Subgroups — C₄²×C₅×C₁₆×C₁₉ #72201614

Order 24320.

1116 conjugacy classes of subgroups.

The subgroup lattice has 156 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	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₆×C₁₉	C₂	1	6	yes	maximal
3	C₄³×C₅×C₁₉	C₄	1	1	yes	characteristic
4	C₂×C₄×C₅×C₈×C₁₉	C₄	1	3	yes	—
5	C₂×C₄×C₅×C₈×C₁₉	C₂²	1	3	yes	—
6	C₂²×C₅×C₁₆×C₁₉	C₂²	1	4	yes	—
7	C₄×C₅×C₁₆×C₁₉	C₄	1	24	yes	direct product factor
8	C₂×C₄²×C₅×C₁₉	C₈	1	4	yes	—
9	C₂×C₄²×C₅×C₁₉	C₂×C₄	1	3	yes	—
10	C₂²×C₅×C₈×C₁₉	C₂×C₄	1	3	yes	—
11	C₂²×C₅×C₈×C₁₉	C₂³	1	1	yes	characteristic
12	C₄×C₅×C₈×C₁₉	C₈	1	12	yes	—
13	C₄×C₅×C₈×C₁₉	C₂×C₄	1	12	yes	—
14	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	24	yes	—
15	C₄²×C₁₆×C₁₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₄×C₁₆×C₁₉	C₂×C₅	1	6	yes	—
17	C₄²×C₈×C₁₉	C₂×C₅	1	1	yes	characteristic
18	C₂²×C₄×C₅×C₁₉	C₂×C₈	1	6	yes	—
19	C₂²×C₄×C₅×C₁₉	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₅×C₈×C₁₉	C₄²	1	6	yes	—
21	C₂×C₅×C₈×C₁₉	C₂×C₈	1	12	yes	—
22	C₂×C₅×C₈×C₁₉	C₂²×C₄	1	6	yes	—
23	C₄²×C₅×C₁₉	C₁₆	1	16	yes	direct product factor
24	C₄²×C₅×C₁₉	C₄²	1	6	yes	—
25	C₄²×C₅×C₁₉	C₂×C₈	1	6	yes	—
26	C₅×C₁₆×C₁₉	C₄²	1	16	yes	direct product factor
27	C₄²×C₅×C₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂²×C₁₆×C₁₉	C₂²×C₅	1	4	yes	—
29	C₂×C₄×C₈×C₁₉	C₄×C₅	1	3	yes	—
30	C₂×C₄×C₈×C₁₉	C₂²×C₅	1	3	yes	—
31	C₄³×C₁₉	C₄×C₅	1	1	yes	characteristic
32	C₄×C₁₆×C₁₉	C₄×C₅	1	24	yes	direct product factor
33	C₂³×C₅×C₁₉	C₂²×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅×C₁₉	C₄×C₈	1	12	yes	—
35	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	24	yes	—
36	C₂×C₄×C₅×C₁₉	C₂×C₄²	1	3	yes	—
37	C₂×C₄×C₅×C₁₉	C₂²×C₈	1	3	yes	—
38	C₅×C₈×C₁₉	C₄×C₈	1	12	yes	—
39	C₅×C₈×C₁₉	C₂×C₄²	1	4	yes	—
40	C₂×C₄×C₅×C₁₆	C₂×C₁₉	1	6	yes	—
41	C₄²×C₅×C₈	C₂×C₁₉	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	C₂×C₄×C₅	1	3	yes	—
43	C₂²×C₈×C₁₉	C₂³×C₅	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₅×C₈	1	4	yes	—
45	C₂×C₄²×C₁₉	C₂×C₄×C₅	1	3	yes	—
46	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	24	yes	—
47	C₄×C₈×C₁₉	C₅×C₈	1	12	yes	—
48	C₄×C₈×C₁₉	C₂×C₄×C₅	1	12	yes	—
49	C₂²×C₅×C₁₉	C₂×C₄×C₈	1	3	yes	—
50	C₂²×C₅×C₁₉	C₂²×C₁₆	1	4	yes	—
51	C₄×C₅×C₁₉	C₄×C₁₆	1	24	yes	direct product factor
52	C₄×C₅×C₁₉	C₄³	1	1	yes	characteristic
53	C₄×C₅×C₁₉	C₂×C₄×C₈	1	3	yes	—
54	C₂²×C₅×C₁₆	C₂²×C₁₉	1	4	yes	—
55	C₂×C₄×C₅×C₈	C₄×C₁₉	1	3	yes	—
56	C₂×C₄×C₅×C₈	C₂²×C₁₉	1	3	yes	—
57	C₄³×C₅	C₄×C₁₉	1	1	yes	characteristic
