Subgroups — C₁₉×#8916 #18014875

Order 4864.

354 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2995	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1569	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1524	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1495	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1302	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#529	C₂²	1	1	yes	characteristic
9	C₁₉×#446	—	2	1	no	—
10	C₁₉×#436	—	2	1	no	—
11	C₁₉×#435	—	2	1	no	—
12	C₁₉×#435	—	2	1	no	—
13	C₁₉×#433	—	2	1	no	—
14	C₁₉×#432	C₂²	1	1	yes	characteristic
15	C₁₉×#431	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#108	—	2	1	no	—
17	C₂×C₁₉×#108	—	2	1	no	—
18	C₁₉×#424	—	2	1	no	—
19	C₁₉×#423	C₂²	1	1	yes	characteristic
20	C₁₉×#423	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
23	C₁₉×#335	—	2	1	no	—
24	C₁₉×#334	—	2	1	no	—
25	C₁₉×#330	C₄	1	1	yes	characteristic
26	C₁₉×#330	C₄	1	1	yes	characteristic
27	C₁₉×#329	C₄	1	1	yes	characteristic
28	C₁₉×#329	C₄	1	1	yes	characteristic
29	C₁₉×Q₈⋊C₈	—	2	1	no	—
30	C₁₉×D₄⋊C₈	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	—	2	1	no	—
34	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	—	2	1	no	—
37	C₂²×C₈×C₁₉	—	2	1	no	—
38	C₁₉×#126	—	2	1	no	—
39	C₁₉×#125	C₂×C₄	1	1	yes	characteristic
40	C₁₉×#124	C₂×C₄	1	1	yes	characteristic
41	C₁₉×#123	—	2	1	no	—
42	C₁₉×C₂²⋊Q₈	Q₈	1	1	yes	characteristic
43	C₁₉×#121	Q₈	1	1	yes	characteristic
44	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
45	C₄×D₄×C₁₉	—	2	1	no	—
46	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
47	C₁₉×#117	—	2	1	no	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₁₉×#108	—	4	2	no	—
50	C₁₉×#108	—	4	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₁₉×C₂²⋊C₈	—	2	1	no	—
61	C₁₉×C₂²⋊C₈	—	2	1	no	—
62	C₁₉×C₂²⋊C₈	—	2	1	no	—
63	C₁₉×C₂²⋊C₈	—	2	1	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₁₉×#97	—	2	1	no	—
67	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
68	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
69	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
70	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₁₉	—	2	1	no	—
72	C₄×C₈×C₁₉	—	2	1	no	—
73	C₄×C₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	—	2	1	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	M₁₆×C₁₉	—	4	2	no	—
82	M₁₆×C₁₉	—	4	1	no	—
83	M₁₆×C₁₉	—	4	1	no	—
84	M₁₆×C₁₉	—	4	1	no	—
85	M₁₆×C₁₉	—	4	1	no	—
86	M₁₆×C₁₉	—	8	1	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	2	1	no	—
91	C₂×C₈×C₁₉	—	2	1	no	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	4	1	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	8	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₉	—	2	1	no	—
107	C₂²⋊C₄×C₁₉	—	2	1	no	—
108	C₂²⋊C₄×C₁₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₉	—	2	1	no	—
110	C₂²⋊C₄×C₁₉	—	2	1	no	—
111	C₄²×C₁₉	—	2	1	no	—
112	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₉	#41	1	1	yes	characteristic
116	#8916	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₁₉	—	2	1	no	—
118	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
119	Q₈×C₁₉	—	4	1	no	—
120	D₄×C₁₉	—	4	1	no	—
121	D₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
137	C₈×C₁₉	—	2	2	no	—
138	C₈×C₁₉	—	2	2	no	—
139	C₈×C₁₉	—	2	2	no	—
140	C₈×C₁₉	—	4	1	no	—
141	C₈×C₁₉	—	4	1	no	—
142	C₈×C₁₉	—	4	1	no	—
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	—	4	1	no	—
145	C₈×C₁₉	—	8	1	no	—
146	C₈×C₁₉	—	8	1	no	—
147	#2995	C₂×C₁₉	1	1	yes	characteristic
148	#1849	C₂×C₁₉	1	1	yes	characteristic, complemented
149	#1569	C₂×C₁₉	1	1	yes	characteristic, complemented
150	#1524	C₂×C₁₉	1	1	yes	characteristic
151	#1495	C₂×C₁₉	1	1	yes	characteristic
152	#1302	C₂×C₁₉	1	1	yes	characteristic, complemented
153	#1299	C₂×C₁₉	1	1	yes	characteristic, complemented
