Subgroups — C₁₉×#9870 #18015829

Order 4864.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1279	C₂	1	4	yes	maximal
5	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
7	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
8	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₉	—	2	1	no	—
11	C₂×C₄×C₈×C₁₉	—	2	1	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
13	C₁₉×#391	—	2	1	no	—
14	C₁₉×#389	—	2	1	no	—
15	C₁₉×#389	C₄	1	2	yes	—
16	C₁₉×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	C₈	1	4	yes	complemented
25	C₄×Q₈×C₁₉	—	4	2	no	—
26	C₄×Q₈×C₁₉	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₁₉×C₄⋊C₈	—	2	4	no	—
42	C₁₉×C₄⋊C₈	—	4	2	no	—
43	C₁₉×C₄⋊C₈	—	4	2	no	—
44	C₄×C₈×C₁₉	—	2	4	no	—
45	C₄×C₈×C₁₉	—	2	4	no	—
46	C₄×C₈×C₁₉	—	2	2	no	—
47	C₄×C₈×C₁₉	Q₈	1	4	yes	complemented
48	C₁₉×#95	—	2	2	no	—
49	C₁₉×#95	—	2	2	no	—
50	C₂×Q₈×C₁₉	—	2	4	no	—
51	C₂×Q₈×C₁₉	—	4	2	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₉	—	2	4	no	—
63	C₂×C₈×C₁₉	—	2	4	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	4	2	no	—
67	C₂×C₈×C₁₉	—	4	2	no	—
68	C₄⋊C₄×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	—	2	2	no	—
70	C₄⋊C₄×C₁₉	—	2	4	no	—
71	C₄⋊C₄×C₁₉	—	4	4	no	—
72	C₄⋊C₄×C₁₉	—	4	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	4	2	no	—
79	C₄²×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	C₂×C₈	1	2	yes	—
81	C₄²×C₁₉	M₁₆	1	2	yes	—
82	C₄²×C₁₉	D₈	1	2	yes	—
83	C₄²×C₁₉	SD₁₆	1	2	yes	—
84	C₄²×C₁₉	SD₁₆	1	2	yes	—
85	C₄²×C₁₉	Q₁₆	1	2	yes	—
86	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
87	C₄²×C₁₉	#41	1	2	yes	—
88	#9870	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₁₉	—	4	8	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₁₉	#102	1	2	yes	—
106	C₂×C₄×C₁₉	#102	1	2	yes	—
107	C₂×C₄×C₁₉	#103	1	2	yes	—
108	C₂×C₄×C₁₉	#103	1	2	yes	—
109	C₂×C₄×C₁₉	#104	1	2	yes	—
110	C₂×C₄×C₁₉	#104	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#131	1	2	yes	—
118	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
122	C₈×C₁₉	—	2	8	no	—
123	C₈×C₁₉	—	8	2	no	—
124	C₄×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
125	#1842	C₂×C₁₉	1	1	yes	characteristic
126	C₄²×C₈	C₂×C₁₉	1	1	yes	characteristic
127	#1279	C₂×C₁₉	1	4	yes	—
128	C₂²×C₁₉	#391	1	1	yes	characteristic
129	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₁₉	#408	1	1	yes	characteristic
131	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
132	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
133	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
134	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	2	no	—
137	C₄×C₁₉	—	2	2	no	—
138	C₄×C₁₉	—	4	4	no	—
139	C₄×C₁₉	—	8	2	no	—
140	C₄×C₁₉	D₄⋊C₈	1	4	yes	—
141	C₄×C₁₉	Q₈⋊C₈	1	4	yes	—
142	C₄×C₁₉	C₈×Q₈	1	2	yes	—
143	C₄×C₁₉	#446	1	2	yes	—
144	C₄×C₁₉	#486	1	1	yes	characteristic
145	C₄×C₁₉	#487	1	1	yes	characteristic
146	C₄×C₁₉	#498	1	1	yes	characteristic
147	C₄×C₁₉	#500	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	C₄×C₁₉	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	C₂×C₄×C₈	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
156	#391	—	2	1	no	—
157	#389	—	2	1	no	—
158	#389	C₄×C₁₉	1	2	yes	—
