Subgroups — C₁₉×#9337 #18015296

Order 4864.

412 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3218	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2995	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1558	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1529	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1307	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1291	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₁₉×#529	C₂²	1	1	yes	characteristic
10	C₁₉×#497	—	2	1	no	—
11	C₁₉×#491	—	2	1	no	—
12	C₁₉×#462	—	2	1	no	—
13	C₁₉×#461	—	2	1	no	—
14	C₁₉×#451	C₄	1	2	yes	—
15	C₁₉×#448	C₄	1	2	yes	—
16	C₁₉×#446	—	2	1	no	—
17	C₁₉×#436	—	2	1	no	—
18	C₁₉×#435	—	2	1	no	—
19	C₁₉×#435	—	2	1	no	—
20	C₁₉×#433	—	2	1	no	—
21	C₁₉×#432	C₂²	1	1	yes	characteristic
22	C₁₉×#432	C₂²	1	1	yes	characteristic
23	C₁₉×#432	C₂²	1	1	yes	characteristic
24	C₁₉×#424	—	2	1	no	—
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#343	—	2	1	no	—
28	C₁₉×#343	—	2	1	no	—
29	C₁₉×#333	—	2	1	no	—
30	C₁₉×#333	—	2	1	no	—
31	C₁₉×#330	—	2	1	no	—
32	C₁₉×#329	—	2	1	no	—
33	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₁₉	—	2	2	no	—
35	C₂×SD₁₆×C₁₉	—	2	2	no	—
36	C₂×M₁₆×C₁₉	—	2	1	no	—
37	C₂×M₁₆×C₁₉	—	2	1	no	—
38	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	—	2	1	no	—
43	C₁₉×C₄⋊Q₈	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	—	2	1	no	—
45	C₁₉×#126	—	2	1	no	—
46	C₁₉×#125	D₄	1	1	yes	characteristic
47	C₁₉×#124	D₄	1	1	yes	characteristic
48	C₁₉×#123	—	2	1	no	—
49	C₁₉×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₁₉×#121	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
52	C₄×D₄×C₁₉	—	2	1	no	—
53	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
54	C₁₉×#117	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₄⋊C₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	—	2	1	no	—
63	C₁₉×C₄⋊C₈	—	2	1	no	—
64	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₉×#103	—	2	2	no	—
67	C₁₉×#103	—	2	2	no	—
68	C₁₉×C₂²⋊C₈	—	2	1	no	—
69	C₁₉×C₂²⋊C₈	—	2	1	no	—
70	C₁₉×C₂²⋊C₈	—	2	1	no	—
71	C₁₉×C₂²⋊C₈	—	2	1	no	—
72	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₁₉×#97	—	2	1	no	—
74	C₁₉×#97	—	2	1	no	—
75	C₁₉×#97	—	2	1	no	—
76	C₁₉×#97	—	2	1	no	—
77	C₁₉×#97	—	2	1	no	—
78	C₄×C₈×C₁₉	—	2	1	no	—
79	C₄×C₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	8	2	no	—
83	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×D₄×C₁₉	—	2	1	no	—
85	C₂×D₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	—	2	1	no	—
87	C₂²×C₄×C₁₉	—	2	2	no	—
88	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
91	Q₁₆×C₁₉	—	8	2	no	—
92	SD₁₆×C₁₉	—	8	2	no	—
93	M₁₆×C₁₉	—	2	2	no	—
94	M₁₆×C₁₉	—	4	1	no	—
95	M₁₆×C₁₉	—	4	1	no	—
96	M₁₆×C₁₉	—	4	1	no	—
97	M₁₆×C₁₉	—	8	1	no	—
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	—	2	1	no	—
100	C₂×C₈×C₁₉	—	2	1	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	2	1	no	—
103	C₂×C₈×C₁₉	—	4	1	no	—
104	C₂×C₈×C₁₉	—	4	1	no	—
105	C₂×C₈×C₁₉	—	8	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	C₄⋊C₄×C₁₉	—	4	1	no	—
113	C₄⋊C₄×C₁₉	—	4	1	no	—
114	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
116	C₂²⋊C₄×C₁₉	—	2	1	no	—
117	C₂²⋊C₄×C₁₉	—	2	1	no	—
118	C₂²⋊C₄×C₁₉	—	2	1	no	—
119	C₂²⋊C₄×C₁₉	—	2	1	no	—
120	C₂²⋊C₄×C₁₉	—	2	1	no	—
121	C₄²×C₁₉	—	2	1	no	—
122	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	#41	1	1	yes	characteristic
126	#9337	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₁₉	—	2	1	no	—
128	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	4	2	no	—
132	Q₈×C₁₉	—	8	2	no	—
133	D₄×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	2	1	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	2	no	—
145	C₂×C₄×C₁₉	—	8	2	no	—
146	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
152	C₈×C₁₉	—	4	2	no	—
153	C₈×C₁₉	—	4	2	no	—
154	C₈×C₁₉	—	4	1	no	—
155	C₈×C₁₉	—	4	1	no	—
156	C₈×C₁₉	—	4	1	no	—
157	C₈×C₁₉	—	4	1	no	—
158	C₈×C₁₉	—	8	1	no	—
159	#3218	C₂×C₁₉	1	1	yes	characteristic
160	#2995	C₂×C₁₉	1	1	yes	characteristic
161	#1997	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#1558	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1529	C₂×C₁₉	1	1	yes	characteristic
