Subgroups — C₂×C₄×C₅×C₁₉×C₃₂ #72201610

Order 24320.

540 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₁₉×C₃₂	C₂	1	2	yes	maximal
3	C₄×C₅×C₁₉×C₃₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₅×C₈×C₁₉	C₄	1	1	yes	characteristic
5	C₂²×C₅×C₁₆×C₁₉	C₄	1	1	yes	characteristic
6	C₂²×C₅×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₄×C₅×C₁₆×C₁₉	C₄	1	2	yes	—
8	C₄×C₅×C₁₆×C₁₉	C₂²	1	2	yes	—
9	C₂×C₅×C₁₉×C₃₂	C₄	1	8	yes	direct product factor
10	C₂×C₅×C₁₉×C₃₂	C₂²	1	4	yes	—
11	C₂×C₄²×C₅×C₁₉	C₈	1	1	yes	characteristic
12	C₂²×C₅×C₈×C₁₉	C₈	1	1	yes	characteristic
13	C₂²×C₅×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
14	C₄×C₅×C₈×C₁₉	C₈	1	2	yes	—
15	C₄×C₅×C₈×C₁₉	C₂×C₄	1	2	yes	—
16	C₂×C₅×C₁₆×C₁₉	C₈	1	4	yes	—
17	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	4	yes	—
20	C₂×C₅×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
21	C₅×C₁₉×C₃₂	C₂×C₄	1	8	yes	direct product factor
22	C₂×C₄×C₁₉×C₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₁₉×C₃₂	C₂×C₅	1	2	yes	—
24	C₂×C₄×C₁₆×C₁₉	C₂×C₅	1	1	yes	characteristic
25	C₄×C₁₉×C₃₂	C₂×C₅	1	4	yes	direct product factor
26	C₂²×C₄×C₅×C₁₉	C₁₆	1	2	yes	—
27	C₂²×C₄×C₅×C₁₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₅×C₈×C₁₉	C₁₆	1	4	yes	—
29	C₂×C₅×C₈×C₁₉	C₄²	1	2	yes	—
30	C₂×C₅×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₅×C₈×C₁₉	C₂×C₈	1	2	yes	—
32	C₂×C₅×C₈×C₁₉	C₂×C₈	1	2	yes	—
33	C₂×C₅×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₅×C₁₉	C₁₆	1	2	yes	—
35	C₄²×C₅×C₁₉	C₂×C₈	1	2	yes	—
36	C₅×C₁₆×C₁₉	C₄²	1	2	yes	—
37	C₅×C₁₆×C₁₉	C₂×C₈	1	4	yes	—
38	C₅×C₁₆×C₁₉	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₅×C₃₂	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂²×C₁₆×C₁₉	C₄×C₅	1	1	yes	characteristic
41	C₂²×C₁₆×C₁₉	C₂²×C₅	1	1	yes	characteristic
42	C₂×C₄×C₈×C₁₉	C₄×C₅	1	1	yes	characteristic
43	C₂×C₁₉×C₃₂	C₄×C₅	1	8	yes	direct product factor
44	C₂×C₁₉×C₃₂	C₂²×C₅	1	4	yes	—
45	C₄×C₁₆×C₁₉	C₄×C₅	1	2	yes	—
46	C₄×C₁₆×C₁₉	C₂²×C₅	1	2	yes	—
47	C₂³×C₅×C₁₉	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅×C₁₉	C₃₂	1	8	yes	direct product factor
49	C₂×C₄×C₅×C₁₉	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	4	yes	—
53	C₂×C₄×C₅×C₁₉	C₂²×C₈	1	1	yes	characteristic
54	C₅×C₈×C₁₉	C₄×C₈	1	2	yes	—
