Subgroups — C₁₉×#9327 #18015286

Order 4864.

492 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3110	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2319	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1986	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1506	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1490	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1300	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#484	—	2	1	no	—
11	C₁₉×#474	C₄	1	2	yes	complemented
12	C₁₉×#461	C₄	1	2	yes	complemented
13	C₁₉×#450	—	2	1	no	—
14	C₁₉×#440	—	2	2	no	—
15	C₁₉×#432	C₂²	1	1	yes	characteristic
16	C₁₉×#432	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#390	—	2	1	no	—
19	C₁₉×#389	—	2	1	no	—
20	C₁₉×#388	—	2	2	no	—
21	C₁₉×#381	—	2	1	no	—
22	C₁₉×#380	—	2	1	no	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#376	C₂²	1	1	yes	characteristic
25	C₁₉×#339	—	2	2	no	—
26	C₁₉×#328	—	2	1	no	—
27	C₁₉×#328	—	2	1	no	—
28	C₁₉×D₄⋊C₈	—	2	2	no	—
29	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	—	2	1	no	—
31	C₂×SD₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#125	—	2	1	no	—
36	C₁₉×C₂²⋊Q₈	—	2	1	no	—
37	C₁₉×C₂²⋊Q₈	—	2	1	no	—
38	C₄×Q₈×C₁₉	—	2	1	no	—
39	C₄×Q₈×C₁₉	—	2	1	no	—
40	C₄×D₄×C₁₉	—	2	1	no	—
41	C₄×D₄×C₁₉	—	2	1	no	—
42	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
43	C₄×D₄×C₁₉	D₄	1	2	yes	—
44	C₁₉×#117	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	D₄	1	2	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₁₉×#107	—	2	2	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₁₉×#103	—	2	2	no	—
58	C₁₉×#102	—	2	2	no	—
59	C₁₉×C₂²⋊C₈	—	2	1	no	—
60	C₁₉×C₂²⋊C₈	—	2	1	no	—
61	C₁₉×#97	—	2	1	no	—
62	C₁₉×#97	—	2	1	no	—
63	C₁₉×#97	—	2	1	no	—
64	C₄×C₈×C₁₉	—	2	1	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	1	no	—
67	C₁₉×#95	—	2	1	no	—
68	C₁₉×#95	D₄	1	1	yes	characteristic
69	C₁₉×#95	D₄	1	1	yes	characteristic
70	C₂⁴×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×D₄×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	2	1	no	—
80	C₂²×C₄×C₁₉	—	2	1	no	—
81	C₂²×C₄×C₁₉	—	2	2	no	—
82	C₂²×C₄×C₁₉	—	8	2	no	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
86	SD₁₆×C₁₉	—	4	4	no	—
87	M₁₆×C₁₉	—	2	2	no	—
88	M₁₆×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	8	2	no	—
101	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₁₉	—	2	2	no	—
103	C₂²⋊C₄×C₁₉	—	2	2	no	—
104	C₂²⋊C₄×C₁₉	—	4	1	no	—
105	C₂²⋊C₄×C₁₉	—	8	2	no	—
106	C₂²⋊C₄×C₁₉	—	8	2	no	—
107	C₄²×C₁₉	—	4	2	no	—
108	C₄²×C₁₉	—	4	2	no	—
109	C₄²×C₁₉	—	4	1	no	—
110	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₉	#41	1	1	yes	characteristic
113	C₄²×C₁₉	#41	1	1	yes	characteristic
114	#9327	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₁₉	—	2	2	no	—
116	C₂³×C₁₉	—	4	2	no	—
117	C₂³×C₁₉	—	4	2	no	—
118	C₂³×C₁₉	—	4	1	no	—
119	C₂³×C₁₉	—	4	1	no	—
120	C₂³×C₁₉	#120	1	1	yes	characteristic
121	Q₈×C₁₉	—	4	2	no	—
122	Q₈×C₁₉	—	8	1	no	—
123	D₄×C₁₉	—	2	2	no	—
124	D₄×C₁₉	—	2	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	8	2	no	—
144	C₂×C₄×C₁₉	—	8	2	no	—
145	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
151	C₈×C₁₉	—	4	2	no	—
152	C₈×C₁₉	—	4	2	no	—
153	C₈×C₁₉	—	8	1	no	—
154	#3110	C₂×C₁₉	1	1	yes	characteristic
155	#2319	C₂×C₁₉	1	1	yes	characteristic
156	#1986	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#1506	C₂×C₁₉	1	1	yes	characteristic, complemented
158	#1490	C₂×C₁₉	1	1	yes	characteristic
159	#1300	C₂×C₁₉	1	1	yes	characteristic, complemented
160	#1285	C₂×C₁₉	1	1	yes	characteristic, complemented
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	4	2	no	—
164	C₂²×C₁₉	—	4	2	no	—
165	C₂²×C₁₉	—	4	2	no	—
166	C₂²×C₁₉	—	8	2	no	—
167	C₂²×C₁₉	—	8	2	no	—
168	C₂²×C₁₉	#386	1	1	yes	characteristic
169	C₂²×C₁₉	#455	1	1	yes	characteristic
170	C₂²×C₁₉	#456	1	1	yes	characteristic
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	4	2	no	—
177	C₄×C₁₉	—	8	1	no	—
178	C₄×C₁₉	—	8	2	no	—
179	C₄×C₁₉	—	8	1	no	—
180	C₄×C₁₉	—	8	1	no	—
181	C₄×C₁₉	—	8	1	no	—
182	C₄×C₁₉	#421	1	2	yes	—
