Subgroups — C₂×C₅×C₈×C₁₆×C₁₉ #72201613

Order 24320.

796 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₆×C₁₉	C₂	1	2	yes	maximal
3	C₅×C₈×C₁₆×C₁₉	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₅×C₈×C₁₉	C₄	1	2	yes	—
5	C₂×C₄×C₅×C₈×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₁₆×C₁₉	C₄	1	4	yes	—
7	C₅×C₈²×C₁₉	C₄	1	2	yes	—
8	C₅×C₈²×C₁₉	C₂²	1	2	yes	—
9	C₄×C₅×C₁₆×C₁₉	C₄	1	4	yes	—
10	C₄×C₅×C₁₆×C₁₉	C₂²	1	4	yes	—
11	C₂×C₄²×C₅×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₅×C₈×C₁₉	C₈	1	4	yes	—
13	C₂²×C₅×C₈×C₁₉	C₂×C₄	1	2	yes	—
14	C₄×C₅×C₈×C₁₉	C₈	1	4	yes	—
15	C₄×C₅×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₄×C₅×C₈×C₁₉	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₈×C₁₉	C₂×C₄	1	4	yes	—
18	C₄×C₅×C₈×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×C₅×C₁₆×C₁₉	C₈	1	16	yes	direct product factor
20	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	8	yes	—
21	C₂×C₈×C₁₆×C₁₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₄×C₁₆×C₁₉	C₂×C₅	1	2	yes	—
23	C₂×C₈²×C₁₉	C₂×C₅	1	1	yes	characteristic
24	C₈×C₁₆×C₁₉	C₂×C₅	1	4	yes	direct product factor
25	C₂²×C₄×C₅×C₁₉	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅×C₁₉	C₂×C₈	1	2	yes	—
27	C₂×C₅×C₈×C₁₉	C₁₆	1	16	yes	direct product factor
28	C₂×C₅×C₈×C₁₉	C₄²	1	2	yes	—
29	C₂×C₅×C₈×C₁₉	C₂×C₈	1	8	yes	—
30	C₂×C₅×C₈×C₁₉	C₂×C₈	1	8	yes	—
31	C₂×C₅×C₈×C₁₉	C₂²×C₄	1	2	yes	—
32	C₄²×C₅×C₁₉	C₄²	1	1	yes	characteristic
33	C₄²×C₅×C₁₉	C₂×C₈	1	2	yes	—
34	C₄²×C₅×C₁₉	C₂²×C₄	1	1	yes	characteristic
35	C₅×C₁₆×C₁₉	C₂×C₈	1	16	yes	direct product factor
36	C₂×C₅×C₈×C₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₂²×C₁₆×C₁₉	C₄×C₅	1	4	yes	—
38	C₂×C₄×C₈×C₁₉	C₄×C₅	1	2	yes	—
39	C₂×C₄×C₈×C₁₉	C₂²×C₅	1	1	yes	characteristic
40	C₄×C₁₆×C₁₉	C₄×C₅	1	4	yes	—
41	C₄×C₁₆×C₁₉	C₂²×C₅	1	4	yes	—
42	C₈²×C₁₉	C₄×C₅	1	2	yes	—
43	C₈²×C₁₉	C₂²×C₅	1	2	yes	—
44	C₂³×C₅×C₁₉	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅×C₁₉	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅×C₁₉	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₅×C₁₉	C₄×C₈	1	4	yes	—