58	C₄×C₅×C₁₆	C₄×C₁₉	1	24	yes	direct product factor
59	C₂²×C₄×C₁₉	C₂×C₅×C₈	1	6	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄×C₅	1	1	yes	characteristic
61	C₂×C₈×C₁₉	C₄²×C₅	1	6	yes	—
62	C₂×C₈×C₁₉	C₂×C₅×C₈	1	12	yes	—
63	C₂×C₈×C₁₉	C₂²×C₄×C₅	1	6	yes	—
64	C₄²×C₁₉	C₅×C₁₆	1	16	yes	direct product factor
65	C₄²×C₁₉	C₄²×C₅	1	6	yes	—
66	C₄²×C₁₉	C₂×C₅×C₈	1	6	yes	—
67	C₁₆×C₁₉	C₄²×C₅	1	16	yes	direct product factor
68	C₄²×C₁₆	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₅×C₁₉	C₄²×C₈	1	1	yes	characteristic
70	C₂×C₅×C₁₉	C₂×C₄×C₁₆	1	6	yes	—
71	C₂²×C₅×C₈	C₂×C₄×C₁₉	1	3	yes	—
72	C₂²×C₅×C₈	C₂³×C₁₉	1	1	yes	characteristic
73	C₂×C₄²×C₅	C₈×C₁₉	1	4	yes	—
74	C₂×C₄²×C₅	C₂×C₄×C₁₉	1	3	yes	—
75	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	24	yes	—
76	C₄×C₅×C₈	C₈×C₁₉	1	12	yes	—
77	C₄×C₅×C₈	C₂×C₄×C₁₉	1	12	yes	—
78	C₂³×C₁₉	C₂²×C₅×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₄×C₅×C₈	1	12	yes	—
80	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	24	yes	—
81	C₂×C₄×C₁₉	C₂×C₄²×C₅	1	3	yes	—
82	C₂×C₄×C₁₉	C₂²×C₅×C₈	1	3	yes	—
83	C₈×C₁₉	C₄×C₅×C₈	1	12	yes	—
84	C₈×C₁₉	C₂×C₄²×C₅	1	4	yes	—
85	C₂×C₄×C₁₆	C₂×C₅×C₁₉	1	6	yes	—
86	C₄²×C₈	C₂×C₅×C₁₉	1	1	yes	characteristic
87	C₅×C₁₉	C₄²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
88	C₂²×C₄×C₅	C₂×C₈×C₁₉	1	6	yes	—
89	C₂²×C₄×C₅	C₂²×C₄×C₁₉	1	1	yes	characteristic
90	C₂×C₅×C₈	C₄²×C₁₉	1	6	yes	—
91	C₂×C₅×C₈	C₂×C₈×C₁₉	1	12	yes	—
92	C₂×C₅×C₈	C₂²×C₄×C₁₉	1	6	yes	—
93	C₄²×C₅	C₁₆×C₁₉	1	16	yes	direct product factor
94	C₄²×C₅	C₄²×C₁₉	1	6	yes	—
95	C₄²×C₅	C₂×C₈×C₁₉	1	6	yes	—
96	C₅×C₁₆	C₄²×C₁₉	1	16	yes	direct product factor
97	C₂²×C₁₉	C₂×C₄×C₅×C₈	1	3	yes	—
98	C₂²×C₁₉	C₂²×C₅×C₁₆	1	4	yes	—
99	C₄×C₁₉	C₄×C₅×C₁₆	1	24	yes	direct product factor
100	C₄×C₁₉	C₄³×C₅	1	1	yes	characteristic
101	C₄×C₁₉	C₂×C₄×C₅×C₈	1	3	yes	—
102	C₂²×C₁₆	C₂²×C₅×C₁₉	1	4	yes	—
103	C₂×C₄×C₈	C₄×C₅×C₁₉	1	3	yes	—
104	C₂×C₄×C₈	C₂²×C₅×C₁₉	1	3	yes	—
105	C₄³	C₄×C₅×C₁₉	1	1	yes	characteristic
106	C₄×C₁₆	C₄×C₅×C₁₉	1	24	yes	direct product factor
107	C₂³×C₅	C₂²×C₈×C₁₉	1	1	yes	characteristic
108	C₂×C₄×C₅	C₄×C₈×C₁₉	1	12	yes	—
109	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	24	yes	—
110	C₂×C₄×C₅	C₂×C₄²×C₁₉	1	3	yes	—
111	C₂×C₄×C₅	C₂²×C₈×C₁₉	1	3	yes	—
112	C₅×C₈	C₄×C₈×C₁₉	1	12	yes	—
113	C₅×C₈	C₂×C₄²×C₁₉	1	4	yes	—
114	C₂×C₁₉	C₄²×C₅×C₈	1	1	yes	characteristic
115	C₂×C₁₉	C₂×C₄×C₅×C₁₆	1	6	yes	—
116	C₂²×C₈	C₂×C₄×C₅×C₁₉	1	3	yes	—
117	C₂²×C₈	C₂³×C₅×C₁₉	1	1	yes	characteristic
118	C₂×C₄²	C₅×C₈×C₁₉	1	4	yes	—
119	C₂×C₄²	C₂×C₄×C₅×C₁₉	1	3	yes	—
