Subgroups — C₁₉×#18397 #18024356

Order 4864.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3501	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3230	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2542	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1614	C₂	1	8	yes	maximal, complemented
5	C₁₉×#1503	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1460	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#554	—	2	2	no	—
8	C₁₉×#554	—	2	4	no	—
9	D₄×Q₈×C₁₉	—	2	2	no	—
10	C₁₉×#544	—	2	1	no	—
11	C₁₉×#543	C₂²	1	4	yes	—
12	C₁₉×#533	C₂²	1	2	yes	—
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
15	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
16	C₁₉×#518	—	2	1	no	—
17	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#500	—	2	2	no	—
19	C₁₉×#487	—	2	2	no	—
20	C₁₉×#484	C₂²	1	8	yes	—
21	C₁₉×#477	—	2	4	no	—
22	C₁₉×#431	C₂²	1	2	yes	—
23	C₁₉×#399	C₂²	1	4	yes	—
24	C₁₉×#390	—	2	1	no	—
25	C₁₉×#384	—	2	1	no	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#328	C₂²	1	8	yes	—
28	C₁₉×#323	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₉	—	2	2	no	—
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₉	D₄	1	2	yes	—
33	C₁₉×C₄⋊Q₈	—	2	4	no	—
34	C₁₉×C₄⋊Q₈	—	2	2	no	—
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
37	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
38	C₁₉×#125	—	2	4	no	—
39	C₁₉×#125	—	2	2	no	—
40	C₁₉×C₂²⋊Q₈	—	2	4	no	—
41	C₁₉×C₂²⋊Q₈	—	2	4	no	—
42	C₁₉×C₂²⋊Q₈	—	2	8	no	—
43	C₁₉×C₂²⋊Q₈	—	2	2	no	—
44	C₁₉×C₂²⋊Q₈	—	2	4	no	—
45	C₁₉×C₂²⋊Q₈	—	2	8	no	—
46	C₁₉×C₂²⋊Q₈	—	4	8	no	—
47	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
48	C₄×Q₈×C₁₉	—	2	2	no	—
49	C₄×Q₈×C₁₉	D₄	1	4	yes	—
50	C₄×D₄×C₁₉	—	2	4	no	—
51	C₄×D₄×C₁₉	—	2	4	no	—
52	C₄×D₄×C₁₉	—	2	2	no	—
53	C₄×D₄×C₁₉	—	2	2	no	—
54	C₁₉×#117	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
60	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
64	C₂×C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
65	C₂×C₄²×C₁₉	—	2	1	no	—
66	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
67	C₁₉×#107	—	2	8	no	—
68	C₁₉×C₄⋊C₈	—	2	2	no	—
69	C₁₉×#103	—	2	8	no	—
70	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
71	C₄×C₈×C₁₉	—	2	2	no	—
72	C₁₉×#95	C₂³	1	2	yes	—
73	C₂⁴×C₁₉	C₂×D₄	1	2	yes	—
74	C₂×Q₈×C₁₉	—	2	4	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	4	4	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂×Q₈×C₁₉	—	8	4	no	—
80	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
81	C₂×D₄×C₁₉	—	2	2	no	—
82	C₂×D₄×C₁₉	—	2	2	no	—
83	C₂×D₄×C₁₉	Q₁₆	1	4	yes	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	—	2	4	no	—
86	C₂²×C₄×C₁₉	—	2	4	no	—
87	C₂²×C₄×C₁₉	—	2	2	no	—
88	C₂²×C₄×C₁₉	—	2	2	no	—
89	C₂²×C₄×C₁₉	—	2	2	no	—
90	C₂²×C₄×C₁₉	—	4	4	no	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	8	no	—
98	C₄⋊C₄×C₁₉	—	2	4	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	4	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	2	4	no	—
106	C₄⋊C₄×C₁₉	—	4	8	no	—
107	C₄⋊C₄×C₁₉	—	4	2	no	—
108	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₁₉	—	2	8	no	—
110	C₂²⋊C₄×C₁₉	—	2	4	no	—
111	C₂²⋊C₄×C₁₉	—	2	4	no	—
112	C₂²⋊C₄×C₁₉	—	4	8	no	—
113	C₄²×C₁₉	—	2	4	no	—
114	C₄²×C₁₉	—	4	2	no	—
115	C₄²×C₁₉	C₂×D₄	1	2	yes	—
116	C₄²×C₁₉	C₂×D₄	1	2	yes	—
117	#18397	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₁₉	—	2	2	no	—
119	C₂³×C₁₉	—	2	2	no	—
120	C₂³×C₁₉	—	4	2	no	—
121	C₂³×C₁₉	—	4	2	no	—
122	C₂³×C₁₉	C₂×Q₁₆	1	4	yes	—
123	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₁₉	—	4	4	no	—
125	Q₈×C₁₉	—	4	4	no	—
126	Q₈×C₁₉	—	8	2	no	—
127	Q₈×C₁₉	—	8	4	no	—
128	Q₈×C₁₉	—	8	2	no	—
129	D₄×C₁₉	—	2	8	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	2	4	no	—
133	C₂×C₄×C₁₉	—	2	4	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	4	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	4	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	8	2	no	—
143	C₂×C₄×C₁₉	—	8	4	no	—
144	C₂×C₄×C₁₉	—	8	4	no	—
145	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#120	1	2	yes	—
147	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
149	C₂×C₄×C₁₉	#137	1	2	yes	—
150	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
151	C₈×C₁₉	—	4	4	no	—
152	#3501	C₂×C₁₉	1	1	yes	characteristic
153	#3230	C₂×C₁₉	1	2	yes	complemented
154	#2542	C₂×C₁₉	1	1	yes	characteristic
155	#1614	C₂×C₁₉	1	8	yes	complemented
156	#1503	C₂×C₁₉	1	2	yes	complemented
157	#1460	C₂×C₁₉	1	1	yes	characteristic