48	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	8	yes	—
49	C₂×C₄×C₅×C₁₉	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₅×C₁₉	C₂²×C₈	1	2	yes	—
51	C₅×C₈×C₁₉	C₄×C₈	1	4	yes	—
52	C₅×C₈×C₁₉	C₂×C₁₆	1	16	yes	direct product factor
53	C₅×C₈×C₁₉	C₂²×C₈	1	4	yes	—
54	C₂×C₄×C₅×C₁₆	C₂×C₁₉	1	2	yes	—
55	C₂×C₅×C₈²	C₂×C₁₉	1	1	yes	characteristic
56	C₅×C₈×C₁₆	C₂×C₁₉	1	4	yes	direct product factor
57	C₂²×C₈×C₁₉	C₅×C₈	1	4	yes	—
58	C₂²×C₈×C₁₉	C₂×C₄×C₅	1	2	yes	—
59	C₂×C₄²×C₁₉	C₂×C₄×C₅	1	1	yes	characteristic
60	C₂×C₁₆×C₁₉	C₅×C₈	1	16	yes	direct product factor
61	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	8	yes	—
62	C₄×C₈×C₁₉	C₅×C₈	1	4	yes	—
63	C₄×C₈×C₁₉	C₂×C₄×C₅	1	1	yes	characteristic
64	C₄×C₈×C₁₉	C₂×C₄×C₅	1	2	yes	—
65	C₄×C₈×C₁₉	C₂×C₄×C₅	1	4	yes	—
66	C₄×C₈×C₁₉	C₂³×C₅	1	1	yes	characteristic
67	C₂²×C₅×C₁₉	C₈²	1	2	yes	—
68	C₂²×C₅×C₁₉	C₄×C₁₆	1	4	yes	—
69	C₂²×C₅×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₅×C₁₉	C₈²	1	2	yes	—
71	C₄×C₅×C₁₉	C₄×C₁₆	1	4	yes	—
72	C₄×C₅×C₁₉	C₂×C₄×C₈	1	2	yes	—
73	C₄×C₅×C₁₉	C₂²×C₁₆	1	4	yes	—
74	C₂²×C₅×C₁₆	C₄×C₁₉	1	4	yes	—
75	C₂×C₄×C₅×C₈	C₄×C₁₉	1	2	yes	—
76	C₂×C₄×C₅×C₈	C₂²×C₁₉	1	1	yes	characteristic
77	C₄×C₅×C₁₆	C₄×C₁₉	1	4	yes	—
78	C₄×C₅×C₁₆	C₂²×C₁₉	1	4	yes	—
79	C₅×C₈²	C₄×C₁₉	1	2	yes	—
80	C₅×C₈²	C₂²×C₁₉	1	2	yes	—
81	C₂²×C₄×C₁₉	C₄²×C₅	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×C₅×C₈	1	2	yes	—
83	C₂×C₈×C₁₉	C₅×C₁₆	1	16	yes	direct product factor
84	C₂×C₈×C₁₉	C₄²×C₅	1	2	yes	—
85	C₂×C₈×C₁₉	C₂×C₅×C₈	1	8	yes	—
86	C₂×C₈×C₁₉	C₂×C₅×C₈	1	8	yes	—
87	C₂×C₈×C₁₉	C₂²×C₄×C₅	1	2	yes	—
88	C₄²×C₁₉	C₄²×C₅	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×C₅×C₈	1	2	yes	—
90	C₄²×C₁₉	C₂²×C₄×C₅	1	1	yes	characteristic
91	C₁₆×C₁₉	C₂×C₅×C₈	1	16	yes	direct product factor
92	C₂×C₈×C₁₆	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
93	C₂×C₅×C₁₉	C₈×C₁₆	1	4	yes	direct product factor
94	C₂×C₅×C₁₉	C₂×C₈²	1	1	yes	characteristic
95	C₂×C₅×C₁₉	C₂×C₄×C₁₆	1	2	yes	—
96	C₂²×C₅×C₈	C₈×C₁₉	1	4	yes	—
97	C₂²×C₅×C₈	C₂×C₄×C₁₉	1	2	yes	—
98	C₂×C₄²×C₅	C₂×C₄×C₁₉	1	1	yes	characteristic
