1	6	yes	—
6	C₅×C₈²×C₁₉	C₄	1	16	yes	direct product factor
7	C₂×C₄²×C₅×C₁₉	C₂×C₄	1	6	yes	—
8	C₂×C₄²×C₅×C₁₉	C₂³	1	1	yes	characteristic
9	C₂²×C₅×C₈×C₁₉	C₂×C₄	1	12	yes	—
10	C₄×C₅×C₈×C₁₉	C₈	1	48	yes	direct product factor
11	C₄×C₅×C₈×C₁₉	C₂×C₄	1	24	yes	—
12	C₄×C₈²×C₁₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₄²×C₈×C₁₉	C₂×C₅	1	3	yes	—
14	C₂×C₈²×C₁₉	C₂×C₅	1	4	yes	—
15	C₂²×C₄×C₅×C₁₉	C₄²	1	4	yes	—
16	C₂²×C₄×C₅×C₁₉	C₂²×C₄	1	3	yes	—
17	C₂×C₅×C₈×C₁₉	C₄²	1	24	yes	—
18	C₂×C₅×C₈×C₁₉	C₂×C₈	1	48	yes	—
19	C₄²×C₅×C₁₉	C₂×C₈	1	24	yes	—
20	C₄²×C₅×C₁₉	C₂²×C₄	1	4	yes	—
21	C₄×C₅×C₈²	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₄×C₈×C₁₉	C₄×C₅	1	12	yes	—
23	C₂×C₄×C₈×C₁₉	C₂²×C₅	1	6	yes	—
24	C₄³×C₁₉	C₂²×C₅	1	1	yes	characteristic
25	C₈²×C₁₉	C₄×C₅	1	16	yes	direct product factor
26	C₂³×C₅×C₁₉	C₂×C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅×C₁₉	C₄×C₈	1	24	yes	—
28	C₂×C₄×C₅×C₁₉	C₂×C₄²	1	6	yes	—
29	C₂×C₄×C₅×C₁₉	C₂²×C₈	1	12	yes	—
30	C₅×C₈×C₁₉	C₄×C₈	1	48	yes	direct product factor
31	C₄²×C₅×C₈	C₂×C₁₉	1	3	yes	—
32	C₂×C₅×C₈²	C₂×C₁₉	1	4	yes	—
33	C₂²×C₈×C₁₉	C₂×C₄×C₅	1	12	yes	—
34	C₂×C₄²×C₁₉	C₂×C₄×C₅	1	6	yes	—
35	C₂×C₄²×C₁₉	C₂³×C₅	1	1	yes	characteristic
36	C₄×C₈×C₁₉	C₅×C₈	1	48	yes	direct product factor
37	C₄×C₈×C₁₉	C₂×C₄×C₅	1	24	yes	—
38	C₂²×C₅×C₁₉	C₄³	1	1	yes	characteristic
39	C₂²×C₅×C₁₉	C₂×C₄×C₈	1	6	yes	—
40	C₄×C₅×C₁₉	C₈²	1	16	yes	direct product factor
41	C₄×C₅×C₁₉	C₂×C₄×C₈	1	12	yes	—
42	C₂×C₄×C₅×C₈	C₄×C₁₉	1	12	yes	—
43	C₂×C₄×C₅×C₈	C₂²×C₁₉	1	6	yes	—
44	C₄³×C₅	C₂²×C₁₉	1	1	yes	characteristic
45	C₅×C₈²	C₄×C₁₉	1	16	yes	direct product factor
46	C₂²×C₄×C₁₉	C₄²×C₅	1	4	yes	—
47	C₂²×C₄×C₁₉	C₂²×C₄×C₅	1	3	yes	—
48	C₂×C₈×C₁₉	C₄²×C₅	1	24	yes	—
49	C₂×C₈×C₁₉	C₂×C₅×C₈	1	48	yes	—
50	C₄²×C₁₉	C₂×C₅×C₈	1	24	yes	—
51	C₄²×C₁₉	C₂²×C₄×C₅	1	4	yes	—
52	C₄×C₈²	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₅×C₁₉	C₂×C₈²	1	4	yes	—