164	#1307	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1291	C₂×C₁₉	1	1	yes	characteristic, complemented
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	—	4	1	no	—
170	C₂²×C₁₉	—	4	1	no	—
171	C₂²×C₁₉	—	4	1	no	—
172	C₂²×C₁₉	#386	1	1	yes	characteristic
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	8	2	no	—
182	C₄×C₁₉	#455	1	1	yes	characteristic
183	C₄×C₁₉	#456	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
185	#529	C₂²×C₁₉	1	1	yes	characteristic
186	#497	—	2	1	no	—
187	#491	—	2	1	no	—
188	#462	—	2	1	no	—
189	#461	—	2	1	no	—
190	#451	C₄×C₁₉	1	2	yes	—
191	#448	C₄×C₁₉	1	2	yes	—
192	#446	—	2	1	no	—
193	#436	—	2	1	no	—
194	#435	—	2	1	no	—
195	#435	—	2	1	no	—
196	#433	—	2	1	no	—
197	#432	C₂²×C₁₉	1	1	yes	characteristic
198	#432	C₂²×C₁₉	1	1	yes	characteristic
199	#432	C₂²×C₁₉	1	1	yes	characteristic
200	#424	—	2	1	no	—
201	#423	C₂²×C₁₉	1	1	yes	characteristic
202	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
203	#343	—	2	1	no	—
204	#343	—	2	1	no	—
205	#333	—	2	1	no	—
206	#333	—	2	1	no	—
207	#330	—	2	1	no	—
208	#329	—	2	1	no	—
209	C₂×C₁₉	—	4	1	no	—
210	C₂×C₁₉	—	8	1	no	—
211	C₂×C₁₉	#1882	1	1	yes	characteristic
212	C₂×C₁₉	#1902	1	1	yes	characteristic
213	C₂×C₁₉	#1914	1	1	yes	characteristic
214	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
220	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
221	C₂²×C₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	#126	—	2	1	no	—
227	#125	D₄×C₁₉	1	1	yes	characteristic
228	#124	D₄×C₁₉	1	1	yes	characteristic
229	#123	—	2	1	no	—
230	C₂²⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
231	#121	C₂×C₄×C₁₉	1	1	yes	characteristic
232	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
235	#117	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
246	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
247	#103	—	2	2	no	—
248	#103	—	2	2	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	C₂²⋊C₈	—	2	1	no	—
253	C₂²⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
254	#97	—	2	1	no	—
255	#97	—	2	1	no	—
256	#97	—	2	1	no	—
257	#97	—	2	1	no	—
258	#97	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₁₉	#9337	1	1	yes	characteristic, direct product factor, Sylow
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	8	2	no	—
265	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
272	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
273	Q₁₆	—	8	2	no	—
274	SD₁₆	—	8	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	1	no	—
279	M₁₆	—	8	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	8	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
297	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
305	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
307	C₄²	#41×C₁₉	1	1	yes	characteristic
308	C₂³	—	2	1	no	—
309	C₂³	C₄×D₄×C₁₉	1	1	yes	characteristic
310	Q₈	—	4	1	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
328	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
330	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
331	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
332	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	8	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	C₁₉×#386	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	8	2	no	—
356	C₄	C₁₉×#455	1	1	yes	characteristic
357	C₄	C₁₉×#456	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	8	1	no	—
360	C₂	C₁₉×#1882	1	1	yes	characteristic
361	C₂	C₁₉×#1902	1	1	yes	characteristic
362	C₂	C₁₉×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1997	C₂	semidirect
4. C₁₉×#1558	C₂	semidirect
6. C₁₉×#1307	C₂	semidirect
7. C₁₉×#1291	C₂	semidirect
126. #9337	C₁₉	direct (N × H)
161. #1997	C₂×C₁₉	semidirect
162. #1558	C₂×C₁₉	semidirect
164. #1307	C₂×C₁₉	semidirect
165. #1291	C₂×C₁₉	semidirect
261. C₁₉	#9337	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×#3218	2
C₁₉×#2995	2
C₁₉×#1997	2
C₁₉×#1558	2
C₁₉×#1529	2
C₁₉×#1307	2
C₁₉×#1291	2
#9337	19

Subgroups — C19×#9337 #18015296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9337 #18015296