Subgroups — C₁₉×#11147 #18017106

Order 4864.

1206 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	C₂²×C₁₉×C₂²⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#336	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₁₉	C₂²	1	4	yes	—
6	C₂²×C₂²⋊C₄×C₁₉	C₄	1	1	yes	characteristic
7	C₂²×C₂²⋊C₄×C₁₉	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
9	C₂×C₁₉×C₂²⋊C₈	—	2	4	no	—
10	C₂×C₁₉×C₂²⋊C₈	—	2	8	no	—
11	C₄×C₂²⋊C₄×C₁₉	—	2	4	no	—
12	C₂×C₁₉×#95	C₄	1	2	yes	—
13	C₁₉×#336	—	2	8	no	—
14	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	—	2	4	no	—
16	C₂³×C₄×C₁₉	—	2	4	no	—
17	C₂³×C₄×C₁₉	—	2	4	no	—
18	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₉	D₄	1	4	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	—	2	4	no	—
24	C₂²×C₈×C₁₉	—	2	4	no	—
25	C₂²×C₈×C₁₉	—	2	4	no	—
26	C₂²×C₈×C₁₉	—	2	4	no	—
27	C₂²×C₈×C₁₉	—	2	4	no	—
28	C₂²×C₈×C₁₉	—	2	4	no	—
29	C₂²×C₈×C₁₉	D₄	1	8	yes	—
30	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₁₉	C₈	1	4	yes	complemented
35	C₂×C₄²×C₁₉	—	2	2	no	—
36	C₂×C₄²×C₁₉	—	2	2	no	—
37	C₂×C₄²×C₁₉	—	2	4	no	—
38	C₂×C₄²×C₁₉	—	2	4	no	—
39	C₁₉×C₂²⋊C₈	—	2	16	no	—
40	C₁₉×#95	—	2	4	no	—
41	C₂⁴×C₁₉	—	2	2	no	—
42	C₂⁴×C₁₉	—	2	4	no	—
43	C₂⁴×C₁₉	—	2	2	no	—
44	C₂⁴×C₁₉	—	2	4	no	—
45	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
46	C₂⁴×C₁₉	C₂×C₈	1	2	yes	—
47	C₂⁴×C₁₉	M₁₆	1	2	yes	—
48	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	2	4	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	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	8	no	—
61	C₂²×C₄×C₁₉	—	2	8	no	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	4	4	no	—
64	C₂²×C₄×C₁₉	—	4	4	no	—
65	C₂²×C₄×C₁₉	—	4	4	no	—
66	C₂²×C₄×C₁₉	—	4	4	no	—
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₉	#41	1	4	yes	—
73	C₂²×C₄×C₁₉	#41	1	4	yes	—
74	C₂×C₈×C₁₉	—	2	8	no	—
75	C₂×C₈×C₁₉	—	2	8	no	—
76	C₂×C₈×C₁₉	—	2	8	no	—
77	C₂×C₈×C₁₉	—	4	4	no	—
78	C₂×C₈×C₁₉	—	4	4	no	—
79	C₂×C₈×C₁₉	—	4	4	no	—
80	C₂×C₈×C₁₉	—	4	4	no	—
81	C₂²⋊C₄×C₁₉	—	2	8	no	—
82	C₂²⋊C₄×C₁₉	—	2	8	no	—
83	C₄²×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	—	4	4	no	—
85	C₄²×C₁₉	—	4	2	no	—
86	#11147	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	—	2	2	no	—
88	C₂³×C₁₉	—	2	4	no	—
89	C₂³×C₁₉	—	2	2	no	—
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	—	2	4	no	—
92	C₂³×C₁₉	—	2	4	no	—
93	C₂³×C₁₉	—	2	8	no	—
94	C₂³×C₁₉	—	2	8	no	—
95	C₂³×C₁₉	—	4	4	no	—
96	C₂³×C₁₉	—	4	4	no	—
97	C₂³×C₁₉	—	4	4	no	—
98	C₂³×C₁₉	—	4	4	no	—
99	C₂³×C₁₉	C₂²⋊C₈	1	4	yes	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	#117	1	2	yes	—
102	C₂³×C₁₉	C₄×D₄	1	4	yes	—
103	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