54	C₂×C₅×C₁₉	C₄²×C₈	1	3	yes	—
55	C₂²×C₅×C₈	C₂×C₄×C₁₉	1	12	yes	—
56	C₂×C₄²×C₅	C₂×C₄×C₁₉	1	6	yes	—
57	C₂×C₄²×C₅	C₂³×C₁₉	1	1	yes	characteristic
58	C₄×C₅×C₈	C₈×C₁₉	1	48	yes	direct product factor
59	C₄×C₅×C₈	C₂×C₄×C₁₉	1	24	yes	—
60	C₂³×C₁₉	C₂×C₄²×C₅	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₄×C₅×C₈	1	24	yes	—
62	C₂×C₄×C₁₉	C₂×C₄²×C₅	1	6	yes	—
63	C₂×C₄×C₁₉	C₂²×C₅×C₈	1	12	yes	—
64	C₈×C₁₉	C₄×C₅×C₈	1	48	yes	direct product factor
65	C₄²×C₈	C₂×C₅×C₁₉	1	3	yes	—
66	C₂×C₈²	C₂×C₅×C₁₉	1	4	yes	—
67	C₅×C₁₉	C₄×C₈²	1	1	yes	characteristic, direct product factor, Hall
68	C₂²×C₄×C₅	C₄²×C₁₉	1	4	yes	—
69	C₂²×C₄×C₅	C₂²×C₄×C₁₉	1	3	yes	—
70	C₂×C₅×C₈	C₄²×C₁₉	1	24	yes	—
71	C₂×C₅×C₈	C₂×C₈×C₁₉	1	48	yes	—
72	C₄²×C₅	C₂×C₈×C₁₉	1	24	yes	—
73	C₄²×C₅	C₂²×C₄×C₁₉	1	4	yes	—
74	C₂²×C₁₉	C₄³×C₅	1	1	yes	characteristic
75	C₂²×C₁₉	C₂×C₄×C₅×C₈	1	6	yes	—
76	C₄×C₁₉	C₅×C₈²	1	16	yes	direct product factor
77	C₄×C₁₉	C₂×C₄×C₅×C₈	1	12	yes	—
78	C₂×C₄×C₈	C₄×C₅×C₁₉	1	12	yes	—
79	C₂×C₄×C₈	C₂²×C₅×C₁₉	1	6	yes	—
80	C₄³	C₂²×C₅×C₁₉	1	1	yes	characteristic
81	C₈²	C₄×C₅×C₁₉	1	16	yes	direct product factor
82	C₂³×C₅	C₂×C₄²×C₁₉	1	1	yes	characteristic
83	C₂×C₄×C₅	C₄×C₈×C₁₉	1	24	yes	—
84	C₂×C₄×C₅	C₂×C₄²×C₁₉	1	6	yes	—
85	C₂×C₄×C₅	C₂²×C₈×C₁₉	1	12	yes	—
86	C₅×C₈	C₄×C₈×C₁₉	1	48	yes	direct product factor
87	C₂×C₁₉	C₂×C₅×C₈²	1	4	yes	—
88	C₂×C₁₉	C₄²×C₅×C₈	1	3	yes	—
89	C₂²×C₈	C₂×C₄×C₅×C₁₉	1	12	yes	—
90	C₂×C₄²	C₂×C₄×C₅×C₁₉	1	6	yes	—
91	C₂×C₄²	C₂³×C₅×C₁₉	1	1	yes	characteristic
92	C₄×C₈	C₅×C₈×C₁₉	1	48	yes	direct product factor
93	C₄×C₈	C₂×C₄×C₅×C₁₉	1	24	yes	—
94	C₂²×C₅	C₄³×C₁₉	1	1	yes	characteristic
95	C₂²×C₅	C₂×C₄×C₈×C₁₉	1	6	yes	—
96	C₄×C₅	C₈²×C₁₉	1	16	yes	direct product factor
97	C₄×C₅	C₂×C₄×C₈×C₁₉	1	12	yes	—
98	C₁₉	C₄×C₅×C₈²	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×C₄	C₄²×C₅×C₁₉	1	4	yes	—
