Subgroups — C₁₉×#11162 #18017121

Order 4864.

898 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#101	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2310	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#391	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1736	C₂	1	2	yes	maximal
5	C₂×C₁₉×#339	C₂	1	2	yes	maximal
6	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
8	C₂×C₁₉×#117	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#101	—	2	2	no	—
12	C₂×C₁₉×#101	—	2	4	no	—
13	C₁₉×#391	—	2	2	no	—
14	C₁₉×#391	—	2	4	no	—
15	C₁₉×#391	C₄	1	4	yes	complemented
16	C₁₉×#381	—	2	2	no	—
17	C₂×C₁₉×#95	—	2	2	no	—
18	C₁₉×#340	—	2	4	no	—
19	C₁₉×#339	—	2	4	no	—
20	C₂²×Q₈×C₁₉	—	2	2	no	—
21	C₂²×Q₈×C₁₉	—	2	2	no	—
22	C₂²×Q₈×C₁₉	—	2	2	no	—
23	C₂²×Q₈×C₁₉	—	2	2	no	—
24	C₂²×Q₈×C₁₉	—	2	2	no	—
25	C₂²×Q₈×C₁₉	—	2	2	no	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×Q₈×C₁₉	D₄	1	2	yes	—
28	C₂³×C₄×C₁₉	—	2	1	no	—
29	C₂³×C₄×C₁₉	—	2	1	no	—
30	C₂³×C₄×C₁₉	D₄	1	2	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂×M₁₆×C₁₉	—	2	2	no	—
34	C₂×M₁₆×C₁₉	—	2	2	no	—
35	C₂×M₁₆×C₁₉	—	2	2	no	—
36	C₂×M₁₆×C₁₉	—	2	2	no	—
37	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
38	C₂²×C₈×C₁₉	—	2	2	no	—
39	C₁₉×#117	—	2	2	no	—
40	C₁₉×#117	—	2	2	no	—
41	C₁₉×#117	—	2	2	no	—
42	C₁₉×#117	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	D₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₄²×C₁₉	—	2	2	no	—
50	C₂×C₄²×C₁₉	—	2	2	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	—	2	2	no	—
53	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
54	C₁₉×#101	—	2	8	no	—
55	C₁₉×#95	—	2	8	no	—
56	C₁₉×#95	—	2	8	no	—
57	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	2	2	no	—
60	C₂×Q₈×C₁₉	—	2	4	no	—
61	C₂×Q₈×C₁₉	—	2	4	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	2	4	no	—
64	C₂×Q₈×C₁₉	—	2	4	no	—
65	C₂×Q₈×C₁₉	—	2	2	no	—
66	C₂×Q₈×C₁₉	—	4	2	no	—
67	C₂×Q₈×C₁₉	—	4	2	no	—
68	C₂×Q₈×C₁₉	—	4	2	no	—
69	C₂×Q₈×C₁₉	—	4	2	no	—
70	C₂×Q₈×C₁₉	—	4	2	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂×Q₈×C₁₉	—	4	2	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂²×C₄×C₁₉	—	2	4	no	—
75	C₂²×C₄×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	4	no	—
79	C₂²×C₄×C₁₉	—	2	2	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	2	4	no	—
82	C₂²×C₄×C₁₉	—	4	2	no	—
83	C₂²×C₄×C₁₉	—	4	2	no	—
84	C₂²×C₄×C₁₉	—	4	4	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
88	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	#41	1	2	yes	—
93	C₂²×C₄×C₁₉	#41	1	2	yes	—
94	C₂²×C₄×C₁₉	#41	1	2	yes	—
95	C₂²×C₄×C₁₉	#41	1	2	yes	—
96	M₁₆×C₁₉	—	2	4	no	—
97	M₁₆×C₁₉	—	2	4	no	—
98	M₁₆×C₁₉	—	4	2	no	—
99	M₁₆×C₁₉	—	4	2	no	—
100	C₂×C₈×C₁₉	—	2	4	no	—
101	C₂×C₈×C₁₉	—	4	2	no	—
102	C₂×C₈×C₁₉	—	4	2	no	—
103	C₄⋊C₄×C₁₉	—	2	4	no	—
104	C₄⋊C₄×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	2	4	no	—
106	C₄⋊C₄×C₁₉	—	4	2	no	—
107	C₂²⋊C₄×C₁₉	—	4	2	no	—
108	C₂²⋊C₄×C₁₉	—	4	2	no	—
109	C₄²×C₁₉	—	2	4	no	—
110	C₄²×C₁₉	—	4	1	no	—
111	C₄²×C₁₉	—	4	2	no	—
112	C₄²×C₁₉	—	4	1	no	—
