Subgroups — C₁₉×#9300 #18015259

Order 4864.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3226	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2327	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1963	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1517	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1274	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#532	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
11	C₁₉×#500	C₄	1	2	yes	complemented
12	C₁₉×#486	C₄	1	2	yes	complemented
13	C₁₉×#480	—	2	2	no	—
14	C₁₉×#474	—	2	2	no	—
15	C₁₉×#431	C₂²	1	1	yes	characteristic
16	C₁₉×#431	C₂²	1	1	yes	characteristic
17	C₁₉×#423	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#104	—	2	1	no	—
19	C₂×C₁₉×#104	—	2	1	no	—
20	C₁₉×#391	—	2	1	no	—
21	C₁₉×#387	—	2	2	no	—
22	C₁₉×#387	—	2	2	no	—
23	C₁₉×#382	C₂²	1	1	yes	characteristic
24	C₁₉×#376	C₂²	1	1	yes	characteristic
25	C₁₉×#332	—	2	2	no	—
26	C₁₉×#327	—	2	2	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
35	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
36	C₁₉×#124	—	2	1	no	—
37	C₁₉×#121	—	2	1	no	—
38	C₁₉×#121	—	2	1	no	—
39	C₄×Q₈×C₁₉	—	4	2	no	—
40	C₄×Q₈×C₁₉	—	4	2	no	—
41	C₄×D₄×C₁₉	—	2	1	no	—
42	C₄×D₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
47	C₂×C₄²×C₁₉	—	2	2	no	—
48	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#107	—	2	4	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×#104	—	4	2	no	—
57	C₁₉×#104	—	4	2	no	—
58	C₁₉×#102	—	2	4	no	—
59	C₁₉×C₂²⋊C₈	—	2	1	no	—
60	C₁₉×C₂²⋊C₈	—	2	1	no	—
61	C₄×C₈×C₁₉	—	2	1	no	—
62	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
63	C₁₉×#95	—	2	4	no	—
64	C₁₉×#95	—	2	1	no	—
65	C₁₉×#95	—	2	1	no	—
66	C₁₉×#95	—	2	1	no	—
67	C₁₉×#95	D₄	1	2	yes	—
68	#41×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	—	2	1	no	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	—	2	2	no	—
86	C₂²×C₄×C₁₉	—	2	1	no	—
87	C₂²×C₄×C₁₉	—	4	1	no	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	M₁₆×C₁₉	—	4	1	no	—
92	M₁₆×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	4	4	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
105	C₂²⋊C₄×C₁₉	—	4	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₄²×C₁₉	—	2	1	no	—
108	C₄²×C₁₉	—	2	1	no	—
109	C₄²×C₁₉	—	2	1	no	—
110	C₄²×C₁₉	—	4	4	no	—
111	C₄²×C₁₉	D₈	1	2	yes	—
112	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₉	#41	1	1	yes	characteristic
115	C₄²×C₁₉	#41	1	1	yes	characteristic
116	#9300	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₁₉	—	4	1	no	—
118	C₂³×C₁₉	#120	1	1	yes	characteristic
119	Q₈×C₁₉	—	4	2	no	—
120	Q₈×C₁₉	—	4	2	no	—
121	Q₈×C₁₉	—	4	2	no	—
122	Q₈×C₁₉	—	4	2	no	—
123	Q₈×C₁₉	—	4	1	no	—
124	Q₈×C₁₉	—	4	1	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	1	no	—
127	D₄×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	8	1	no	—
129	D₄×C₁₉	—	8	1	no	—
130	D₄×C₁₉	—	8	1	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	2	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	8	1	no	—
147	C₂×C₄×C₁₉	—	8	1	no	—
148	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
154	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
156	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₈×C₁₉	—	4	2	no	—
159	C₈×C₁₉	—	8	1	no	—
160	C₈×C₁₉	—	8	1	no	—
161	#3226	C₂×C₁₉	1	1	yes	characteristic
162	#2327	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1963	C₂×C₁₉	1	1	yes	characteristic
164	#1546	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1517	C₂×C₁₉	1	1	yes	characteristic
166	#1286	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#1274	C₂×C₁₉	1	1	yes	characteristic, complemented
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	8	1	no	—
171	C₂²×C₁₉	—	8	1	no	—
172	C₂²×C₁₉	—	8	1	no	—
173	C₂²×C₁₉	#386	1	1	yes	characteristic
174	C₂²×C₁₉	#447	1	1	yes	characteristic
175	C₂²×C₁₉	#449	1	1	yes	characteristic
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	4	2	no	—
181	C₄×C₁₉	—	4	4	no	—
182	C₄×C₁₉	—	8	2	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
186	C₄×C₁₉	#442	1	1	yes	characteristic
187	C₄×C₁₉	#459	1	1	yes	characteristic
188	C₄×C₁₉	#463	1	1	yes	characteristic