99	C₂×C₅×C₁₆	C₈×C₁₉	1	16	yes	direct product factor
100	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	8	yes	—
101	C₄×C₅×C₈	C₈×C₁₉	1	4	yes	—
102	C₄×C₅×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
103	C₄×C₅×C₈	C₂×C₄×C₁₉	1	2	yes	—
104	C₄×C₅×C₈	C₂×C₄×C₁₉	1	4	yes	—
105	C₄×C₅×C₈	C₂³×C₁₉	1	1	yes	characteristic
106	C₂³×C₁₉	C₄×C₅×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×C₅×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₄×C₅×C₈	1	2	yes	—
109	C₂×C₄×C₁₉	C₄×C₅×C₈	1	4	yes	—
110	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	8	yes	—
111	C₂×C₄×C₁₉	C₂×C₄²×C₅	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂²×C₅×C₈	1	2	yes	—
113	C₈×C₁₉	C₄×C₅×C₈	1	4	yes	—
114	C₈×C₁₉	C₂×C₅×C₁₆	1	16	yes	direct product factor
115	C₈×C₁₉	C₂²×C₅×C₈	1	4	yes	—
116	C₂×C₄×C₁₆	C₂×C₅×C₁₉	1	2	yes	—
117	C₂×C₈²	C₂×C₅×C₁₉	1	1	yes	characteristic
118	C₈×C₁₆	C₂×C₅×C₁₉	1	4	yes	direct product factor
119	C₅×C₁₉	C₂×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Hall
120	C₂²×C₄×C₅	C₄²×C₁₉	1	1	yes	characteristic
121	C₂²×C₄×C₅	C₂×C₈×C₁₉	1	2	yes	—
122	C₂×C₅×C₈	C₁₆×C₁₉	1	16	yes	direct product factor
123	C₂×C₅×C₈	C₄²×C₁₉	1	2	yes	—
124	C₂×C₅×C₈	C₂×C₈×C₁₉	1	8	yes	—
125	C₂×C₅×C₈	C₂×C₈×C₁₉	1	8	yes	—
126	C₂×C₅×C₈	C₂²×C₄×C₁₉	1	2	yes	—
127	C₄²×C₅	C₄²×C₁₉	1	1	yes	characteristic
128	C₄²×C₅	C₂×C₈×C₁₉	1	2	yes	—
129	C₄²×C₅	C₂²×C₄×C₁₉	1	1	yes	characteristic
130	C₅×C₁₆	C₂×C₈×C₁₉	1	16	yes	direct product factor
131	C₂²×C₁₉	C₅×C₈²	1	2	yes	—
132	C₂²×C₁₉	C₄×C₅×C₁₆	1	4	yes	—
133	C₂²×C₁₉	C₂×C₄×C₅×C₈	1	1	yes	characteristic
134	C₄×C₁₉	C₅×C₈²	1	2	yes	—
135	C₄×C₁₉	C₄×C₅×C₁₆	1	4	yes	—
136	C₄×C₁₉	C₂×C₄×C₅×C₈	1	2	yes	—
137	C₄×C₁₉	C₂²×C₅×C₁₆	1	4	yes	—
138	C₂²×C₁₆	C₄×C₅×C₁₉	1	4	yes	—
139	C₂×C₄×C₈	C₄×C₅×C₁₉	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₅×C₁₉	1	1	yes	characteristic
141	C₄×C₁₆	C₄×C₅×C₁₉	1	4	yes	—
142	C₄×C₁₆	C₂²×C₅×C₁₉	1	4	yes	—
143	C₈²	C₄×C₅×C₁₉	1	2	yes	—
144	C₈²	C₂²×C₅×C₁₉	1	2	yes	—
145	C₂³×C₅	C₄×C₈×C₁₉	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×C₈×C₁₉	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×C₈×C₁₉	1	2	yes	—
