Subgroups — C₁₉×#17632 #18023591

Order 4864.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2983	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2356	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1518	C₂	1	2	yes	maximal
5	C₁₉×#1515	C₂	1	4	yes	maximal
6	C₁₉×#1506	C₂	1	4	yes	maximal
7	C₁₉×#1503	C₂	1	2	yes	maximal
8	Q₈²×C₁₉	C₄	1	4	yes	complemented
9	C₁₉×#557	—	2	4	no	—
10	C₁₉×#557	C₄	1	4	yes	complemented
11	C₁₉×#544	—	2	1	no	—
12	C₁₉×#544	—	2	2	no	—
13	C₁₉×#533	C₂²	1	2	yes	—
14	C₁₉×#533	C₂²	1	4	yes	—
15	C₁₉×#519	—	2	1	no	—
16	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#446	—	2	4	no	—
18	C₁₉×#432	C₂²	1	4	yes	—
19	C₁₉×#431	C₂²	1	2	yes	—
20	C₁₉×#391	—	2	1	no	—
21	C₁₉×#387	—	2	2	no	—
22	C₁₉×#383	C₂²	1	2	yes	—
23	C₁₉×#382	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#376	C₂²	1	2	yes	—
27	C₁₉×#328	C₂²	1	4	yes	—
28	C₁₉×#328	C₂²	1	4	yes	—
29	C₁₉×#328	C₂²	1	8	yes	—
30	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
31	C₁₉×C₄⋊Q₈	—	2	8	no	—
32	C₁₉×C₄⋊Q₈	—	2	4	no	—
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
35	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
37	C₁₉×#125	—	2	8	no	—
38	C₁₉×#125	—	2	4	no	—
39	C₁₉×#125	—	2	2	no	—
40	C₁₉×#125	—	2	4	no	—
41	C₁₉×C₂²⋊Q₈	—	2	4	no	—
42	C₁₉×C₂²⋊Q₈	—	2	2	no	—
43	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
44	C₄×Q₈×C₁₉	—	2	4	no	—
45	C₄×Q₈×C₁₉	—	2	4	no	—
46	C₄×Q₈×C₁₉	—	2	2	no	—
47	C₄×Q₈×C₁₉	—	2	4	no	—
48	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₁₉	C₂×C₄	1	8	yes	—
50	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₁₉	D₄	1	4	yes	—
52	C₁₉×#117	—	2	4	no	—
53	C₁₉×#117	—	2	2	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
58	C₂×C₄²×C₁₉	—	2	2	no	—
59	C₂×C₄²×C₁₉	—	2	1	no	—
60	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
62	C₁₉×C₄⋊C₈	—	2	2	no	—
63	C₁₉×C₄⋊C₈	—	2	4	no	—
64	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
65	C₁₉×#97	—	2	4	no	—
66	C₄×C₈×C₁₉	—	2	2	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₁₉×#95	—	2	2	no	—
69	C₁₉×#95	—	2	1	no	—
70	C₁₉×#95	—	2	2	no	—
71	C₁₉×#95	C₂³	1	1	yes	characteristic
72	C₁₉×#95	C₂³	1	2	yes	—
73	C₁₉×#95	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	4	no	—
76	C₂×Q₈×C₁₉	—	2	2	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
79	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₉	—	2	1	no	—
81	C₂²×C₄×C₁₉	—	2	2	no	—
82	C₂²×C₄×C₁₉	—	2	1	no	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₁₉	—	2	4	no	—
89	C₄⋊C₄×C₁₉	—	2	4	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	4	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	8	no	—
97	C₄⋊C₄×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	4	2	no	—
99	C₄⋊C₄×C₁₉	—	4	2	no	—
100	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
101	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
102	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₁₉	—	2	4	no	—
104	C₄²×C₁₉	—	2	8	no	—
105	C₄²×C₁₉	—	2	4	no	—
106	C₄²×C₁₉	—	4	4	no	—
107	C₄²×C₁₉	—	4	2	no	—
108	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
109	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₉	C₂×D₄	1	2	yes	—
111	C₄²×C₁₉	C₂×D₄	1	4	yes	—
112	#17632	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₉	—	2	8	no	—
115	Q₈×C₁₉	—	4	4	no	—
116	Q₈×C₁₉	—	8	4	no	—
117	Q₈×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₉	#137	1	2	yes	—
139	C₂×C₄×C₁₉	#137	1	2	yes	—
140	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₉	—	4	4	no	—
143	#3502	C₂×C₁₉	1	1	yes	characteristic
144	#2983	C₂×C₁₉	1	1	yes	characteristic
145	#2356	C₂×C₁₉	1	1	yes	characteristic
146	#1518	C₂×C₁₉	1	2	yes	—
147	#1515	C₂×C₁₉	1	4	yes	—
148	#1506	C₂×C₁₉	1	4	yes	—
149	#1503	C₂×C₁₉	1	2	yes	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	4	no	—
