Subgroups — C₁₉×#8397 #18014356

Order 4864.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#555	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1800	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1549	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1486	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1460	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#555	—	2	4	no	—
7	C₁₉×#555	—	2	4	no	—
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#124	—	2	2	no	—
10	C₂×C₁₉×#121	—	2	1	no	—
11	C₂×C₁₉×#121	C₄	1	2	yes	—
12	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#117	—	2	1	no	—
15	C₂×C₁₉×#117	C₄	1	2	yes	—
16	C₁₉×#431	C₂²	1	2	yes	—
17	C₁₉×#423	C₂²	1	2	yes	—
18	C₂×C₁₉×#100	—	2	1	no	—
19	C₁₉×#343	—	2	1	no	—
20	C₁₉×#332	—	2	4	no	—
21	C₁₉×D₄⋊C₈	—	2	4	no	—
22	C₁₉×#323	—	2	2	no	—
23	C₂²×Q₈×C₁₉	—	2	1	no	—
24	C₂²×D₄×C₁₉	—	2	2	no	—
25	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	—	2	1	no	—
28	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₁₉×C₄⋊Q₈	—	2	2	no	—
31	C₁₉×C₄⋊Q₈	D₄	1	4	yes	—
32	C₁₉×#124	—	2	16	no	—
33	C₁₉×#121	—	2	8	no	—
34	C₁₉×#121	—	2	4	no	—
35	C₁₉×#121	—	2	4	no	—
36	C₄×D₄×C₁₉	—	2	2	no	—
37	C₄×D₄×C₁₉	—	2	2	no	—
38	C₄×D₄×C₁₉	—	2	2	no	—
39	C₄×D₄×C₁₉	D₄	1	4	yes	—
40	C₁₉×#117	—	2	8	no	—
41	C₁₉×#117	—	2	4	no	—
42	C₁₉×#117	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
51	C₂×C₄²×C₁₉	—	2	2	no	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	—	2	2	no	—
54	C₁₉×C₄⋊C₈	—	2	2	no	—
55	C₁₉×C₄⋊C₈	—	2	2	no	—
56	C₁₉×#100	—	4	2	no	—
57	C₁₉×C₂²⋊C₈	—	2	2	no	—
58	C₄×C₈×C₁₉	—	2	2	no	—
59	C₂⁴×C₁₉	—	2	1	no	—
60	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂×Q₈×C₁₉	—	2	4	no	—
62	C₂×Q₈×C₁₉	—	8	2	no	—
63	C₂×D₄×C₁₉	—	2	8	no	—
64	C₂×D₄×C₁₉	—	2	1	no	—
65	C₂×D₄×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	2	2	no	—
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	—	2	1	no	—
69	C₂×D₄×C₁₉	—	4	2	no	—
70	C₂×D₄×C₁₉	—	8	4	no	—
71	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	—	2	4	no	—
77	C₂²×C₄×C₁₉	—	4	4	no	—
78	C₂²×C₄×C₁₉	—	4	2	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₁₉	—	4	2	no	—
86	M₁₆×C₁₉	—	4	2	no	—
87	C₂×C₈×C₁₉	—	4	2	no	—
88	C₂×C₈×C₁₉	—	4	2	no	—
89	C₄⋊C₄×C₁₉	—	2	1	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	C₂²⋊C₄	1	2	yes	—
94	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
95	C₂²⋊C₄×C₁₉	—	2	8	no	—
96	C₂²⋊C₄×C₁₉	—	2	8	no	—
97	C₂²⋊C₄×C₁₉	—	2	4	no	—
98	C₂²⋊C₄×C₁₉	—	2	4	no	—
99	C₄²×C₁₉	—	2	8	no	—
100	C₄²×C₁₉	C₂×D₄	1	2	yes	—
101	C₄²×C₁₉	C₂×D₄	1	2	yes	—
102	#8397	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	—	2	2	no	—
105	C₂³×C₁₉	—	2	2	no	—
106	C₂³×C₁₉	—	2	2	no	—
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	—	4	2	no	—
110	C₂³×C₁₉	—	4	2	no	—
111	C₂³×C₁₉	—	4	1	no	—
112	C₂³×C₁₉	—	4	1	no	—
113	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	Q₈×C₁₉	—	8	4	no	—
115	D₄×C₁₉	—	2	4	no	—
116	D₄×C₁₉	—	4	2	no	—
117	D₄×C₁₉	—	4	2	no	—
118	D₄×C₁₉	—	4	2	no	—
119	D₄×C₁₉	—	8	8	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	4	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	—	4	4	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#120	1	2	yes	—
136	C₂×C₄×C₁₉	#120	1	2	yes	—
137	C₂×C₄×C₁₉	#135	1	2	yes	—
138	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
140	C₈×C₁₉	—	8	2	no	—
141	C₈×C₁₉	—	8	2	no	—
142	C₂×#555	C₂×C₁₉	1	1	yes	characteristic
143	#1800	C₂×C₁₉	1	1	yes	characteristic, complemented
144	#1549	C₂×C₁₉	1	2	yes	complemented
145	#1486	C₂×C₁₉	1	2	yes	complemented
146	#1460	C₂×C₁₉	1	1	yes	characteristic, complemented
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	4	2	no	—
150	C₂²×C₁₉	—	4	2	no	—
151	C₂²×C₁₉	—	4	2	no	—
152	C₂²×C₁₉	—	4	2	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	—	4	2	no	—
155	C₂²×C₁₉	—	8	2	no	—