148	C₂×C₄×C₅	C₄×C₈×C₁₉	1	4	yes	—
149	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	8	yes	—
150	C₂×C₄×C₅	C₂×C₄²×C₁₉	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×C₈×C₁₉	1	2	yes	—
152	C₅×C₈	C₄×C₈×C₁₉	1	4	yes	—
153	C₅×C₈	C₂×C₁₆×C₁₉	1	16	yes	direct product factor
154	C₅×C₈	C₂²×C₈×C₁₉	1	4	yes	—
155	C₂×C₁₉	C₅×C₈×C₁₆	1	4	yes	direct product factor
156	C₂×C₁₉	C₂×C₅×C₈²	1	1	yes	characteristic
157	C₂×C₁₉	C₂×C₄×C₅×C₁₆	1	2	yes	—
158	C₂²×C₈	C₅×C₈×C₁₉	1	4	yes	—
159	C₂²×C₈	C₂×C₄×C₅×C₁₉	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₅×C₁₉	1	1	yes	characteristic
161	C₂×C₁₆	C₅×C₈×C₁₉	1	16	yes	direct product factor
162	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	8	yes	—
163	C₄×C₈	C₅×C₈×C₁₉	1	4	yes	—
164	C₄×C₈	C₂×C₄×C₅×C₁₉	1	1	yes	characteristic
165	C₄×C₈	C₂×C₄×C₅×C₁₉	1	2	yes	—
166	C₄×C₈	C₂×C₄×C₅×C₁₉	1	4	yes	—
167	C₄×C₈	C₂³×C₅×C₁₉	1	1	yes	characteristic
168	C₂²×C₅	C₈²×C₁₉	1	2	yes	—
169	C₂²×C₅	C₄×C₁₆×C₁₉	1	4	yes	—
170	C₂²×C₅	C₂×C₄×C₈×C₁₉	1	1	yes	characteristic
171	C₄×C₅	C₈²×C₁₉	1	2	yes	—
172	C₄×C₅	C₄×C₁₆×C₁₉	1	4	yes	—
173	C₄×C₅	C₂×C₄×C₈×C₁₉	1	2	yes	—
174	C₄×C₅	C₂²×C₁₆×C₁₉	1	4	yes	—
175	C₁₉	C₂×C₅×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×C₄	C₄²×C₅×C₁₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₅×C₈×C₁₉	1	2	yes	—
178	C₂×C₈	C₅×C₁₆×C₁₉	1	16	yes	direct product factor
179	C₂×C₈	C₄²×C₅×C₁₉	1	2	yes	—
180	C₂×C₈	C₂×C₅×C₈×C₁₉	1	8	yes	—
181	C₂×C₈	C₂×C₅×C₈×C₁₉	1	8	yes	—
182	C₂×C₈	C₂²×C₄×C₅×C₁₉	1	2	yes	—
183	C₄²	C₄²×C₅×C₁₉	1	1	yes	characteristic
184	C₄²	C₂×C₅×C₈×C₁₉	1	2	yes	—
185	C₄²	C₂²×C₄×C₅×C₁₉	1	1	yes	characteristic
186	C₁₆	C₂×C₅×C₈×C₁₉	1	16	yes	direct product factor
187	C₂×C₅	C₈×C₁₆×C₁₉	1	4	yes	direct product factor
188	C₂×C₅	C₂×C₈²×C₁₉	1	1	yes	characteristic
189	C₂×C₅	C₂×C₄×C₁₆×C₁₉	1	2	yes	—
190	C₂³	C₄×C₅×C₈×C₁₉	1	1	yes	characteristic
191	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	8	yes	—
192	C₂×C₄	C₄×C₅×C₈×C₁₉	1	1	yes	characteristic
193	C₂×C₄	C₄×C₅×C₈×C₁₉	1	2	yes	—
194	C₂×C₄	C₄×C₅×C₈×C₁₉	1	4	yes	—
195	C₂×C₄	C₂²×C₅×C₈×C₁₉	1	2	yes	—