113	C₄²×C₁₉	—	4	2	no	—
114	#11162	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₁₉	—	2	2	no	—
116	C₂³×C₁₉	—	2	2	no	—
117	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂³×C₁₉	#120	1	1	yes	characteristic
119	C₂³×C₁₉	#121	1	2	yes	—
120	C₂³×C₁₉	#123	1	2	yes	—
121	C₂³×C₁₉	#123	1	1	yes	characteristic
122	Q₈×C₁₉	—	4	4	no	—
123	Q₈×C₁₉	—	4	4	no	—
124	Q₈×C₁₉	—	4	4	no	—
125	Q₈×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	2	4	no	—
127	C₂×C₄×C₁₉	—	2	4	no	—
128	C₂×C₄×C₁₉	—	2	4	no	—
129	C₂×C₄×C₁₉	—	2	4	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	8	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	4	no	—
134	C₂×C₄×C₁₉	—	4	4	no	—
135	C₂×C₄×C₁₉	—	4	4	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	4	no	—
138	C₂×C₄×C₁₉	—	4	4	no	—
139	C₂×C₄×C₁₉	—	4	4	no	—
140	C₂×C₄×C₁₉	—	8	2	no	—
141	C₂×C₄×C₁₉	—	8	2	no	—
142	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#121	1	2	yes	—
148	C₂×C₄×C₁₉	#121	1	2	yes	—
149	C₂×C₄×C₁₉	#121	1	2	yes	—
150	C₂×C₄×C₁₉	#123	1	2	yes	—
151	C₂×C₄×C₁₉	#123	1	2	yes	—
152	C₂×C₄×C₁₉	#123	1	2	yes	—
153	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#124	1	2	yes	—
155	C₂×C₄×C₁₉	#124	1	2	yes	—
156	C₂×C₄×C₁₉	#126	1	2	yes	—
157	C₂×C₄×C₁₉	#126	1	2	yes	—
158	C₈×C₁₉	—	4	4	no	—
159	C₂²×#101	C₂×C₁₉	1	1	yes	characteristic
160	#2310	C₂×C₁₉	1	1	yes	characteristic
161	C₂×#391	C₂×C₁₉	1	1	yes	characteristic
162	#1736	C₂×C₁₉	1	2	yes	—
163	C₂×#339	C₂×C₁₉	1	2	yes	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	4	no	—
169	C₂²×C₁₉	—	2	4	no	—
170	C₂²×C₁₉	#381	1	1	yes	characteristic
171	C₂²×C₁₉	#386	1	1	yes	characteristic
172	C₂²×C₁₉	#386	1	1	yes	characteristic
173	C₂²×C₁₉	#394	1	2	yes	—
174	C₂²×C₁₉	#397	1	2	yes	—
175	C₂²×C₁₉	#454	1	2	yes	—
176	C₂²×C₁₉	#456	1	2	yes	—
177	C₄×C₁₉	—	4	4	no	—
178	C₄×C₁₉	—	4	4	no	—
179	C₄×C₁₉	—	8	2	no	—
180	C₄×C₁₉	—	8	4	no	—
181	C₄×C₁₉	#388	1	2	yes	—
182	C₄×C₁₉	#388	1	2	yes	—
183	C₄×C₁₉	#394	1	2	yes	—
184	C₄×C₁₉	#397	1	2	yes	—
185	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
186	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
187	C₂×#117	C₂²×C₁₉	1	2	yes	—
188	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
189	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×#101	—	2	2	no	—
191	C₂×#101	—	2	4	no	—
192	#391	—	2	2	no	—
193	#391	—	2	4	no	—
194	#391	C₄×C₁₉	1	4	yes	complemented
195	#381	—	2	2	no	—
196	C₂×#95	—	2	2	no	—
197	#340	—	2	4	no	—
198	#339	—	2	4	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	2	no	—
202	C₂×C₁₉	#1440	1	1	yes	characteristic
203	C₂×C₁₉	#1886	1	1	yes	characteristic
204	C₂×C₁₉	#1888	1	1	yes	characteristic
205	C₂×C₁₉	#2038	1	4	yes	—
206	C₂²×Q₈	—	2	2	no	—
207	C₂²×Q₈	—	2	2	no	—
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	—	2	2	no	—
210	C₂²×Q₈	—	2	2	no	—
211	C₂²×Q₈	—	2	2	no	—
212	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
213	C₂²×Q₈	D₄×C₁₉	1	2	yes	—
214	C₂³×C₄	—	2	1	no	—
215	C₂³×C₄	—	2	1	no	—
216	C₂³×C₄	D₄×C₁₉	1	2	yes	—
217	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	D₄×C₁₉	1	4	yes	—