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	4	no	—
161	C₂²×C₁₉	—	4	4	no	—
162	C₂²×C₁₉	—	4	2	no	—
163	C₂²×C₁₉	—	4	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
166	C₂²×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	4	4	no	—
171	C₄×C₁₉	—	4	2	no	—
172	C₄×C₁₉	—	4	4	no	—
173	C₄×C₁₉	—	4	4	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	#451	1	4	yes	—
176	C₄×C₁₉	#453	1	2	yes	—
177	#554	—	2	2	no	—
178	#554	—	2	4	no	—
179	D₄×Q₈	—	2	2	no	—
180	#544	—	2	1	no	—
181	#543	C₂²×C₁₉	1	4	yes	—
182	#533	C₂²×C₁₉	1	2	yes	—
183	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×C₂²⋊Q₈	—	2	2	no	—
185	C₂×C₂²⋊Q₈	—	2	2	no	—
186	#518	—	2	1	no	—
187	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
188	#500	—	2	2	no	—
189	#487	—	2	2	no	—
190	#484	C₂²×C₁₉	1	8	yes	—
191	#477	—	2	4	no	—
192	#431	C₂²×C₁₉	1	2	yes	—
193	#399	C₂²×C₁₉	1	4	yes	—
194	#390	—	2	1	no	—
195	#384	—	2	1	no	—
196	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
197	#328	C₂²×C₁₉	1	8	yes	—
198	#323	C₂²×C₁₉	1	4	yes	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	4	4	no	—
202	C₂×C₁₉	C₂×#451	1	1	yes	characteristic
203	C₂×C₁₉	C₂×#453	1	1	yes	characteristic
204	C₂×C₁₉	#3028	1	1	yes	characteristic
205	C₂²×Q₈	—	2	2	no	—
206	C₂²×Q₈	—	2	1	no	—
207	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
208	C₂³×C₄	D₄×C₁₉	1	2	yes	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
213	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
214	#125	—	2	4	no	—
215	#125	—	2	2	no	—
216	C₂²⋊Q₈	—	2	4	no	—
217	C₂²⋊Q₈	—	2	4	no	—
218	C₂²⋊Q₈	—	2	8	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	2	4	no	—
221	C₂²⋊Q₈	—	2	8	no	—
222	C₂²⋊Q₈	—	4	8	no	—
223	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	D₄×C₁₉	1	4	yes	—
226	C₄×D₄	—	2	4	no	—
227	C₄×D₄	—	2	4	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	#117	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	4	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	4	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
238	C₂×C₂²⋊C₄	—	2	4	no	—
239	C₂×C₂²⋊C₄	—	2	4	no	—
240	C₂×C₂²⋊C₄	C₂³×C₁₉	1	2	yes	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
243	#107	—	2	8	no	—
244	C₄⋊C₈	—	2	2	no	—
245	#103	—	2	8	no	—
246	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
247	C₄×C₈	—	2	2	no	—
248	#95	C₂³×C₁₉	1	2	yes	—
249	C₁₉	#18397	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	C₂×D₄×C₁₉	1	2	yes	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	8	4	no	—
257	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	Q₁₆×C₁₉	1	4	yes	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	4	4	no	—
268	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
272	C₂×C₈	—	2	4	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	4	8	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
286	C₂²⋊C₄	—	2	8	no	—
287	C₂²⋊C₄	—	2	4	no	—
288	C₂²⋊C₄	—	2	4	no	—
289	C₂²⋊C₄	—	4	8	no	—
290	C₄²	—	2	4	no	—
291	C₄²	—	4	2	no	—
292	C₄²	C₂×D₄×C₁₉	1	2	yes	—
293	C₄²	C₂×D₄×C₁₉	1	2	yes	—
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	C₂×Q₁₆×C₁₉	1	4	yes	—
299	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	Q₈	—	8	2	no	—
303	Q₈	—	8	4	no	—
304	Q₈	—	8	2	no	—
305	D₄	—	2	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
322	C₂×C₄	C₁₉×#120	1	2	yes	—
323	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
325	C₂×C₄	C₁₉×#137	1	2	yes	—
326	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
327	C₈	—	4	4	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
336	C₂²	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
337	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	4	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₄	C₁₉×#451	1	4	yes	—
346	C₄	C₁₉×#453	1	2	yes	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	4	no	—
350	C₂	C₂×C₁₉×#451	1	1	yes	characteristic
351	C₂	C₂×C₁₉×#453	1	1	yes	characteristic
352	C₂	C₁₉×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	D₄
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
2. C₁₉×#3230	C₂	semidirect
4. C₁₉×#1614	C₂	semidirect
5. C₁₉×#1503	C₂	semidirect
117. #18397	C₁₉	direct (N × H)
153. #3230	C₂×C₁₉	semidirect
155. #1614	C₂×C₁₉	semidirect
156. #1503	C₂×C₁₉	semidirect
249. C₁₉	#18397	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×#3501	2
C₁₉×#3230	2
C₁₉×#2542	2
C₁₉×#1614	2
C₁₉×#1503	2
C₁₉×#1460	2
#18397	19

Subgroups — C19×#18397 #18024356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18397 #18024356