Subgroups — C₁₉×#9912 #18015871

Order 4864.

592 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1434	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1294	C₂	1	4	yes	maximal
5	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
6	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
10	C₁₉×#391	—	2	1	no	—
11	C₁₉×#389	—	2	1	no	—
12	C₁₉×#389	C₄	1	2	yes	complemented
13	C₁₉×#384	C₄	1	1	yes	characteristic, complemented
14	C₁₉×#380	—	2	1	no	—
15	C₁₉×#380	C₂²	1	2	yes	—
16	C₂×C₁₉×#95	—	2	1	no	—
17	C₂×C₁₉×#95	—	2	1	no	—
18	C₂×C₁₉×#95	C₂²	1	2	yes	—
19	C₁₉×#336	—	2	1	no	—
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	—	2	2	no	—
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₁₉×C₄⋊Q₈	—	4	2	no	—
27	C₁₉×C₂²⋊Q₈	—	4	2	no	—
28	C₁₉×C₂²⋊Q₈	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₄²×C₁₉	—	4	2	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₁₉×C₄⋊C₈	—	4	2	no	—
42	C₁₉×C₂²⋊C₈	—	2	4	no	—
43	C₁₉×#95	—	2	2	no	—
44	C₁₉×#95	—	2	2	no	—
45	C₁₉×#95	—	2	2	no	—
46	C₁₉×#95	—	4	2	no	—
47	C₁₉×#95	—	4	2	no	—
48	C₁₉×#95	Q₈	1	4	yes	complemented
49	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₉	—	4	2	no	—
51	C₂×Q₈×C₁₉	—	4	2	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	—	4	2	no	—
66	C₂²×C₄×C₁₉	—	4	2	no	—
67	C₂²×C₄×C₁₉	—	4	2	no	—
68	C₂²×C₄×C₁₉	—	4	2	no	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	D₈	1	2	yes	—
74	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₉	#41	1	2	yes	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₂×C₈×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄⋊C₄×C₁₉	—	4	2	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₄⋊C₄×C₁₉	—	4	4	no	—
88	C₂²⋊C₄×C₁₉	—	4	2	no	—
89	C₂²⋊C₄×C₁₉	—	4	2	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	2	no	—
92	C₄²×C₁₉	—	4	1	no	—
93	C₄²×C₁₉	—	4	1	no	—
94	#9912	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
102	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂³×C₁₉	#137	1	2	yes	—
104	Q₈×C₁₉	—	8	4	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	4	no	—
132	C₂×C₄×C₁₉	—	8	2	no	—
133	C₂×C₄×C₁₉	—	8	2	no	—
134	C₂×C₄×C₁₉	—	8	2	no	—
135	C₂×C₄×C₁₉	—	8	2	no	—
136	C₂×C₄×C₁₉	#99	1	2	yes	—
137	C₂×C₄×C₁₉	#102	1	2	yes	—
138	C₂×C₄×C₁₉	#102	1	2	yes	—
139	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
142	C₈×C₁₉	—	8	2	no	—
143	#2857	C₂×C₁₉	1	1	yes	characteristic
144	#1797	C₂×C₁₉	1	1	yes	characteristic
145	#1434	C₂×C₁₉	1	1	yes	characteristic
146	#1294	C₂×C₁₉	1	4	yes	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	#391	1	1	yes	characteristic
156	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
157	C₂²×C₁₉	#410	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
159	C₂²×C₁₉	#419	1	1	yes	characteristic
160	C₂²×C₁₉	#421	1	1	yes	characteristic
161	C₂²×C₁₉	#421	1	1	yes	characteristic
162	C₂²×C₁₉	#474	1	2	yes	—
163	C₂²×C₁₉	#475	1	2	yes	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	—	8	4	no	—
172	C₄×C₁₉	—	8	2	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₂×C₄⋊Q₈	C₄×C₁₉	1	1	yes	characteristic, complemented
175	C₂×C₂²⋊Q₈	—	2	1	no	—
176	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×C₄⋊C₈	—	2	1	no	—