120	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	24	yes	—
121	C₄×C₈	C₅×C₈×C₁₉	1	12	yes	—
122	C₄×C₈	C₂×C₄×C₅×C₁₉	1	12	yes	—
123	C₂²×C₅	C₂×C₄×C₈×C₁₉	1	3	yes	—
124	C₂²×C₅	C₂²×C₁₆×C₁₉	1	4	yes	—
125	C₄×C₅	C₄×C₁₆×C₁₉	1	24	yes	direct product factor
126	C₄×C₅	C₄³×C₁₉	1	1	yes	characteristic
127	C₄×C₅	C₂×C₄×C₈×C₁₉	1	3	yes	—
128	C₁₉	C₄²×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	C₂×C₅×C₈×C₁₉	1	6	yes	—
130	C₂²×C₄	C₂²×C₄×C₅×C₁₉	1	1	yes	characteristic
131	C₂×C₈	C₄²×C₅×C₁₉	1	6	yes	—
132	C₂×C₈	C₂×C₅×C₈×C₁₉	1	12	yes	—
133	C₂×C₈	C₂²×C₄×C₅×C₁₉	1	6	yes	—
134	C₄²	C₅×C₁₆×C₁₉	1	16	yes	direct product factor
135	C₄²	C₄²×C₅×C₁₉	1	6	yes	—
136	C₄²	C₂×C₅×C₈×C₁₉	1	6	yes	—
137	C₁₆	C₄²×C₅×C₁₉	1	16	yes	direct product factor
138	C₂×C₅	C₄²×C₈×C₁₉	1	1	yes	characteristic
139	C₂×C₅	C₂×C₄×C₁₆×C₁₉	1	6	yes	—
140	C₂³	C₂²×C₅×C₈×C₁₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	24	yes	—
142	C₂×C₄	C₄×C₅×C₈×C₁₉	1	12	yes	—
143	C₂×C₄	C₂²×C₅×C₈×C₁₉	1	3	yes	—
144	C₂×C₄	C₂×C₄²×C₅×C₁₉	1	3	yes	—
145	C₈	C₄×C₅×C₈×C₁₉	1	12	yes	—
146	C₈	C₂×C₄²×C₅×C₁₉	1	4	yes	—
147	C₅	C₄²×C₁₆×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²	C₂²×C₅×C₁₆×C₁₉	1	4	yes	—
149	C₂²	C₂×C₄×C₅×C₈×C₁₉	1	3	yes	—
150	C₄	C₄×C₅×C₁₆×C₁₉	1	24	yes	direct product factor
151	C₄	C₂×C₄×C₅×C₈×C₁₉	1	3	yes	—
152	C₄	C₄³×C₅×C₁₉	1	1	yes	characteristic
153	C₂	C₂×C₄×C₅×C₁₆×C₁₉	1	6	yes	—
154	C₂	C₄²×C₅×C₈×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈	C₂³×C₅×C₁₉
socle, prime radical	C₂³×C₅×C₁₉	C₂²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₄×C₅×C₁₆×C₁₉	C₄	direct (N × H)
15. C₄²×C₁₆×C₁₉	C₅	direct (N × H)
23. C₄²×C₅×C₁₉	C₁₆	direct (N × H)
26. C₅×C₁₆×C₁₉	C₄²	direct (N × H)
27. C₄²×C₅×C₁₆	C₁₉	direct (N × H)
32. C₄×C₁₆×C₁₉	C₄×C₅	direct (N × H)
51. C₄×C₅×C₁₉	C₄×C₁₆	direct (N × H)
58. C₄×C₅×C₁₆	C₄×C₁₉	direct (N × H)
64. C₄²×C₁₉	C₅×C₁₆	direct (N × H)
67. C₁₆×C₁₉	C₄²×C₅	direct (N × H)
68. C₄²×C₁₆	C₅×C₁₉	direct (N × H)
87. C₅×C₁₉	C₄²×C₁₆	direct (N × H)
93. C₄²×C₅	C₁₆×C₁₉	direct (N × H)
96. C₅×C₁₆	C₄²×C₁₉	direct (N × H)
99. C₄×C₁₉	C₄×C₅×C₁₆	direct (N × H)
106. C₄×C₁₆	C₄×C₅×C₁₉	direct (N × H)
125. C₄×C₅	C₄×C₁₆×C₁₉	direct (N × H)
128. C₁₉	C₄²×C₅×C₁₆	direct (N × H)
134. C₄²	C₅×C₁₆×C₁₉	direct (N × H)
137. C₁₆	C₄²×C₅×C₁₉	direct (N × H)
147. C₅	C₄²×C₁₆×C₁₉	direct (N × H)
150. C₄	C₄×C₅×C₁₆×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×C₄×C₅×C₁₆×C₁₉	2
C₄²×C₁₆×C₁₉	5
C₄²×C₅×C₁₆	19

Subgroups — C42×C5×C16×C19 #72201614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅×C₁₆×C₁₉ #72201614