Subgroups — C₁₉×#8726 #18014685

Order 4864.

514 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2285	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1296	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1296	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#117	—	2	1	no	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
11	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
12	C₁₉×#384	C₂²	1	1	yes	characteristic
13	C₁₉×#384	C₂²	1	1	yes	characteristic
14	C₁₉×#382	—	2	1	no	—
15	C₁₉×#380	—	2	1	no	—
16	C₁₉×#380	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
19	C₁₉×#336	—	2	1	no	—
20	C₁₉×#336	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₁₉×#117	—	4	2	no	—
27	C₁₉×#117	—	4	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	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	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	—	2	4	no	—
36	C₂×C₄²×C₁₉	—	2	2	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₁₉×C₄⋊C₈	—	2	4	no	—
42	C₁₉×C₄⋊C₈	—	2	4	no	—
43	C₁₉×C₂²⋊C₈	—	4	2	no	—
44	C₁₉×C₂²⋊C₈	—	4	2	no	—
45	C₁₉×#95	—	2	2	no	—
46	C₁₉×#95	—	2	2	no	—
47	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	—	2	2	no	—
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	2	no	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	D₈	1	2	yes	—
63	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	—	4	2	no	—
68	C₂×C₈×C₁₉	—	4	2	no	—
69	C₂×C₈×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	1	no	—
81	C₄⋊C₄×C₁₉	—	4	1	no	—
82	C₂²⋊C₄×C₁₉	—	4	4	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	4	4	no	—
87	C₄²×C₁₉	#41	1	4	yes	—
88	#8726	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	2	1	no	—
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	#102	1	2	yes	—
96	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	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	—	4	4	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	#102	1	2	yes	—
119	C₂×C₄×C₁₉	#117	1	2	yes	—
120	C₂×C₄×C₁₉	#123	1	2	yes	—
121	C₂×C₄×C₁₉	#123	1	2	yes	—
122	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#135	1	2	yes	—
125	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#137	1	2	yes	—
127	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
128	C₈×C₁₉	—	8	2	no	—
129	C₈×C₁₉	—	8	2	no	—
130	#2285	C₂×C₁₉	1	1	yes	characteristic
131	#1797	C₂×C₁₉	1	1	yes	characteristic
132	#1797	C₂×C₁₉	1	1	yes	characteristic
133	#1296	C₂×C₁₉	1	2	yes	complemented
134	#1296	C₂×C₁₉	1	2	yes	complemented
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	#381	1	1	yes	characteristic
143	C₂²×C₁₉	#413	1	1	yes	characteristic
144	C₂²×C₁₉	#413	1	1	yes	characteristic
145	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
146	C₂²×C₁₉	#416	1	1	yes	characteristic
147	C₂²×C₁₉	#417	1	1	yes	characteristic
148	C₂²×C₁₉	#419	1	1	yes	characteristic
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	8	2	no	—
154	C₄×C₁₉	—	8	2	no	—
155	C₄×C₁₉	#480	1	2	yes	—
156	C₄×C₁₉	#482	1	2	yes	—
157	C₄×C₁₉	#483	1	2	yes	—
158	C₄×C₁₉	#485	1	2	yes	—
159	C₂×#117	—	2	1	no	—
160	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
163	C₂×C₂²⋊C₈	—	2	1	no	—
164	C₂×C₂²⋊C₈	—	2	1	no	—
165	#384	C₂²×C₁₉	1	1	yes	characteristic
166	#384	C₂²×C₁₉	1	1	yes	characteristic
167	#382	—	2	1	no	—
168	#380	—	2	1	no	—
169	#380	C₄×C₁₉	1	2	yes	—
170	C₄×C₄⋊C₄	—	2	1	no	—
171	C₄×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
172	#336	—	2	1	no	—
173	#336	—	2	1	no	—
174	C₂×C₁₉	—	4	2	no	—
175	C₂×C₁₉	#1540	1	1	yes	characteristic
176	C₂×C₁₉	#1547	1	1	yes	characteristic
177	C₂×C₁₉	#1554	1	1	yes	characteristic
178	C₂×C₁₉	#1559	1	1	yes	characteristic
179	C₂×C₁₉	#1822	1	1	yes	characteristic
180	C₂×C₁₉	#1823	1	1	yes	characteristic
181	C₂×C₁₉	#1825	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
183	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
184	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	#117	—	4	2	no	—
188	#117	—	4	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	4	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₂²⋊C₈	—	4	2	no	—
205	C₂²⋊C₈	—	4	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	C₁₉	#8726	1	1	yes	characteristic, direct product factor, Sylow
209	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	D₈×C₁₉	1	2	yes	—
225	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	4	no	—
249	C₄²	#41×C₁₉	1	4	yes	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₁₉×#102	1	2	yes	—
257	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	C₁₉×#102	1	2	yes	—
280	C₂×C₄	C₁₉×#117	1	2	yes	—
281	C₂×C₄	C₁₉×#123	1	2	yes	—
282	C₂×C₄	C₁₉×#123	1	2	yes	—
283	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
285	C₂×C₄	C₁₉×#135	1	2	yes	—
286	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₂×C₄	C₁₉×#137	1	2	yes	—
288	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
289	C₈	—	8	2	no	—
290	C₈	—	8	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₁₉×#381	1	1	yes	characteristic
299	C₂²	C₁₉×#413	1	1	yes	characteristic
300	C₂²	C₁₉×#413	1	1	yes	characteristic
301	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
302	C₂²	C₁₉×#416	1	1	yes	characteristic
303	C₂²	C₁₉×#417	1	1	yes	characteristic
304	C₂²	C₁₉×#419	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₁₉×#480	1	2	yes	—
312	C₄	C₁₉×#482	1	2	yes	—
313	C₄	C₁₉×#483	1	2	yes	—
314	C₄	C₁₉×#485	1	2	yes	—
315	C₂	—	4	2	no	—
316	C₂	C₁₉×#1540	1	1	yes	characteristic
317	C₂	C₁₉×#1547	1	1	yes	characteristic
318	C₂	C₁₉×#1554	1	1	yes	characteristic
319	C₂	C₁₉×#1559	1	1	yes	characteristic
320	C₂	C₁₉×#1822	1	1	yes	characteristic
321	C₂	C₁₉×#1823	1	1	yes	characteristic
322	C₂	C₁₉×#1825	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
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
4. C₁₉×#1296	C₂	semidirect
5. C₁₉×#1296	C₂	semidirect
88. #8726	C₁₉	direct (N × H)
133. #1296	C₂×C₁₉	semidirect
134. #1296	C₂×C₁₉	semidirect
208. C₁₉	#8726	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×#2285	2
C₁₉×#1797	2
C₁₉×#1296	2
#8726	19

Subgroups — C19×#8726 #18014685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8726 #18014685