Subgroups — C₁₉×#9723 #18015682

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3167	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1526	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1525	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1485	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1484	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₁₉×#498	C₄	1	1	yes	characteristic
11	C₁₉×#492	C₄	1	1	yes	characteristic
12	C₁₉×#487	C₄	1	1	yes	characteristic
13	C₁₉×#486	C₄	1	1	yes	characteristic
14	C₁₉×#465	—	2	1	no	—
15	C₁₉×#465	—	2	1	no	—
16	C₄×SD₁₆×C₁₉	—	2	1	no	—
17	C₄×SD₁₆×C₁₉	—	2	1	no	—
18	C₁₉×#431	C₂²	1	1	yes	characteristic
19	C₂×C₁₉×#108	—	2	1	no	—
20	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#341	—	2	1	no	—
25	C₁₉×#330	—	2	2	no	—
26	C₁₉×#329	—	2	2	no	—
27	C₁₉×Q₈⋊C₈	—	2	2	no	—
28	C₁₉×D₄⋊C₈	—	2	2	no	—
29	C₁₉×#322	—	2	1	no	—
30	C₈²×C₁₉	—	2	1	no	—
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₂×SD₁₆×C₁₉	—	2	1	no	—
33	C₂×SD₁₆×C₁₉	—	2	1	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₂×M₁₆×C₁₉	—	2	1	no	—
36	C₂×M₁₆×C₁₉	—	2	1	no	—
37	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
39	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
40	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
43	C₁₉×#125	—	2	1	no	—
44	C₁₉×#124	—	2	1	no	—
45	C₁₉×C₂²⋊Q₈	—	2	1	no	—
46	C₁₉×C₂²⋊Q₈	—	2	1	no	—
47	C₁₉×#121	—	2	1	no	—
48	C₁₉×#121	—	2	1	no	—
49	C₄×Q₈×C₁₉	—	2	1	no	—
50	C₄×Q₈×C₁₉	—	2	1	no	—
51	C₄×D₄×C₁₉	—	2	1	no	—
52	C₄×D₄×C₁₉	—	2	1	no	—
53	C₁₉×#117	—	2	1	no	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#108	—	2	4	no	—
56	C₁₉×#106	—	2	1	no	—
57	C₁₉×#106	—	2	1	no	—
58	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
59	C₁₉×#103	—	2	1	no	—
60	C₁₉×#103	—	2	1	no	—
61	C₁₉×#102	—	2	1	no	—
62	C₁₉×#102	—	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	D₄	1	2	yes	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	C₄×C₈×C₁₉	—	2	1	no	—
69	C₄×C₈×C₁₉	—	2	1	no	—
70	C₄×C₈×C₁₉	—	2	1	no	—
71	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
75	#41×C₁₉	—	8	2	no	—
76	C₂×Q₈×C₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	—	2	1	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	—	4	1	no	—
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
87	SD₁₆×C₁₉	—	4	2	no	—
88	SD₁₆×C₁₉	—	4	2	no	—
89	M₁₆×C₁₉	—	2	4	no	—
90	M₁₆×C₁₉	—	2	2	no	—
91	M₁₆×C₁₉	—	2	2	no	—
92	M₁₆×C₁₉	—	4	1	no	—
93	M₁₆×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×C₁₉	—	2	2	no	—
97	C₂×C₈×C₁₉	—	4	1	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	2	1	no	—
106	C₄⋊C₄×C₁₉	—	2	1	no	—
107	C₄⋊C₄×C₁₉	—	2	1	no	—
108	C₄⋊C₄×C₁₉	—	2	1	no	—
109	C₄⋊C₄×C₁₉	—	2	1	no	—
110	C₄⋊C₄×C₁₉	—	4	1	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₂²⋊C₄×C₁₉	—	4	1	no	—
113	C₂²⋊C₄×C₁₉	—	4	1	no	—
114	C₂²⋊C₄×C₁₉	—	4	1	no	—
115	C₄²×C₁₉	—	4	1	no	—
116	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
120	#9723	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₁₉	—	4	1	no	—
122	C₂³×C₁₉	#124	1	1	yes	characteristic
123	Q₈×C₁₉	—	4	1	no	—
124	Q₈×C₁₉	—	4	1	no	—
125	Q₈×C₁₉	—	4	2	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	8	1	no	—
131	D₄×C₁₉	—	8	2	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	8	2	no	—
140	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
146	C₈×C₁₉	—	2	4	no	—
147	C₈×C₁₉	—	2	1	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₈×C₁₉	—	2	1	no	—
150	C₈×C₁₉	—	4	2	no	—
151	C₈×C₁₉	—	4	2	no	—
152	C₈×C₁₉	—	4	2	no	—
153	C₈×C₁₉	#104	1	2	yes	—
154	#3167	C₂×C₁₉	1	1	yes	characteristic
155	C₄×#108	C₂×C₁₉	1	1	yes	characteristic, complemented
156	#1526	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#1525	C₂×C₁₉	1	1	yes	characteristic
158	#1485	C₂×C₁₉	1	1	yes	characteristic, complemented
159	#1484	C₂×C₁₉	1	1	yes	characteristic
160	C₈×M₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	8	1	no	—
165	C₂²×C₁₉	—	8	1	no	—
166	C₂²×C₁₉	—	8	1	no	—
167	C₂²×C₁₉	#390	1	1	yes	characteristic
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	8	1	no	—
172	C₄×C₁₉	—	8	1	no	—
173	C₄×C₁₉	—	8	1	no	—
174	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
175	C₄×C₁₉	#421	1	1	yes	characteristic
176	C₄×C₁₉	#443	1	2	yes	—
177	C₄×C₁₉	#471	1	2	yes	—