224	C₂²×C₈	—	2	2	no	—
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
233	C₂×C₄⋊C₄	D₄×C₁₉	1	4	yes	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	—	2	1	no	—
239	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
240	#101	—	2	8	no	—
241	#95	—	2	8	no	—
242	#95	—	2	8	no	—
243	C₁₉	#11162	1	1	yes	characteristic, direct product factor, Sylow
244	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
275	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
277	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
279	C₂²×C₄	#41×C₁₉	1	2	yes	—
280	C₂²×C₄	#41×C₁₉	1	2	yes	—
281	C₂²×C₄	#41×C₁₉	1	2	yes	—
282	C₂²×C₄	#41×C₁₉	1	2	yes	—
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	—	4	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	4	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₄²	—	2	4	no	—
297	C₄²	—	4	1	no	—
298	C₄²	—	4	2	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	4	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
304	C₂³	C₁₉×#120	1	1	yes	characteristic
305	C₂³	C₁₉×#121	1	2	yes	—
306	C₂³	C₁₉×#123	1	2	yes	—
307	C₂³	C₁₉×#123	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	8	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	8	2	no	—
328	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
330	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
331	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
332	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
333	C₂×C₄	C₁₉×#121	1	2	yes	—
334	C₂×C₄	C₁₉×#121	1	2	yes	—
335	C₂×C₄	C₁₉×#121	1	2	yes	—
336	C₂×C₄	C₁₉×#123	1	2	yes	—
337	C₂×C₄	C₁₉×#123	1	2	yes	—
338	C₂×C₄	C₁₉×#123	1	2	yes	—
339	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
340	C₂×C₄	C₁₉×#124	1	2	yes	—
341	C₂×C₄	C₁₉×#124	1	2	yes	—
342	C₂×C₄	C₁₉×#126	1	2	yes	—
343	C₂×C₄	C₁₉×#126	1	2	yes	—
344	C₈	—	4	4	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	4	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	C₁₉×#381	1	1	yes	characteristic
352	C₂²	C₁₉×#386	1	1	yes	characteristic
353	C₂²	C₁₉×#386	1	1	yes	characteristic
354	C₂²	C₁₉×#394	1	2	yes	—
355	C₂²	C₁₉×#397	1	2	yes	—
356	C₂²	C₁₉×#454	1	2	yes	—
357	C₂²	C₁₉×#456	1	2	yes	—
358	C₄	—	4	4	no	—
359	C₄	—	4	4	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	4	no	—
362	C₄	C₁₉×#388	1	2	yes	—
363	C₄	C₁₉×#388	1	2	yes	—
364	C₄	C₁₉×#394	1	2	yes	—
365	C₄	C₁₉×#397	1	2	yes	—
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	—	2	2	no	—
369	C₂	C₁₉×#1440	1	1	yes	characteristic
370	C₂	C₁₉×#1886	1	1	yes	characteristic
371	C₂	C₁₉×#1888	1	1	yes	characteristic
372	C₂	C₁₉×#2038	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	#120
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₉×#391	C₄	semidirect
114. #11162	C₁₉	direct (N × H)
194. #391	C₄×C₁₉	semidirect
243. C₁₉	#11162	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×#101	2
C₁₉×#2310	2
C₂×C₁₉×#391	2
C₁₉×#1736	2
C₂×C₁₉×#339	2
#11162	19

Subgroups — C19×#11162 #18017121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11162 #18017121