100	C₂²×C₄	C₂²×C₄×C₅×C₁₉	1	3	yes	—
101	C₂×C₈	C₄²×C₅×C₁₉	1	24	yes	—
102	C₂×C₈	C₂×C₅×C₈×C₁₉	1	48	yes	—
103	C₄²	C₂×C₅×C₈×C₁₉	1	24	yes	—
104	C₄²	C₂²×C₄×C₅×C₁₉	1	4	yes	—
105	C₂×C₅	C₂×C₈²×C₁₉	1	4	yes	—
106	C₂×C₅	C₄²×C₈×C₁₉	1	3	yes	—
107	C₂³	C₂×C₄²×C₅×C₁₉	1	1	yes	characteristic
108	C₂×C₄	C₄×C₅×C₈×C₁₉	1	24	yes	—
109	C₂×C₄	C₂²×C₅×C₈×C₁₉	1	12	yes	—
110	C₂×C₄	C₂×C₄²×C₅×C₁₉	1	6	yes	—
111	C₈	C₄×C₅×C₈×C₁₉	1	48	yes	direct product factor
112	C₅	C₄×C₈²×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²	C₂×C₄×C₅×C₈×C₁₉	1	6	yes	—
114	C₂²	C₄³×C₅×C₁₉	1	1	yes	characteristic
115	C₄	C₅×C₈²×C₁₉	1	16	yes	direct product factor
116	C₄	C₂×C₄×C₅×C₈×C₁₉	1	12	yes	—
117	C₂	C₂×C₅×C₈²×C₁₉	1	4	yes	—
118	C₂	C₄²×C₅×C₈×C₁₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄²	C₂³×C₅×C₁₉
socle, prime radical	C₂³×C₅×C₁₉	C₂×C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×C₈²×C₁₉	C₄	direct (N × H)
10. C₄×C₅×C₈×C₁₉	C₈	direct (N × H)
12. C₄×C₈²×C₁₉	C₅	direct (N × H)
21. C₄×C₅×C₈²	C₁₉	direct (N × H)
25. C₈²×C₁₉	C₄×C₅	direct (N × H)
30. C₅×C₈×C₁₉	C₄×C₈	direct (N × H)
36. C₄×C₈×C₁₉	C₅×C₈	direct (N × H)
40. C₄×C₅×C₁₉	C₈²	direct (N × H)
45. C₅×C₈²	C₄×C₁₉	direct (N × H)
52. C₄×C₈²	C₅×C₁₉	direct (N × H)
58. C₄×C₅×C₈	C₈×C₁₉	direct (N × H)
64. C₈×C₁₉	C₄×C₅×C₈	direct (N × H)
67. C₅×C₁₉	C₄×C₈²	direct (N × H)
76. C₄×C₁₉	C₅×C₈²	direct (N × H)
81. C₈²	C₄×C₅×C₁₉	direct (N × H)
86. C₅×C₈	C₄×C₈×C₁₉	direct (N × H)
92. C₄×C₈	C₅×C₈×C₁₉	direct (N × H)
96. C₄×C₅	C₈²×C₁₉	direct (N × H)
98. C₁₉	C₄×C₅×C₈²	direct (N × H)
111. C₈	C₄×C₅×C₈×C₁₉	direct (N × H)
112. C₅	C₄×C₈²×C₁₉	direct (N × H)
115. C₄	C₅×C₈²×C₁₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₈×C₁₉	2
C₂×C₅×C₈²×C₁₉	2
C₄×C₈²×C₁₉	5
C₄×C₅×C₈²	19

Subgroups — C4×C5×C82×C19 #72201617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₈²×C₁₉ #72201617