189	#532	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×C₄×Q₈	—	2	1	no	—
192	#500	C₄×C₁₉	1	2	yes	complemented
193	#486	C₄×C₁₉	1	2	yes	complemented
194	#480	—	2	2	no	—
195	#474	—	2	2	no	—
196	#431	C₂²×C₁₉	1	1	yes	characteristic
197	#431	C₂²×C₁₉	1	1	yes	characteristic
198	#423	C₂²×C₁₉	1	1	yes	characteristic
199	C₂×#104	—	2	1	no	—
200	C₂×#104	—	2	1	no	—
201	#391	—	2	1	no	—
202	#387	—	2	2	no	—
203	#387	—	2	2	no	—
204	#382	C₂²×C₁₉	1	1	yes	characteristic
205	#376	C₂²×C₁₉	1	1	yes	characteristic
206	#332	—	2	2	no	—
207	#327	—	2	2	no	—
208	C₂×C₁₉	—	2	1	no	—
209	C₂×C₁₉	—	2	1	no	—
210	C₂×C₁₉	—	16	1	no	—
211	C₂×C₁₉	#1879	1	1	yes	characteristic
212	C₂×C₁₉	#1905	1	1	yes	characteristic
213	C₂×C₁₉	#1912	1	1	yes	characteristic
214	C₂×#41	—	2	1	no	—
215	C₂²×Q₈	—	2	1	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
220	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
221	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
222	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
223	#124	—	2	1	no	—
224	#121	—	2	1	no	—
225	#121	—	2	1	no	—
226	C₄×Q₈	—	4	2	no	—
227	C₄×Q₈	—	4	2	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
236	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
238	#107	—	2	4	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	#104	—	4	2	no	—
244	#104	—	4	2	no	—
245	#102	—	2	4	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
250	#95	—	2	4	no	—
251	#95	—	2	1	no	—
252	#95	—	2	1	no	—
253	#95	—	2	1	no	—
254	#95	D₄×C₁₉	1	2	yes	—
255	C₁₉	#9300	1	1	yes	characteristic, direct product factor, Sylow
256	#41	—	4	1	no	—
257	#41	—	4	1	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
279	M₁₆	—	4	1	no	—
280	M₁₆	—	4	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	4	4	no	—
299	C₄²	D₈×C₁₉	1	2	yes	—
300	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
302	C₄²	#41×C₁₉	1	1	yes	characteristic
303	C₄²	#41×C₁₉	1	1	yes	characteristic
304	C₂³	—	4	1	no	—
305	C₂³	C₁₉×#120	1	1	yes	characteristic
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	1	no	—
316	D₄	—	8	1	no	—
317	D₄	—	8	1	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	8	1	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
336	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
337	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
338	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
339	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
340	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
341	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
342	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
343	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
344	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
345	C₈	—	4	2	no	—
346	C₈	—	8	1	no	—
347	C₈	—	8	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	8	1	no	—
351	C₂²	—	8	1	no	—
352	C₂²	—	8	1	no	—
353	C₂²	C₁₉×#386	1	1	yes	characteristic
354	C₂²	C₁₉×#447	1	1	yes	characteristic
355	C₂²	C₁₉×#449	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	4	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
366	C₄	C₁₉×#442	1	1	yes	characteristic
367	C₄	C₁₉×#459	1	1	yes	characteristic
368	C₄	C₁₉×#463	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	16	1	no	—
372	C₂	C₁₉×#1879	1	1	yes	characteristic
373	C₂	C₁₉×#1905	1	1	yes	characteristic
374	C₂	C₁₉×#1912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ 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×#2327	C₂	semidirect
4. C₁₉×#1546	C₂	semidirect
6. C₁₉×#1286	C₂	semidirect
7. C₁₉×#1274	C₂	semidirect
11. C₁₉×#500	C₄	semidirect
12. C₁₉×#486	C₄	semidirect
116. #9300	C₁₉	direct (N × H)
162. #2327	C₂×C₁₉	semidirect
164. #1546	C₂×C₁₉	semidirect
166. #1286	C₂×C₁₉	semidirect
167. #1274	C₂×C₁₉	semidirect
192. #500	C₄×C₁₉	semidirect
193. #486	C₄×C₁₉	semidirect
255. C₁₉	#9300	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×#3226	2
C₁₉×#2327	2
C₁₉×#1963	2
C₁₉×#1546	2
C₁₉×#1517	2
C₁₉×#1286	2
C₁₉×#1274	2
#9300	19

Subgroups — C19×#9300 #18015259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9300 #18015259