55	C₅×C₈×C₁₉	C₂×C₁₆	1	4	yes	—
56	C₅×C₈×C₁₉	C₂×C₄²	1	1	yes	characteristic
57	C₅×C₈×C₁₉	C₂²×C₈	1	1	yes	characteristic
58	C₂²×C₅×C₃₂	C₂×C₁₉	1	2	yes	—
59	C₂×C₄×C₅×C₁₆	C₂×C₁₉	1	1	yes	characteristic
60	C₄×C₅×C₃₂	C₂×C₁₉	1	4	yes	direct product factor
61	C₂²×C₈×C₁₉	C₅×C₈	1	1	yes	characteristic
62	C₂²×C₈×C₁₉	C₂×C₄×C₅	1	1	yes	characteristic
63	C₂×C₄²×C₁₉	C₅×C₈	1	1	yes	characteristic
64	C₂×C₁₆×C₁₉	C₅×C₈	1	4	yes	—
65	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	1	yes	characteristic
66	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	2	yes	—
67	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	4	yes	—
68	C₂×C₁₆×C₁₉	C₂³×C₅	1	1	yes	characteristic
69	C₄×C₈×C₁₉	C₅×C₈	1	2	yes	—
70	C₄×C₈×C₁₉	C₂×C₄×C₅	1	2	yes	—
71	C₁₉×C₃₂	C₂×C₄×C₅	1	8	yes	direct product factor
72	C₂²×C₅×C₁₉	C₄×C₁₆	1	2	yes	—
73	C₂²×C₅×C₁₉	C₂×C₃₂	1	4	yes	—
74	C₂²×C₅×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
75	C₄×C₅×C₁₉	C₄×C₁₆	1	2	yes	—
76	C₄×C₅×C₁₉	C₂×C₃₂	1	8	yes	direct product factor
77	C₄×C₅×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₅×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
79	C₂²×C₅×C₁₆	C₄×C₁₉	1	1	yes	characteristic
80	C₂²×C₅×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
81	C₂×C₄×C₅×C₈	C₄×C₁₉	1	1	yes	characteristic
82	C₂×C₅×C₃₂	C₄×C₁₉	1	8	yes	direct product factor
83	C₂×C₅×C₃₂	C₂²×C₁₉	1	4	yes	—
84	C₄×C₅×C₁₆	C₄×C₁₉	1	2	yes	—
85	C₄×C₅×C₁₆	C₂²×C₁₉	1	2	yes	—
86	C₂²×C₄×C₁₉	C₅×C₁₆	1	2	yes	—
87	C₂²×C₄×C₁₉	C₂×C₅×C₈	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₅×C₁₆	1	4	yes	—
89	C₂×C₈×C₁₉	C₄²×C₅	1	2	yes	—
90	C₂×C₈×C₁₉	C₂×C₅×C₈	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂×C₅×C₈	1	2	yes	—
92	C₂×C₈×C₁₉	C₂×C₅×C₈	1	2	yes	—
93	C₂×C₈×C₁₉	C₂²×C₄×C₅	1	1	yes	characteristic
94	C₄²×C₁₉	C₅×C₁₆	1	2	yes	—
95	C₄²×C₁₉	C₂×C₅×C₈	1	2	yes	—
96	C₁₆×C₁₉	C₄²×C₅	1	2	yes	—
97	C₁₆×C₁₉	C₂×C₅×C₈	1	4	yes	—
98	C₁₆×C₁₉	C₂²×C₄×C₅	1	2	yes	—
99	C₂×C₄×C₃₂	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₅×C₁₉	C₄×C₃₂	1	4	yes	direct product factor
101	C₂×C₅×C₁₉	C₂×C₄×C₁₆	1	1	yes	characteristic
102	C₂×C₅×C₁₉	C₂²×C₃₂	1	2	yes	—
103	C₂²×C₅×C₈	C₈×C₁₉	1	1	yes	characteristic
104	C₂²×C₅×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
