Subgroups — C₁₉×#19118 #18025077

Order 4864.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3160	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3104	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1676	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1674	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1672	C₂	1	4	yes	maximal, complemented
7	C₁₉×#1564	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#1542	C₂	1	2	yes	maximal
9	C₁₉×#533	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#500	C₂²	1	1	yes	characteristic
12	C₁₉×#500	C₂²	1	1	yes	characteristic
13	C₁₉×#499	—	2	1	no	—
14	C₁₉×#499	—	2	1	no	—
15	C₁₉×#498	C₂²	1	2	yes	—
16	C₁₉×#484	—	2	2	no	—
17	C₁₉×#477	C₂²	1	4	yes	—
18	C₁₉×#462	C₂²	1	4	yes	—
19	C₁₉×#451	—	2	2	no	—
20	C₈×Q₈×C₁₉	—	2	1	no	—
21	C₈×Q₈×C₁₉	—	2	1	no	—
22	C₈×Q₈×C₁₉	C₂²	1	2	yes	—
23	C₈×Q₈×C₁₉	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₁₉	—	2	4	no	—
25	C₁₉×#431	C₂²	1	2	yes	—
26	C₁₉×#427	—	2	1	no	—
27	C₁₉×#427	—	2	1	no	—
28	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
29	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₉	—	2	1	no	—
31	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
32	C₁₉×#335	C₂²	1	2	yes	—
33	C₁₉×#334	C₂²	1	2	yes	—
34	C₁₉×#333	C₂²	1	4	yes	—
35	C₁₉×#332	C₂²	1	4	yes	—
36	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₁₉	—	2	4	no	—
38	C₂²×C₈×C₁₉	—	2	1	no	—
39	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
43	C₁₉×#125	—	2	2	no	—
44	C₁₉×C₂²⋊Q₈	—	2	2	no	—
45	C₁₉×C₂²⋊Q₈	—	2	2	no	—
46	C₄×Q₈×C₁₉	—	2	1	no	—
47	C₄×Q₈×C₁₉	—	2	1	no	—
48	C₄×Q₈×C₁₉	—	2	2	no	—
49	C₄×Q₈×C₁₉	—	2	2	no	—
50	C₄×Q₈×C₁₉	D₄	1	2	yes	—
51	C₄×Q₈×C₁₉	C₂³	1	2	yes	—
52	C₁₉×#117	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₉	—	2	1	no	—
56	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×#107	—	2	1	no	—
58	C₁₉×#107	—	2	1	no	—
59	C₁₉×#107	—	2	1	no	—
60	C₁₉×#107	—	2	1	no	—
61	C₁₉×#107	—	2	4	no	—
62	C₁₉×#106	—	2	2	no	—
63	C₁₉×#106	—	2	2	no	—
64	C₁₉×C₄⋊C₈	—	2	2	no	—
65	C₁₉×C₄⋊C₈	—	2	2	no	—
66	C₁₉×C₄⋊C₈	—	2	1	no	—
67	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
69	C₁₉×#103	—	2	4	no	—
70	C₁₉×#103	—	2	4	no	—
71	C₁₉×C₂²⋊C₈	—	2	2	no	—
72	C₄×C₈×C₁₉	—	2	2	no	—
73	C₄×C₈×C₁₉	—	2	2	no	—
74	C₄×C₈×C₁₉	—	2	2	no	—
75	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₉	—	2	4	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	2	2	no	—
83	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₉	—	2	1	no	—
85	C₂²×C₄×C₁₉	—	2	1	no	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
89	Q₁₆×C₁₉	—	4	8	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	2	4	no	—
93	C₂×C₈×C₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	Q₁₆	1	4	yes	complemented
95	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	1	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	2	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	4	2	no	—
107	C₄⋊C₄×C₁₉	—	4	2	no	—
108	C₄⋊C₄×C₁₉	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₁₉	—	4	2	no	—
110	C₄²×C₁₉	—	2	2	no	—
111	C₄²×C₁₉	—	2	2	no	—
112	C₄²×C₁₉	—	4	2	no	—
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	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
117	#19118	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₁₉	#121	1	1	yes	characteristic
119	Q₈×C₁₉	—	2	4	no	—
120	Q₈×C₁₉	—	2	2	no	—
121	Q₈×C₁₉	—	4	4	no	—
122	Q₈×C₁₉	—	8	2	no	—
123	Q₈×C₁₉	#135	1	4	yes	—
124	C₂×C₄×C₁₉	—	2	4	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₁₉	#137	1	2	yes	—
137	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₈×C₁₉	—	4	4	no	—
143	C₈×C₁₉	—	4	2	no	—
144	C₈×C₁₉	C₂×Q₁₆	1	2	yes	—
145	C₈×C₁₉	C₂×Q₁₆	1	2	yes	—
146	#3160	C₂×C₁₉	1	1	yes	characteristic
147	#3104	C₂×C₁₉	1	2	yes	—
148	C₂×C₈×Q₈	C₂×C₁₉	1	1	yes	characteristic
149	#1676	C₂×C₁₉	1	2	yes	complemented
150	#1674	C₂×C₁₉	1	2	yes	complemented
151	#1672	C₂×C₁₉	1	4	yes	complemented
152	#1564	C₂×C₁₉	1	1	yes	characteristic
153	#1542	C₂×C₁₉	1	2	yes	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	#462	1	2	yes	—