178	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
179	#391	—	2	1	no	—
180	#389	—	2	1	no	—
181	#389	C₄×C₁₉	1	2	yes	complemented
182	#384	C₄×C₁₉	1	1	yes	characteristic, complemented
183	#380	—	2	1	no	—
184	#380	C₂²×C₁₉	1	2	yes	—
185	C₂×#95	—	2	1	no	—
186	C₂×#95	—	2	1	no	—
187	C₂×#95	C₂²×C₁₉	1	2	yes	—
188	#336	—	2	1	no	—
189	C₂×C₁₉	—	2	2	no	—
190	C₂×C₁₉	—	2	2	no	—
191	C₂×C₁₉	#1511	1	1	yes	characteristic
192	C₂×C₁₉	#1515	1	1	yes	characteristic
193	C₂×C₁₉	#1517	1	1	yes	characteristic
194	C₂×C₁₉	#1519	1	1	yes	characteristic
195	C₂×C₁₉	#1988	1	1	yes	characteristic
196	C₂×C₁₉	#1995	1	1	yes	characteristic
197	C₂×C₁₉	#1996	1	1	yes	characteristic
198	C₂²×Q₈	—	2	1	no	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
203	C₂²×C₈	—	2	2	no	—
204	C₄⋊Q₈	—	4	2	no	—
205	C₂²⋊Q₈	—	4	2	no	—
206	C₂²⋊Q₈	—	4	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	4	2	no	—
215	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
219	C₄⋊C₈	—	4	2	no	—
220	C₂²⋊C₈	—	2	4	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	4	2	no	—
225	#95	—	4	2	no	—
226	#95	Q₈×C₁₉	1	4	yes	complemented
227	C₁₉	#9912	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₁₉	1	2	yes	—
253	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
257	C₂²×C₄	#41×C₁₉	1	2	yes	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
278	C₂³	C₁₉×C₂²⋊Q₈	1	2	yes	—
279	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
280	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
281	C₂³	C₁₉×#137	1	2	yes	—
282	Q₈	—	8	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	C₁₉×#99	1	2	yes	—
315	C₂×C₄	C₁₉×#102	1	2	yes	—
316	C₂×C₄	C₁₉×#102	1	2	yes	—
317	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
318	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
319	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
320	C₈	—	8	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	C₁₉×#391	1	1	yes	characteristic
330	C₂²	C₂×C₁₉×#99	1	1	yes	characteristic
331	C₂²	C₁₉×#410	1	1	yes	characteristic
332	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
333	C₂²	C₁₉×#419	1	1	yes	characteristic
334	C₂²	C₁₉×#421	1	1	yes	characteristic
335	C₂²	C₁₉×#421	1	1	yes	characteristic
336	C₂²	C₁₉×#474	1	2	yes	—
337	C₂²	C₁₉×#475	1	2	yes	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	8	4	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	C₁₉×#1511	1	1	yes	characteristic
351	C₂	C₁₉×#1515	1	1	yes	characteristic
352	C₂	C₁₉×#1517	1	1	yes	characteristic
353	C₂	C₁₉×#1519	1	1	yes	characteristic
354	C₂	C₁₉×#1988	1	1	yes	characteristic
355	C₂	C₁₉×#1995	1	1	yes	characteristic
356	C₂	C₁₉×#1996	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
12. C₁₉×#389	C₄	semidirect
13. C₁₉×#384	C₄	semidirect
48. C₁₉×#95	Q₈	semidirect
94. #9912	C₁₉	direct (N × H)
174. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
181. #389	C₄×C₁₉	semidirect
182. #384	C₄×C₁₉	semidirect
226. #95	Q₈×C₁₉	semidirect
227. C₁₉	#9912	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×#2857	2
C₁₉×#1797	2
C₁₉×#1434	2
C₁₉×#1294	2
#9912	19

Subgroups — C19×#9912 #18015871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9912 #18015871