105	C₂×C₄²×C₅	C₈×C₁₉	1	1	yes	characteristic
106	C₂×C₅×C₁₆	C₈×C₁₉	1	4	yes	—
107	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
108	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
109	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	4	yes	—
110	C₂×C₅×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
111	C₄×C₅×C₈	C₈×C₁₉	1	2	yes	—
112	C₄×C₅×C₈	C₂×C₄×C₁₉	1	2	yes	—
113	C₅×C₃₂	C₂×C₄×C₁₉	1	8	yes	direct product factor
114	C₂³×C₁₉	C₂×C₅×C₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₅×C₃₂	1	8	yes	direct product factor
116	C₂×C₄×C₁₉	C₄×C₅×C₈	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	4	yes	—
120	C₂×C₄×C₁₉	C₂²×C₅×C₈	1	1	yes	characteristic
121	C₈×C₁₉	C₄×C₅×C₈	1	2	yes	—
122	C₈×C₁₉	C₂×C₅×C₁₆	1	4	yes	—
123	C₈×C₁₉	C₂×C₄²×C₅	1	1	yes	characteristic
124	C₈×C₁₉	C₂²×C₅×C₈	1	1	yes	characteristic
125	C₂²×C₃₂	C₂×C₅×C₁₉	1	2	yes	—
126	C₂×C₄×C₁₆	C₂×C₅×C₁₉	1	1	yes	characteristic
127	C₄×C₃₂	C₂×C₅×C₁₉	1	4	yes	direct product factor
128	C₅×C₁₉	C₂×C₄×C₃₂	1	1	yes	characteristic, direct product factor, Hall
129	C₂²×C₄×C₅	C₁₆×C₁₉	1	2	yes	—
130	C₂²×C₄×C₅	C₂×C₈×C₁₉	1	1	yes	characteristic
131	C₂×C₅×C₈	C₁₆×C₁₉	1	4	yes	—
132	C₂×C₅×C₈	C₄²×C₁₉	1	2	yes	—
133	C₂×C₅×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
134	C₂×C₅×C₈	C₂×C₈×C₁₉	1	2	yes	—
135	C₂×C₅×C₈	C₂×C₈×C₁₉	1	2	yes	—
136	C₂×C₅×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
137	C₄²×C₅	C₁₆×C₁₉	1	2	yes	—
138	C₄²×C₅	C₂×C₈×C₁₉	1	2	yes	—
139	C₅×C₁₆	C₄²×C₁₉	1	2	yes	—
140	C₅×C₁₆	C₂×C₈×C₁₉	1	4	yes	—
141	C₅×C₁₆	C₂²×C₄×C₁₉	1	2	yes	—
142	C₂²×C₁₉	C₄×C₅×C₁₆	1	2	yes	—
143	C₂²×C₁₉	C₂×C₅×C₃₂	1	4	yes	—
144	C₂²×C₁₉	C₂²×C₅×C₁₆	1	1	yes	characteristic
145	C₄×C₁₉	C₄×C₅×C₁₆	1	2	yes	—
146	C₄×C₁₉	C₂×C₅×C₃₂	1	8	yes	direct product factor
147	C₄×C₁₉	C₂×C₄×C₅×C₈	1	1	yes	characteristic
148	C₄×C₁₉	C₂²×C₅×C₁₆	1	1	yes	characteristic
149	C₂²×C₁₆	C₄×C₅×C₁₉	1	1	yes	characteristic
150	C₂²×C₁₆	C₂²×C₅×C₁₉	1	1	yes	characteristic
151	C₂×C₄×C₈	C₄×C₅×C₁₉	1	1	yes	characteristic
152	C₂×C₃₂	C₄×C₅×C₁₉	1	8	yes	direct product factor
153	C₂×C₃₂	C₂²×C₅×C₁₉	1	4	yes	—
154	C₄×C₁₆	C₄×C₅×C₁₉	1	2	yes	—
155	C₄×C₁₆	C₂²×C₅×C₁₉	1	2	yes	—
156	C₂³×C₅	C₂×C₁₆×C₁₉	1	1	yes	characteristic
157	C₂×C₄×C₅	C₁₉×C₃₂	1	8	yes	direct product factor
