Subgroups — C₁₉×#20482 #18026441

Order 4864.

950 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1770	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₁₉	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1753	C₂	1	4	yes	maximal
7	C₁₉×#1746	C₂	1	2	yes	maximal, complemented
8	C₁₉×#1726	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#431	C₄	1	8	yes	complemented
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₄×M₁₆×C₁₉	C₄	1	8	yes	complemented
16	C₂×C₁₉×#97	C₂²	1	4	yes	—
17	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₉	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
22	C₁₉×#383	C₄	1	2	yes	—
23	C₁₉×#382	C₄	1	2	yes	—
24	C₁₉×#381	C₄	1	2	yes	—
25	C₁₉×#380	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₁₉×#376	C₂²	1	2	yes	—
29	C₁₉×#336	C₂²	1	4	yes	—
30	C₁₉×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₁₉	—	2	4	no	—
35	C₂×M₁₆×C₁₉	—	2	4	no	—
36	C₂×M₁₆×C₁₉	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
50	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₉×C₂²⋊C₈	C₂×C₄	1	8	yes	—
53	C₁₉×#97	—	2	8	no	—
54	C₁₉×#97	C₂×C₄	1	4	yes	—
55	C₁₉×#97	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₁₉	—	2	8	no	—
57	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	—
59	C₁₉×#95	C₂×C₄	1	4	yes	—
60	C₁₉×#95	C₂×C₄	1	4	yes	—
61	C₁₉×#95	C₂×C₄	1	4	yes	—
62	C₁₉×#95	C₂×C₄	1	4	yes	—
63	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	4	2	no	—
68	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
69	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₁₉	—	2	16	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	4	no	—
83	C₂×C₈×C₁₉	—	2	4	no	—
84	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₉	#41	1	4	yes	—
89	C₂×C₈×C₁₉	#41	1	4	yes	—
90	C₄⋊C₄×C₁₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₉	—	4	4	no	—
92	C₄²×C₁₉	—	2	4	no	—
93	C₄²×C₁₉	—	2	4	no	—
94	C₄²×C₁₉	C₄²	1	4	yes	—
95	C₄²×C₁₉	C₄²	1	4	yes	—
96	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
100	#20482	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	—	2	2	no	—
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
105	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	8	no	—
114	C₂×C₄×C₁₉	—	2	8	no	—
115	C₂×C₄×C₁₉	—	2	8	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	4	4	no	—
118	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₁₉	#131	1	4	yes	—
124	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
128	C₈×C₁₉	—	2	4	no	—
129	C₈×C₁₉	—	2	4	no	—
130	C₈×C₁₉	—	2	4	no	—
131	C₈×C₁₉	#117	1	8	yes	—
132	C₂×#431	C₂×C₁₉	1	1	yes	characteristic
133	C₂×C₄×M₁₆	C₂×C₁₉	1	1	yes	characteristic
134	#2283	C₂×C₁₉	1	1	yes	characteristic
135	#1770	C₂×C₁₉	1	2	yes	complemented
136	C₈×C₄⋊C₄	C₂×C₁₉	1	2	yes	complemented
137	#1753	C₂×C₁₉	1	4	yes	—
138	#1746	C₂×C₁₉	1	2	yes	complemented
139	#1726	C₂×C₁₉	1	2	yes	complemented
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	4	4	no	—
144	C₂²×C₁₉	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₁₉	C₄×#41	1	2	yes	—
146	C₂²×C₁₉	C₂×#131	1	2	yes	—
147	C₂²×C₁₉	#568	1	2	yes	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	4	8	no	—
153	C₄×C₁₉	#405	1	4	yes	—
154	C₄×C₁₉	C₂×#117	1	2	yes	—
155	C₄×C₁₉	C₄×#41	1	1	yes	characteristic
156	C₄×C₁₉	C₄×#41	1	1	yes	characteristic
157	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
158	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
159	#431	C₄×C₁₉	1	8	yes	complemented
160	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
163	C₄×M₁₆	C₄×C₁₉	1	8	yes	complemented
164	C₂×#97	C₂²×C₁₉	1	4	yes	—
165	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₉	1	4	yes	—
168	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