183	#533	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
185	#484	—	2	1	no	—
186	#474	C₄×C₁₉	1	2	yes	complemented
187	#461	C₄×C₁₉	1	2	yes	complemented
188	#450	—	2	1	no	—
189	#440	—	2	2	no	—
190	#432	C₂²×C₁₉	1	1	yes	characteristic
191	#432	C₂²×C₁₉	1	1	yes	characteristic
192	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
193	#390	—	2	1	no	—
194	#389	—	2	1	no	—
195	#388	—	2	2	no	—
196	#381	—	2	1	no	—
197	#380	—	2	1	no	—
198	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
199	#376	C₂²×C₁₉	1	1	yes	characteristic
200	#339	—	2	2	no	—
201	#328	—	2	1	no	—
202	#328	—	2	1	no	—
203	D₄⋊C₈	—	2	2	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	8	2	no	—
207	C₂×C₁₉	#1882	1	1	yes	characteristic
208	C₂×C₁₉	#1897	1	1	yes	characteristic
209	C₂×C₁₉	#1904	1	1	yes	characteristic
210	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
211	C₂³×C₄	—	2	1	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
215	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
216	#125	—	2	1	no	—
217	C₂²⋊Q₈	—	2	1	no	—
218	C₂²⋊Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	C₂×C₄×C₁₉	1	2	yes	—
224	C₄×D₄	D₄×C₁₉	1	2	yes	—
225	#117	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	D₄×C₁₉	1	2	yes	—
234	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
235	#107	—	2	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
238	#103	—	2	2	no	—
239	#102	—	2	2	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	#97	—	2	1	no	—
243	#97	—	2	1	no	—
244	#97	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	#95	—	2	2	no	—
247	#95	—	2	1	no	—
248	#95	—	2	1	no	—
249	#95	D₄×C₁₉	1	1	yes	characteristic
250	#95	D₄×C₁₉	1	1	yes	characteristic
251	C₁₉	#9327	1	1	yes	characteristic, direct product factor, Sylow
252	C₂⁴	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	8	2	no	—
265	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
268	SD₁₆	—	4	4	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	4	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	8	2	no	—
283	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	8	2	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₄²	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	—	4	1	no	—
292	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
294	C₄²	#41×C₁₉	1	1	yes	characteristic
295	C₄²	#41×C₁₉	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	1	no	—
301	C₂³	C₁₉×#120	1	1	yes	characteristic
302	Q₈	—	4	2	no	—
303	Q₈	—	8	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
327	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
328	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
330	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
331	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	8	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	C₁₉×#386	1	1	yes	characteristic
343	C₂²	C₁₉×#455	1	1	yes	characteristic
344	C₂²	C₁₉×#456	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	1	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	1	no	—
354	C₄	—	8	1	no	—
355	C₄	—	8	1	no	—
356	C₄	C₁₉×#421	1	2	yes	—
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	8	2	no	—
360	C₂	C₁₉×#1882	1	1	yes	characteristic
361	C₂	C₁₉×#1897	1	1	yes	characteristic
362	C₂	C₁₉×#1904	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	C₂²×D₄×C₁₉	D₄
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
3. C₁₉×#1986	C₂	semidirect
4. C₁₉×#1506	C₂	semidirect
6. C₁₉×#1300	C₂	semidirect
7. C₁₉×#1285	C₂	semidirect
11. C₁₉×#474	C₄	semidirect
12. C₁₉×#461	C₄	semidirect
114. #9327	C₁₉	direct (N × H)
156. #1986	C₂×C₁₉	semidirect
157. #1506	C₂×C₁₉	semidirect
159. #1300	C₂×C₁₉	semidirect
160. #1285	C₂×C₁₉	semidirect
186. #474	C₄×C₁₉	semidirect
187. #461	C₄×C₁₉	semidirect
251. C₁₉	#9327	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×#3110	2
C₁₉×#2319	2
C₁₉×#1986	2
C₁₉×#1506	2
C₁₉×#1490	2
C₁₉×#1300	2
C₁₉×#1285	2
#9327	19

Subgroups — C19×#9327 #18015286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9327 #18015286