158	C₂×C₄×C₅	C₄×C₈×C₁₉	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	4	yes	—
162	C₂×C₄×C₅	C₂²×C₈×C₁₉	1	1	yes	characteristic
163	C₅×C₈	C₄×C₈×C₁₉	1	2	yes	—
164	C₅×C₈	C₂×C₁₆×C₁₉	1	4	yes	—
165	C₅×C₈	C₂×C₄²×C₁₉	1	1	yes	characteristic
166	C₅×C₈	C₂²×C₈×C₁₉	1	1	yes	characteristic
167	C₂×C₁₉	C₄×C₅×C₃₂	1	4	yes	direct product factor
168	C₂×C₁₉	C₂×C₄×C₅×C₁₆	1	1	yes	characteristic
169	C₂×C₁₉	C₂²×C₅×C₃₂	1	2	yes	—
170	C₂²×C₈	C₅×C₈×C₁₉	1	1	yes	characteristic
171	C₂²×C₈	C₂×C₄×C₅×C₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₅×C₈×C₁₉	1	1	yes	characteristic
173	C₂×C₁₆	C₅×C₈×C₁₉	1	4	yes	—
174	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	1	yes	characteristic
175	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	2	yes	—
176	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	4	yes	—
177	C₂×C₁₆	C₂³×C₅×C₁₉	1	1	yes	characteristic
178	C₄×C₈	C₅×C₈×C₁₉	1	2	yes	—
179	C₄×C₈	C₂×C₄×C₅×C₁₉	1	2	yes	—
180	C₃₂	C₂×C₄×C₅×C₁₉	1	8	yes	direct product factor
181	C₂²×C₅	C₄×C₁₆×C₁₉	1	2	yes	—
182	C₂²×C₅	C₂×C₁₉×C₃₂	1	4	yes	—
183	C₂²×C₅	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
184	C₄×C₅	C₄×C₁₆×C₁₉	1	2	yes	—
185	C₄×C₅	C₂×C₁₉×C₃₂	1	8	yes	direct product factor
186	C₄×C₅	C₂×C₄×C₈×C₁₉	1	1	yes	characteristic
187	C₄×C₅	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
188	C₁₉	C₂×C₄×C₅×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×C₄	C₅×C₁₆×C₁₉	1	2	yes	—
190	C₂²×C₄	C₂×C₅×C₈×C₁₉	1	1	yes	characteristic
191	C₂×C₈	C₅×C₁₆×C₁₉	1	4	yes	—
192	C₂×C₈	C₄²×C₅×C₁₉	1	2	yes	—
193	C₂×C₈	C₂×C₅×C₈×C₁₉	1	1	yes	characteristic
194	C₂×C₈	C₂×C₅×C₈×C₁₉	1	2	yes	—
195	C₂×C₈	C₂×C₅×C₈×C₁₉	1	2	yes	—
196	C₂×C₈	C₂²×C₄×C₅×C₁₉	1	1	yes	characteristic
197	C₄²	C₅×C₁₆×C₁₉	1	2	yes	—
198	C₄²	C₂×C₅×C₈×C₁₉	1	2	yes	—
199	C₁₆	C₄²×C₅×C₁₉	1	2	yes	—
200	C₁₆	C₂×C₅×C₈×C₁₉	1	4	yes	—
201	C₁₆	C₂²×C₄×C₅×C₁₉	1	2	yes	—
202	C₂×C₅	C₄×C₁₉×C₃₂	1	4	yes	direct product factor
203	C₂×C₅	C₂×C₄×C₁₆×C₁₉	1	1	yes	characteristic
204	C₂×C₅	C₂²×C₁₉×C₃₂	1	2	yes	—
205	C₂³	C₂×C₅×C₁₆×C₁₉	1	1	yes	characteristic
206	C₂×C₄	C₅×C₁₉×C₃₂	1	8	yes	direct product factor
207	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	1	yes	characteristic
208	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	2	yes	—
209	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	4	yes	—