170	#383	C₄×C₁₉	1	2	yes	—
171	#382	C₄×C₁₉	1	2	yes	—
172	#381	C₄×C₁₉	1	2	yes	—
173	#380	C₄×C₁₉	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
175	C₄×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
176	#376	C₂²×C₁₉	1	2	yes	—
177	#336	C₂²×C₁₉	1	4	yes	—
178	#336	C₂²×C₁₉	1	4	yes	—
179	C₂×C₁₉	—	4	2	no	—
180	C₂×C₁₉	C₄×#117	1	1	yes	characteristic
181	C₂×C₁₉	C₂×#405	1	1	yes	characteristic
182	C₂×C₁₉	#2873	1	1	yes	characteristic
183	C₂×C₁₉	#2969	1	4	yes	—
184	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	C₂×C₄×C₁₉	1	8	yes	—
190	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
192	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
199	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
200	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
201	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
203	C₄⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
204	C₄⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
205	C₂²⋊C₈	C₂×C₄×C₁₉	1	8	yes	—
206	#97	—	2	8	no	—
207	#97	C₂×C₄×C₁₉	1	4	yes	—
208	#97	C₂×C₄×C₁₉	1	4	yes	—
209	C₄×C₈	—	2	8	no	—
210	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	—
211	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	—
212	#95	C₂×C₄×C₁₉	1	4	yes	—
213	#95	C₂×C₄×C₁₉	1	4	yes	—
214	#95	C₂×C₄×C₁₉	1	4	yes	—
215	#95	C₂×C₄×C₁₉	1	4	yes	—
216	C₁₉	#20482	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₄²×C₁₉	1	4	yes	—
223	C₂²×C₄	C₄²×C₁₉	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
232	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
233	M₁₆	—	2	16	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
242	C₂×C₈	#41×C₁₉	1	4	yes	—
243	C₂×C₈	#41×C₁₉	1	4	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	C₄²×C₁₉	1	4	yes	—
249	C₄²	C₄²×C₁₉	1	4	yes	—
250	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
251	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
252	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
253	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₄²×C₁₉	1	2	yes	—
258	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
274	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
276	C₂×C₄	C₁₉×#131	1	4	yes	—
277	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
280	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	C₁₉×#117	1	8	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂²×C₄²×C₁₉	1	1	yes	characteristic
290	C₂²	C₄×#41×C₁₉	1	2	yes	—
291	C₂²	C₂×C₁₉×#131	1	2	yes	—
292	C₂²	C₁₉×#568	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	8	no	—
298	C₄	C₁₉×#405	1	4	yes	—
299	C₄	C₂×C₁₉×#117	1	2	yes	—
300	C₄	C₄×#41×C₁₉	1	1	yes	characteristic
301	C₄	C₄×#41×C₁₉	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	C₄×C₁₉×#117	1	1	yes	characteristic
304	C₂	C₂×C₁₉×#405	1	1	yes	characteristic
305	C₂	C₁₉×#2873	1	1	yes	characteristic
306	C₂	C₁₉×#2969	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂³×C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1770	C₂	semidirect
5. C₈×C₄⋊C₄×C₁₉	C₂	semidirect
7. C₁₉×#1746	C₂	semidirect
8. C₁₉×#1726	C₂	semidirect
11. C₁₉×#431	C₄	semidirect
15. C₄×M₁₆×C₁₉	C₄	semidirect
100. #20482	C₁₉	direct (N × H)
135. #1770	C₂×C₁₉	semidirect
136. C₈×C₄⋊C₄	C₂×C₁₉	semidirect
138. #1746	C₂×C₁₉	semidirect
139. #1726	C₂×C₁₉	semidirect
159. #431	C₄×C₁₉	semidirect
163. C₄×M₁₆	C₄×C₁₉	semidirect
216. C₁₉	#20482	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×C₁₉×#431	2
C₂×C₄×M₁₆×C₁₉	2
C₁₉×#2283	2
C₁₉×#1770	2
C₈×C₄⋊C₄×C₁₉	2
C₁₉×#1753	2
C₁₉×#1746	2
C₁₉×#1726	2
#20482	19

Subgroups — C19×#20482 #18026441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20482 #18026441