104	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
106	C₂³×C₁₉	#131	1	4	yes	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	8	no	—
112	C₂×C₄×C₁₉	—	2	8	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	8	no	—
121	C₂×C₄×C₁₉	—	4	8	no	—
122	C₂×C₄×C₁₉	#120	1	2	yes	—
123	C₂×C₄×C₁₉	#121	1	4	yes	—
124	C₂×C₄×C₁₉	#121	1	4	yes	—
125	C₂×C₄×C₁₉	#123	1	4	yes	—
126	C₂×C₄×C₁₉	#123	1	4	yes	—
127	C₂×C₄×C₁₉	#123	1	2	yes	—
128	C₂×C₄×C₁₉	#124	1	4	yes	—
129	C₂×C₄×C₁₉	#126	1	4	yes	—
130	C₈×C₁₉	—	4	8	no	—
131	C₂²×C₂²⋊C₈	C₂×C₁₉	1	2	yes	—
132	C₂×C₄×C₂²⋊C₄	C₂×C₁₉	1	1	yes	characteristic
133	C₂×#336	C₂×C₁₉	1	4	yes	complemented
134	C₂²×C₁₉	—	2	4	no	—
135	C₂²×C₁₉	—	2	8	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	—	4	4	no	—
138	C₂²×C₁₉	—	4	8	no	—
139	C₂²×C₁₉	—	4	8	no	—
140	C₂²×C₁₉	—	4	2	no	—
141	C₂²×C₁₉	#381	1	1	yes	characteristic
142	C₂²×C₁₉	#386	1	2	yes	—
143	C₂²×C₁₉	#388	1	4	yes	—
144	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₉	#407	1	1	yes	characteristic
146	C₂²×C₁₉	#408	1	2	yes	—
147	C₂²×C₁₉	#431	1	2	yes	—
148	C₂²×C₁₉	#432	1	2	yes	—
149	C₂²×C₁₉	#433	1	4	yes	—
150	C₂²×C₁₉	C₈×D₄	1	4	yes	—
151	C₂²×C₁₉	#435	1	4	yes	—
152	C₂²×C₁₉	#435	1	4	yes	—
153	C₂²×C₁₉	#436	1	4	yes	—
154	C₄×C₁₉	—	4	4	no	—
155	C₄×C₁₉	—	4	4	no	—
156	C₄×C₁₉	—	4	2	no	—
157	C₄×C₁₉	#394	1	4	yes	—
158	C₄×C₁₉	#397	1	4	yes	—
159	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
160	C₂³×C₈	C₂²×C₁₉	1	4	yes	—
161	C₂²×C₂²⋊C₄	C₄×C₁₉	1	1	yes	characteristic
162	C₂²×C₂²⋊C₄	C₄×C₁₉	1	1	yes	characteristic
163	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
164	C₂×C₂²⋊C₈	—	2	4	no	—
165	C₂×C₂²⋊C₈	—	2	8	no	—
166	C₄×C₂²⋊C₄	—	2	4	no	—
167	C₂×#95	C₄×C₁₉	1	2	yes	—
168	#336	—	2	8	no	—
169	C₂×C₁₉	—	4	4	no	—
170	C₂×C₁₉	#1440	1	1	yes	characteristic
171	C₂×C₁₉	#1817	1	1	yes	characteristic
172	C₂×C₁₉	#1818	1	1	yes	characteristic
173	C₂×C₁₉	#1869	1	2	yes	—
174	C₂×C₁₉	#1870	1	2	yes	—
175	C₂×C₁₉	#1923	1	4	yes	—
176	C₂×C₁₉	#1924	1	4	yes	—
177	C₂⁵	C₂×C₄×C₁₉	1	1	yes	characteristic
178	C₂³×C₄	—	2	4	no	—
179	C₂³×C₄	—	2	4	no	—
180	C₂³×C₄	—	2	4	no	—
181	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
184	C₂³×C₄	D₄×C₁₉	1	4	yes	—
185	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	D₄×C₁₉	1	8	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₂²⋊C₄	C₈×C₁₉	1	4	yes	complemented
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂²⋊C₈	—	2	16	no	—
203	#95	—	2	4	no	—
204	C₁₉	#11147	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	—	2	2	no	—
206	C₂⁴	—	2	4	no	—
207	C₂⁴	—	2	2	no	—
208	C₂⁴	—	2	4	no	—
209	C₂⁴	C₂²⋊C₄×C₁₉	1	2	yes	—
210	C₂⁴	C₂×C₈×C₁₉	1	2	yes	—
211	C₂⁴	M₁₆×C₁₉	1	2	yes	—
212	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