178	#533	C₂²×C₁₉	1	1	yes	characteristic
179	#532	C₂²×C₁₉	1	1	yes	characteristic
180	#498	C₄×C₁₉	1	1	yes	characteristic
181	#492	C₄×C₁₉	1	1	yes	characteristic
182	#487	C₄×C₁₉	1	1	yes	characteristic
183	#486	C₄×C₁₉	1	1	yes	characteristic
184	#465	—	2	1	no	—
185	#465	—	2	1	no	—
186	C₄×SD₁₆	—	2	1	no	—
187	C₄×SD₁₆	—	2	1	no	—
188	#431	C₂²×C₁₉	1	1	yes	characteristic
189	C₂×#108	—	2	1	no	—
190	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
191	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
192	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
194	#341	—	2	1	no	—
195	#330	—	2	2	no	—
196	#329	—	2	2	no	—
197	Q₈⋊C₈	—	2	2	no	—
198	D₄⋊C₈	—	2	2	no	—
199	#322	—	2	1	no	—
200	C₈²	—	2	1	no	—
201	C₂×C₁₉	—	4	1	no	—
202	C₂×C₁₉	—	16	1	no	—
203	C₂×C₁₉	#1974	1	1	yes	characteristic
204	C₂×C₁₉	#1977	1	1	yes	characteristic
205	C₂×C₁₉	#1980	1	1	yes	characteristic
206	C₂×#41	—	2	1	no	—
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×SD₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
214	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
215	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
216	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
217	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
218	#125	—	2	1	no	—
219	#124	—	2	1	no	—
220	C₂²⋊Q₈	—	2	1	no	—
221	C₂²⋊Q₈	—	2	1	no	—
222	#121	—	2	1	no	—
223	#121	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	2	1	no	—
228	#117	—	2	1	no	—
229	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
230	#108	—	2	4	no	—
231	#106	—	2	1	no	—
232	#106	—	2	1	no	—
233	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
234	#103	—	2	1	no	—
235	#103	—	2	1	no	—
236	#102	—	2	1	no	—
237	#102	—	2	1	no	—
238	C₂²⋊C₈	—	2	1	no	—
239	#97	—	2	1	no	—
240	#97	—	2	1	no	—
241	#97	D₄×C₁₉	1	2	yes	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
247	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
248	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
249	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
250	C₁₉	#9723	1	1	yes	characteristic, direct product factor, Sylow
251	#41	—	8	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
261	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
262	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
263	SD₁₆	—	4	2	no	—
264	SD₁₆	—	4	2	no	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
278	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
279	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
296	C₂³	—	4	1	no	—
297	C₂³	C₁₉×#124	1	1	yes	characteristic
298	Q₈	—	4	1	no	—
299	Q₈	—	4	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
316	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
317	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
318	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
319	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
320	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
321	C₈	—	2	4	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₈	C₁₉×#104	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	8	1	no	—
335	C₂²	C₁₉×#390	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	1	no	—
340	C₄	—	8	1	no	—
341	C₄	—	8	1	no	—
342	C₄	C₂×C₁₉×#104	1	1	yes	characteristic
343	C₄	C₁₉×#421	1	1	yes	characteristic
344	C₄	C₁₉×#443	1	2	yes	—
345	C₄	C₁₉×#471	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	16	1	no	—
348	C₂	C₁₉×#1974	1	1	yes	characteristic
349	C₂	C₁₉×#1977	1	1	yes	characteristic
350	C₂	C₁₉×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#121
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#390
prime radical	C₁₉×#532	C₂²
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. C₄×C₁₉×#108	C₂	semidirect
3. C₁₉×#1526	C₂	semidirect
5. C₁₉×#1485	C₂	semidirect
7. C₈×M₁₆×C₁₉	C₂	semidirect
120. #9723	C₁₉	direct (N × H)
155. C₄×#108	C₂×C₁₉	semidirect
156. #1526	C₂×C₁₉	semidirect
158. #1485	C₂×C₁₉	semidirect
160. C₈×M₁₆	C₂×C₁₉	semidirect
250. C₁₉	#9723	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×#3167	2
C₄×C₁₉×#108	2
C₁₉×#1526	2
C₁₉×#1525	2
C₁₉×#1485	2
C₁₉×#1484	2
C₈×M₁₆×C₁₉	2
#9723	19

Subgroups — C19×#9723 #18015682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9723 #18015682