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	#385	1	1	yes	characteristic
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	#453	1	1	yes	characteristic
170	C₄×C₁₉	#456	1	1	yes	characteristic
171	#529	C₂²×C₁₉	1	1	yes	characteristic
172	#446	—	2	1	no	—
173	#436	—	2	1	no	—
174	#435	—	2	1	no	—
175	#435	—	2	1	no	—
176	#433	—	2	1	no	—
177	#432	C₂²×C₁₉	1	1	yes	characteristic
178	#431	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×#108	—	2	1	no	—
180	C₂×#108	—	2	1	no	—
181	#424	—	2	1	no	—
182	#423	C₂²×C₁₉	1	1	yes	characteristic
183	#423	C₂²×C₁₉	1	1	yes	characteristic
184	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
185	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
186	#335	—	2	1	no	—
187	#334	—	2	1	no	—
188	#330	C₄×C₁₉	1	1	yes	characteristic
189	#330	C₄×C₁₉	1	1	yes	characteristic
190	#329	C₄×C₁₉	1	1	yes	characteristic
191	#329	C₄×C₁₉	1	1	yes	characteristic
192	Q₈⋊C₈	—	2	1	no	—
193	D₄⋊C₈	—	2	1	no	—
194	C₂×C₁₉	—	4	1	no	—
195	C₂×C₁₉	—	8	1	no	—
196	C₂×C₁₉	#1858	1	1	yes	characteristic
197	C₂×C₁₉	#1859	1	1	yes	characteristic
198	C₂×C₁₉	#1861	1	1	yes	characteristic
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
203	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	#126	—	2	1	no	—
207	#125	C₂×C₄×C₁₉	1	1	yes	characteristic
208	#124	C₂×C₄×C₁₉	1	1	yes	characteristic
209	#123	—	2	1	no	—
210	C₂²⋊Q₈	Q₈×C₁₉	1	1	yes	characteristic
211	#121	Q₈×C₁₉	1	1	yes	characteristic
212	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
215	#117	—	2	1	no	—
216	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
217	#108	—	4	2	no	—
218	#108	—	4	2	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
227	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	#97	—	2	1	no	—
233	#97	—	2	1	no	—
234	#97	—	2	1	no	—
235	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
236	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
237	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
238	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₁₉	#8916	1	1	yes	characteristic, direct product factor, Sylow
243	C₂×Q₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	8	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	8	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
284	C₄²	#41×C₁₉	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
287	Q₈	—	4	1	no	—
288	D₄	—	4	1	no	—
289	D₄	—	8	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
301	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
302	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
303	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	1	no	—
310	C₈	—	4	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	4	1	no	—
313	C₈	—	8	1	no	—
314	C₈	—	8	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₁₉×#385	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	C₁₉×#453	1	1	yes	characteristic
331	C₄	C₁₉×#456	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	8	1	no	—
334	C₂	C₁₉×#1858	1	1	yes	characteristic
335	C₂	C₁₉×#1859	1	1	yes	characteristic
336	C₂	C₁₉×#1861	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#120
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#385
prime radical	C₁₉×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1849	C₂	semidirect
3. C₁₉×#1569	C₂	semidirect
6. C₁₉×#1302	C₂	semidirect
7. C₁₉×#1299	C₂	semidirect
116. #8916	C₁₉	direct (N × H)
148. #1849	C₂×C₁₉	semidirect
149. #1569	C₂×C₁₉	semidirect
152. #1302	C₂×C₁₉	semidirect
153. #1299	C₂×C₁₉	semidirect
242. C₁₉	#8916	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2995	2
C₁₉×#1849	2
C₁₉×#1569	2
C₁₉×#1524	2
C₁₉×#1495	2
C₁₉×#1302	2
C₁₉×#1299	2
#8916	19

Subgroups — C19×#8916 #18014875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8916 #18014875