232	C₂²×C₄	C₂×C₈×C₁₉	1	4	yes	—
233	C₂²×C₄	M₁₆×C₁₉	1	4	yes	—
234	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
235	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
236	C₂²×C₄	#41×C₁₉	1	4	yes	—
237	C₂²×C₄	#41×C₁₉	1	4	yes	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	2	8	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂²⋊C₄	—	2	8	no	—
246	C₂²⋊C₄	—	2	8	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	4	no	—
249	C₄²	—	4	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	8	no	—
257	C₂³	—	2	8	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	C₁₉×C₂²⋊C₈	1	4	yes	—
263	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
264	C₂³	C₁₉×#117	1	2	yes	—
265	C₂³	C₄×D₄×C₁₉	1	4	yes	—
266	C₂³	C₂²×C₈×C₁₉	1	1	yes	characteristic
267	C₂³	C₂×M₁₆×C₁₉	1	1	yes	characteristic
268	C₂³	C₂×M₁₆×C₁₉	1	2	yes	—
269	C₂³	C₁₉×#131	1	4	yes	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	C₁₉×#120	1	2	yes	—
286	C₂×C₄	C₁₉×#121	1	4	yes	—
287	C₂×C₄	C₁₉×#121	1	4	yes	—
288	C₂×C₄	C₁₉×#123	1	4	yes	—
289	C₂×C₄	C₁₉×#123	1	4	yes	—
290	C₂×C₄	C₁₉×#123	1	2	yes	—
291	C₂×C₄	C₁₉×#124	1	4	yes	—
292	C₂×C₄	C₁₉×#126	1	4	yes	—
293	C₈	—	4	8	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	4	8	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	C₁₉×#381	1	1	yes	characteristic
302	C₂²	C₁₉×#386	1	2	yes	—
303	C₂²	C₁₉×#388	1	4	yes	—
304	C₂²	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
305	C₂²	C₁₉×#407	1	1	yes	characteristic
306	C₂²	C₁₉×#408	1	2	yes	—
307	C₂²	C₁₉×#431	1	2	yes	—
308	C₂²	C₁₉×#432	1	2	yes	—
309	C₂²	C₁₉×#433	1	4	yes	—
310	C₂²	C₈×D₄×C₁₉	1	4	yes	—
311	C₂²	C₁₉×#435	1	4	yes	—
312	C₂²	C₁₉×#435	1	4	yes	—
313	C₂²	C₁₉×#436	1	4	yes	—
314	C₄	—	4	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	C₁₉×#394	1	4	yes	—
318	C₄	C₁₉×#397	1	4	yes	—
319	C₂	—	4	4	no	—
320	C₂	C₁₉×#1440	1	1	yes	characteristic
321	C₂	C₁₉×#1817	1	1	yes	characteristic
322	C₂	C₁₉×#1818	1	1	yes	characteristic
323	C₂	C₁₉×#1869	1	2	yes	—
324	C₂	C₁₉×#1870	1	2	yes	—
325	C₂	C₁₉×#1923	1	4	yes	—
326	C₂	C₁₉×#1924	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₉	C₂³
commutator	C₂³	C₂²×C₈×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
socle	C₂⁴×C₁₉	C₂²×C₄
prime radical	C₂⁵×C₁₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₉×#336	C₂	semidirect
34. C₂×C₂²⋊C₄×C₁₉	C₈	semidirect
86. #11147	C₁₉	direct (N × H)
133. C₂×#336	C₂×C₁₉	semidirect
197. C₂×C₂²⋊C₄	C₈×C₁₉	semidirect
204. C₁₉	#11147	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×C₂²⋊C₈	2
C₂×C₄×C₂²⋊C₄×C₁₉	2
C₂×C₁₉×#336	2
#11147	19

Subgroups — C19×#11147 #18017106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11147 #18017106