196	C₂×C₄	C₂×C₄²×C₅×C₁₉	1	1	yes	characteristic
197	C₈	C₂×C₅×C₁₆×C₁₉	1	16	yes	direct product factor
198	C₈	C₄×C₅×C₈×C₁₉	1	4	yes	—
199	C₈	C₂²×C₅×C₈×C₁₉	1	4	yes	—
200	C₅	C₂×C₈×C₁₆×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	C₄×C₅×C₁₆×C₁₉	1	4	yes	—
202	C₂²	C₅×C₈²×C₁₉	1	2	yes	—
203	C₂²	C₂×C₄×C₅×C₈×C₁₉	1	1	yes	characteristic
204	C₄	C₄×C₅×C₁₆×C₁₉	1	4	yes	—
205	C₄	C₅×C₈²×C₁₉	1	2	yes	—
206	C₄	C₂²×C₅×C₁₆×C₁₉	1	4	yes	—
207	C₄	C₂×C₄×C₅×C₈×C₁₉	1	2	yes	—
208	C₂	C₅×C₈×C₁₆×C₁₉	1	4	yes	direct product factor
209	C₂	C₂×C₄×C₅×C₁₆×C₁₉	1	2	yes	—
210	C₂	C₂×C₅×C₈²×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂³×C₅×C₁₉
socle, prime radical	C₂³×C₅×C₁₉	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×C₈×C₁₆×C₁₉	C₂	direct (N × H)
19. C₂×C₅×C₁₆×C₁₉	C₈	direct (N × H)
21. C₂×C₈×C₁₆×C₁₉	C₅	direct (N × H)
24. C₈×C₁₆×C₁₉	C₂×C₅	direct (N × H)
27. C₂×C₅×C₈×C₁₉	C₁₆	direct (N × H)
35. C₅×C₁₆×C₁₉	C₂×C₈	direct (N × H)
36. C₂×C₅×C₈×C₁₆	C₁₉	direct (N × H)
52. C₅×C₈×C₁₉	C₂×C₁₆	direct (N × H)
56. C₅×C₈×C₁₆	C₂×C₁₉	direct (N × H)
60. C₂×C₁₆×C₁₉	C₅×C₈	direct (N × H)
83. C₂×C₈×C₁₉	C₅×C₁₆	direct (N × H)
91. C₁₆×C₁₉	C₂×C₅×C₈	direct (N × H)
92. C₂×C₈×C₁₆	C₅×C₁₉	direct (N × H)
93. C₂×C₅×C₁₉	C₈×C₁₆	direct (N × H)
99. C₂×C₅×C₁₆	C₈×C₁₉	direct (N × H)
114. C₈×C₁₉	C₂×C₅×C₁₆	direct (N × H)
118. C₈×C₁₆	C₂×C₅×C₁₉	direct (N × H)
119. C₅×C₁₉	C₂×C₈×C₁₆	direct (N × H)
122. C₂×C₅×C₈	C₁₆×C₁₉	direct (N × H)
130. C₅×C₁₆	C₂×C₈×C₁₉	direct (N × H)
153. C₅×C₈	C₂×C₁₆×C₁₉	direct (N × H)
155. C₂×C₁₉	C₅×C₈×C₁₆	direct (N × H)
161. C₂×C₁₆	C₅×C₈×C₁₉	direct (N × H)
175. C₁₉	C₂×C₅×C₈×C₁₆	direct (N × H)
178. C₂×C₈	C₅×C₁₆×C₁₉	direct (N × H)
186. C₁₆	C₂×C₅×C₈×C₁₉	direct (N × H)
187. C₂×C₅	C₈×C₁₆×C₁₉	direct (N × H)
197. C₈	C₂×C₅×C₁₆×C₁₉	direct (N × H)
200. C₅	C₂×C₈×C₁₆×C₁₉	direct (N × H)
208. C₂	C₅×C₈×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
C₂×C₅×C₈²×C₁₉	2
C₂×C₄×C₅×C₁₆×C₁₉	2
C₅×C₈×C₁₆×C₁₉	2
C₂×C₈×C₁₆×C₁₉	5
C₂×C₅×C₈×C₁₆	19

Subgroups — C2×C5×C8×C16×C19 #72201613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₈×C₁₆×C₁₉ #72201613