159	#384	C₄×C₁₉	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	2	1	no	—
161	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
162	C₄³	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×C₁₉	C₂×D₄⋊C₈	1	1	yes	characteristic
164	C₂×C₁₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
165	C₂×C₁₉	#1492	1	1	yes	characteristic
166	C₂×C₁₉	#1493	1	1	yes	characteristic
167	C₂×C₁₉	#1984	1	1	yes	characteristic
168	C₂×C₁₉	#1990	1	1	yes	characteristic
169	C₂×C₁₉	#1994	1	1	yes	characteristic
170	C₂²×Q₈	—	2	1	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₈×C₁₉	1	4	yes	complemented
175	C₄×Q₈	—	4	2	no	—
176	C₄×Q₈	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	4	2	no	—
193	C₄⋊C₈	—	4	2	no	—
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	Q₈×C₁₉	1	4	yes	complemented
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₁₉	#9870	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	C₂×C₈×C₁₉	1	4	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×C₈×C₁₉	1	2	yes	—
232	C₄²	M₁₆×C₁₉	1	2	yes	—
233	C₄²	D₈×C₁₉	1	2	yes	—
234	C₄²	SD₁₆×C₁₉	1	2	yes	—
235	C₄²	SD₁₆×C₁₉	1	2	yes	—
236	C₄²	Q₁₆×C₁₉	1	2	yes	—
237	C₄²	C₂×Q₈×C₁₉	1	2	yes	—
238	C₄²	#41×C₁₉	1	2	yes	—
239	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₁₉×C₂²⋊C₈	1	2	yes	—
254	C₂×C₄	C₁₉×C₂²⋊C₈	1	2	yes	—
255	C₂×C₄	C₁₉×#102	1	2	yes	—
256	C₂×C₄	C₁₉×#102	1	2	yes	—
257	C₂×C₄	C₁₉×#103	1	2	yes	—
258	C₂×C₄	C₁₉×#103	1	2	yes	—
259	C₂×C₄	C₁₉×#104	1	2	yes	—
260	C₂×C₄	C₁₉×#104	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
262	C₂×C₄	C₁₉×C₂²⋊Q₈	1	2	yes	—
263	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
264	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
267	C₂×C₄	C₁₉×#131	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
271	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	8	2	no	—
274	C₂²	C₁₉×#391	1	1	yes	characteristic
275	C₂²	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
276	C₂²	C₁₉×#408	1	1	yes	characteristic
277	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
278	C₂²	C₂×C₁₉×#103	1	1	yes	characteristic
279	C₂²	C₂×C₁₉×#104	1	1	yes	characteristic
280	C₂²	C₂×C₁₉×#104	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₁₉×D₄⋊C₈	1	4	yes	—
287	C₄	C₁₉×Q₈⋊C₈	1	4	yes	—
288	C₄	C₈×Q₈×C₁₉	1	2	yes	—
289	C₄	C₁₉×#446	1	2	yes	—
290	C₄	C₁₉×#486	1	1	yes	characteristic
291	C₄	C₁₉×#487	1	1	yes	characteristic
292	C₄	C₁₉×#498	1	1	yes	characteristic
293	C₄	C₁₉×#500	1	1	yes	characteristic
294	C₂	C₂×C₁₉×D₄⋊C₈	1	1	yes	characteristic
295	C₂	C₂×C₁₉×Q₈⋊C₈	1	1	yes	characteristic
296	C₂	C₁₉×#1492	1	1	yes	characteristic
297	C₂	C₁₉×#1493	1	1	yes	characteristic
298	C₂	C₁₉×#1984	1	1	yes	characteristic
299	C₂	C₁₉×#1990	1	1	yes	characteristic
300	C₂	C₁₉×#1994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₁₉×C₄⋊Q₈	C₈	semidirect
47. C₄×C₈×C₁₉	Q₈	semidirect
88. #9870	C₁₉	direct (N × H)
174. C₄⋊Q₈	C₈×C₁₉	semidirect
197. C₄×C₈	Q₈×C₁₉	semidirect
200. C₁₉	#9870	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₉×C₄⋊Q₈	2
C₁₉×#1842	2
C₄²×C₈×C₁₉	2
C₁₉×#1279	2
#9870	19

Subgroups — C19×#9870 #18015829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9870 #18015829