157	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
158	C₄×C₁₉	—	2	4	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	4	2	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	—	8	2	no	—
165	C₄×C₁₉	#462	1	2	yes	—
166	C₄×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
167	C₄×C₁₉	C₂×#135	1	2	yes	—
168	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
169	#533	C₂²×C₁₉	1	2	yes	—
170	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
171	#500	C₂²×C₁₉	1	1	yes	characteristic
172	#500	C₂²×C₁₉	1	1	yes	characteristic
173	#499	—	2	1	no	—
174	#499	—	2	1	no	—
175	#498	C₂²×C₁₉	1	2	yes	—
176	#484	—	2	2	no	—
177	#477	C₂²×C₁₉	1	4	yes	—
178	#462	C₂²×C₁₉	1	4	yes	—
179	#451	—	2	2	no	—
180	C₈×Q₈	—	2	1	no	—
181	C₈×Q₈	—	2	1	no	—
182	C₈×Q₈	C₂²×C₁₉	1	2	yes	—
183	C₈×Q₈	C₂²×C₁₉	1	2	yes	—
184	C₄×Q₁₆	—	2	4	no	—
185	#431	C₂²×C₁₉	1	2	yes	—
186	#427	—	2	1	no	—
187	#427	—	2	1	no	—
188	C₂×C₄⋊C₈	—	2	1	no	—
189	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×C₄×C₈	—	2	1	no	—
191	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
192	#335	C₂²×C₁₉	1	2	yes	—
193	#334	C₂²×C₁₉	1	2	yes	—
194	#333	C₂²×C₁₉	1	4	yes	—
195	#332	C₂²×C₁₉	1	4	yes	—
196	C₂×C₁₉	—	2	2	no	—
197	C₂×C₁₉	C₂×#462	1	1	yes	characteristic
198	C₂×C₁₉	#3050	1	1	yes	characteristic
199	C₂×C₁₉	#3067	1	1	yes	characteristic
200	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
201	C₂×Q₁₆	—	2	4	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
204	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
205	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
207	#125	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	D₄×C₁₉	1	2	yes	—
215	C₄×Q₈	C₂³×C₁₉	1	2	yes	—
216	#117	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	#107	—	2	4	no	—
226	#106	—	2	2	no	—
227	#106	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
232	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
233	#103	—	2	4	no	—
234	#103	—	2	4	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
240	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
243	C₁₉	#19118	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
253	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
254	Q₁₆	—	4	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	Q₁₆×C₁₉	1	4	yes	complemented
260	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₂×D₄×C₁₉	1	2	yes	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
280	C₄²	#41×C₁₉	1	1	yes	characteristic
281	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
282	C₂³	C₁₉×#121	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	C₁₉×#135	1	4	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
297	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
298	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
300	C₂×C₄	C₁₉×#137	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	4	no	—
307	C₈	—	4	2	no	—
308	C₈	C₂×Q₁₆×C₁₉	1	2	yes	—
309	C₈	C₂×Q₁₆×C₁₉	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	C₁₉×#462	1	2	yes	—
313	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₁₉×#462	1	2	yes	—
322	C₄	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
323	C₄	C₂×C₁₉×#135	1	2	yes	—
324	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×C₁₉×#462	1	1	yes	characteristic
327	C₂	C₁₉×#3050	1	1	yes	characteristic
328	C₂	C₁₉×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×#121
prime radical	C₂³×C₁₉	#121
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
4. C₁₉×#1676	C₂	semidirect
5. C₁₉×#1674	C₂	semidirect
6. C₁₉×#1672	C₂	semidirect
94. C₂×C₈×C₁₉	Q₁₆	semidirect
117. #19118	C₁₉	direct (N × H)
149. #1676	C₂×C₁₉	semidirect
150. #1674	C₂×C₁₉	semidirect
151. #1672	C₂×C₁₉	semidirect
243. C₁₉	#19118	direct (N × H)
259. C₂×C₈	Q₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3160	2
C₁₉×#3104	2
C₂×C₈×Q₈×C₁₉	2
C₁₉×#1676	2
C₁₉×#1674	2
C₁₉×#1672	2
C₁₉×#1564	2
C₁₉×#1542	2
#19118	19

Subgroups — C19×#19118 #18025077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19118 #18025077