156	C₂²×C₁₉	—	8	2	no	—
157	C₂²×C₁₉	—	8	2	no	—
158	C₂²×C₁₉	—	8	2	no	—
159	C₂²×C₁₉	#379	1	1	yes	characteristic
160	C₂²×C₁₉	#416	1	1	yes	characteristic
161	C₂²×C₁₉	#417	1	1	yes	characteristic
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	4	4	no	—
165	C₄×C₁₉	—	4	2	no	—
166	C₄×C₁₉	—	4	2	no	—
167	C₄×C₁₉	#413	1	2	yes	—
168	C₄×C₁₉	#449	1	2	yes	—
169	C₄×C₁₉	#452	1	2	yes	—
170	#555	—	2	4	no	—
171	#555	—	2	4	no	—
172	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×#124	—	2	2	no	—
174	C₂×#121	—	2	1	no	—
175	C₂×#121	C₄×C₁₉	1	2	yes	—
176	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
178	C₂×#117	—	2	1	no	—
179	C₂×#117	C₄×C₁₉	1	2	yes	—
180	#431	C₂²×C₁₉	1	2	yes	—
181	#423	C₂²×C₁₉	1	2	yes	—
182	C₂×#100	—	2	1	no	—
183	#343	—	2	1	no	—
184	#332	—	2	4	no	—
185	D₄⋊C₈	—	2	4	no	—
186	#323	—	2	2	no	—
187	C₂×C₁₉	—	2	1	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	8	2	no	—
190	C₂×C₁₉	—	8	2	no	—
191	C₂×C₁₉	#1785	1	1	yes	characteristic
192	C₂×C₁₉	#1788	1	1	yes	characteristic
193	C₂×C₁₉	#1795	1	1	yes	characteristic
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
197	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
198	C₂³×C₄	—	2	1	no	—
199	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
200	C₂×M₁₆	—	2	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	D₄×C₁₉	1	4	yes	—
203	#124	—	2	16	no	—
204	#121	—	2	8	no	—
205	#121	—	2	4	no	—
206	#121	—	2	4	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	D₄×C₁₉	1	4	yes	—
211	#117	—	2	8	no	—
212	#117	—	2	4	no	—
213	#117	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
217	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	#100	—	4	2	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₄×C₈	—	2	2	no	—
230	C₁₉	#8397	1	1	yes	characteristic, direct product factor, Sylow
231	C₂⁴	—	2	1	no	—
232	C₂⁴	C₂²⋊C₄×C₁₉	1	2	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	8	2	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	8	4	no	—
243	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂²⋊C₄×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×D₄×C₁₉	1	1	yes	characteristic
257	M₁₆	—	4	2	no	—
258	M₁₆	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
266	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₄²	—	2	8	no	—
272	C₄²	C₂×D₄×C₁₉	1	2	yes	—
273	C₄²	C₂×D₄×C₁₉	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
285	Q₈	—	8	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	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	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
306	C₂×C₄	C₁₉×#120	1	2	yes	—
307	C₂×C₄	C₁₉×#120	1	2	yes	—
308	C₂×C₄	C₁₉×#135	1	2	yes	—
309	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
311	C₈	—	8	2	no	—
312	C₈	—	8	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₁₉×#379	1	1	yes	characteristic
326	C₂²	C₁₉×#416	1	1	yes	characteristic
327	C₂²	C₁₉×#417	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	4	no	—
333	C₄	C₁₉×#413	1	2	yes	—
334	C₄	C₁₉×#449	1	2	yes	—
335	C₄	C₁₉×#452	1	2	yes	—
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	8	2	no	—
339	C₂	—	8	2	no	—
340	C₂	C₁₉×#1785	1	1	yes	characteristic
341	C₂	C₁₉×#1788	1	1	yes	characteristic
342	C₂	C₁₉×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂×C₁₉×#555	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1800	C₂	semidirect
3. C₁₉×#1549	C₂	semidirect
4. C₁₉×#1486	C₂	semidirect
5. C₁₉×#1460	C₂	semidirect
102. #8397	C₁₉	direct (N × H)
143. #1800	C₂×C₁₉	semidirect
144. #1549	C₂×C₁₉	semidirect
145. #1486	C₂×C₁₉	semidirect
146. #1460	C₂×C₁₉	semidirect
230. C₁₉	#8397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#555	2
C₁₉×#1800	2
C₁₉×#1549	2
C₁₉×#1486	2
C₁₉×#1460	2
#8397	19

Subgroups — C19×#8397 #18014356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8397 #18014356