158	C₄×C₁₉	—	2	4	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	4	2	no	—
161	C₄×C₁₉	—	4	4	no	—
162	C₄×C₁₉	—	4	4	no	—
163	C₄×C₁₉	—	8	1	no	—
164	C₄×C₁₉	—	8	2	no	—
165	C₄×C₁₉	#419	1	4	yes	—
166	Q₈²	C₄×C₁₉	1	4	yes	complemented
167	#557	—	2	4	no	—
168	#557	C₄×C₁₉	1	4	yes	complemented
169	#544	—	2	1	no	—
170	#544	—	2	2	no	—
171	#533	C₂²×C₁₉	1	2	yes	—
172	#533	C₂²×C₁₉	1	4	yes	—
173	#519	—	2	1	no	—
174	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
175	#446	—	2	4	no	—
176	#432	C₂²×C₁₉	1	4	yes	—
177	#431	C₂²×C₁₉	1	2	yes	—
178	#391	—	2	1	no	—
179	#387	—	2	2	no	—
180	#383	C₂²×C₁₉	1	2	yes	—
181	#382	C₂²×C₁₉	1	1	yes	characteristic
182	C₄×C₄⋊C₄	—	2	1	no	—
183	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
184	#376	C₂²×C₁₉	1	2	yes	—
185	#328	C₂²×C₁₉	1	4	yes	—
186	#328	C₂²×C₁₉	1	4	yes	—
187	#328	C₂²×C₁₉	1	8	yes	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	#2884	1	1	yes	characteristic
191	C₂×C₁₉	C₂×#419	1	1	yes	characteristic
192	C₂×C₁₉	#2901	1	1	yes	characteristic
193	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	8	no	—
195	C₄⋊Q₈	—	2	4	no	—
196	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
197	C₄⋊Q₈	C₂×C₄×C₁₉	1	4	yes	—
198	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
199	C₄⋊Q₈	C₂×C₄×C₁₉	1	4	yes	—
200	#125	—	2	8	no	—
201	#125	—	2	4	no	—
202	#125	—	2	2	no	—
203	#125	—	2	4	no	—
204	C₂²⋊Q₈	—	2	4	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
207	C₄×Q₈	—	2	4	no	—
208	C₄×Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
212	C₄×Q₈	C₂×C₄×C₁₉	1	8	yes	—
213	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
214	C₄×Q₈	D₄×C₁₉	1	4	yes	—
215	#117	—	2	4	no	—
216	#117	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
220	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
224	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	4	no	—
227	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
228	#97	—	2	4	no	—
229	C₄×C₈	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	1	no	—
233	#95	—	2	2	no	—
234	#95	C₂³×C₁₉	1	1	yes	characteristic
235	#95	C₂³×C₁₉	1	2	yes	—
236	#95	C₂³×C₁₉	1	1	yes	characteristic
237	C₁₉	#17632	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₂²⋊C₄×C₁₉	1	4	yes	—
243	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
265	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
266	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₄²	—	2	8	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	4	4	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
273	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
274	C₄²	C₂×D₄×C₁₉	1	2	yes	—
275	C₄²	C₂×D₄×C₁₉	1	4	yes	—
276	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	8	4	no	—
280	Q₈	—	8	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
299	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
301	C₂×C₄	C₁₉×#137	1	2	yes	—
302	C₂×C₄	C₁₉×#137	1	2	yes	—
303	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
305	C₈	—	4	4	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
309	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
310	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	C₁₉×#419	1	4	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	C₁₉×#2884	1	1	yes	characteristic
325	C₂	C₂×C₁₉×#419	1	1	yes	characteristic
326	C₂	C₁₉×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. Q₈²×C₁₉	C₄	semidirect
10. C₁₉×#557	C₄	semidirect
112. #17632	C₁₉	direct (N × H)
166. Q₈²	C₄×C₁₉	semidirect
168. #557	C₄×C₁₉	semidirect
237. C₁₉	#17632	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×#3502	2
C₁₉×#2983	2
C₁₉×#2356	2
C₁₉×#1518	2
C₁₉×#1515	2
C₁₉×#1506	2
C₁₉×#1503	2
#17632	19

Subgroups — C19×#17632 #18023591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17632 #18023591