210	C₂×C₄	C₄×C₅×C₈×C₁₉	1	2	yes	—
211	C₂×C₄	C₂²×C₅×C₈×C₁₉	1	1	yes	characteristic
212	C₈	C₂×C₅×C₁₆×C₁₉	1	4	yes	—
213	C₈	C₄×C₅×C₈×C₁₉	1	2	yes	—
214	C₈	C₂²×C₅×C₈×C₁₉	1	1	yes	characteristic
215	C₈	C₂×C₄²×C₅×C₁₉	1	1	yes	characteristic
216	C₅	C₂×C₄×C₁₉×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	C₂×C₅×C₁₉×C₃₂	1	4	yes	—
218	C₂²	C₄×C₅×C₁₆×C₁₉	1	2	yes	—
219	C₂²	C₂²×C₅×C₁₆×C₁₉	1	1	yes	characteristic
220	C₄	C₂×C₅×C₁₉×C₃₂	1	8	yes	direct product factor
221	C₄	C₄×C₅×C₁₆×C₁₉	1	2	yes	—
222	C₄	C₂²×C₅×C₁₆×C₁₉	1	1	yes	characteristic
223	C₄	C₂×C₄×C₅×C₈×C₁₉	1	1	yes	characteristic
224	C₂	C₄×C₅×C₁₉×C₃₂	1	4	yes	direct product factor
225	C₂	C₂²×C₅×C₁₉×C₃₂	1	2	yes	—
226	C₂	C₂×C₄×C₅×C₁₆×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₆	C₂³×C₅×C₁₉
socle, prime radical	C₂³×C₅×C₁₉	C₂×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×C₁₉×C₃₂	C₂	direct (N × H)
9. C₂×C₅×C₁₉×C₃₂	C₄	direct (N × H)
21. C₅×C₁₉×C₃₂	C₂×C₄	direct (N × H)
22. C₂×C₄×C₁₉×C₃₂	C₅	direct (N × H)
25. C₄×C₁₉×C₃₂	C₂×C₅	direct (N × H)
39. C₂×C₄×C₅×C₃₂	C₁₉	direct (N × H)
43. C₂×C₁₉×C₃₂	C₄×C₅	direct (N × H)
48. C₂×C₄×C₅×C₁₉	C₃₂	direct (N × H)
60. C₄×C₅×C₃₂	C₂×C₁₉	direct (N × H)
71. C₁₉×C₃₂	C₂×C₄×C₅	direct (N × H)
76. C₄×C₅×C₁₉	C₂×C₃₂	direct (N × H)
82. C₂×C₅×C₃₂	C₄×C₁₉	direct (N × H)
99. C₂×C₄×C₃₂	C₅×C₁₉	direct (N × H)
100. C₂×C₅×C₁₉	C₄×C₃₂	direct (N × H)
113. C₅×C₃₂	C₂×C₄×C₁₉	direct (N × H)
115. C₂×C₄×C₁₉	C₅×C₃₂	direct (N × H)
127. C₄×C₃₂	C₂×C₅×C₁₉	direct (N × H)
128. C₅×C₁₉	C₂×C₄×C₃₂	direct (N × H)
146. C₄×C₁₉	C₂×C₅×C₃₂	direct (N × H)
152. C₂×C₃₂	C₄×C₅×C₁₉	direct (N × H)
157. C₂×C₄×C₅	C₁₉×C₃₂	direct (N × H)
167. C₂×C₁₉	C₄×C₅×C₃₂	direct (N × H)
180. C₃₂	C₂×C₄×C₅×C₁₉	direct (N × H)
185. C₄×C₅	C₂×C₁₉×C₃₂	direct (N × H)
188. C₁₉	C₂×C₄×C₅×C₃₂	direct (N × H)
202. C₂×C₅	C₄×C₁₉×C₃₂	direct (N × H)
206. C₂×C₄	C₅×C₁₉×C₃₂	direct (N × H)
216. C₅	C₂×C₄×C₁₉×C₃₂	direct (N × H)
220. C₄	C₂×C₅×C₁₉×C₃₂	direct (N × H)
224. C₂	C₄×C₅×C₁₉×C₃₂	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×C₄×C₅×C₁₆×C₁₉	2
C₂²×C₅×C₁₉×C₃₂	2
C₄×C₅×C₁₉×C₃₂	2
C₂×C₄×C₁₉×C₃₂	5
C₂×C₄×C₅×C₃₂	19

Subgroups — C2×C4×C5×C19×C32 #72201610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×C₁₉×C₃₂ #72201610