Subgroups — C₁₉×#21811 #18027770

Order 4864.

1302 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#432	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1981	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1960	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1787	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
8	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₁₉×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	8	yes	complemented
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#432	—	2	4	no	—
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₁₉×#103	C₂²	1	16	yes	—
20	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×#97	C₂²	1	2	yes	—
22	C₁₉×#383	—	2	1	no	—
23	C₁₉×#382	C₂²	1	2	yes	—
24	C₁₉×#381	—	2	1	no	—
25	C₁₉×#380	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
28	C₁₉×#376	—	2	1	no	—
29	C₂²×Q₈×C₁₉	—	2	4	no	—
30	C₂²×Q₈×C₁₉	—	2	2	no	—
31	C₂²×Q₈×C₁₉	—	4	8	no	—
32	C₂²×Q₈×C₁₉	C₂×C₄	1	8	yes	—
33	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₉	—	2	2	no	—
35	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
39	C₁₉×C₄⋊Q₈	—	2	8	no	—
40	C₁₉×C₄⋊Q₈	—	2	8	no	—
41	C₁₉×C₄⋊Q₈	—	2	4	no	—
42	C₁₉×C₄⋊Q₈	—	2	4	no	—
43	C₁₉×C₄⋊Q₈	C₂×C₄	1	8	yes	—
44	C₁₉×C₄⋊Q₈	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
53	C₂×C₄²×C₁₉	—	2	2	no	—
54	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₉	D₄	1	4	yes	—
58	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₁₉×C₄⋊C₈	—	2	4	no	—
61	C₁₉×#103	—	4	16	no	—
62	C₁₉×C₂²⋊C₈	—	2	4	no	—
63	C₁₉×#97	—	2	4	no	—
64	C₁₉×#95	—	2	2	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	2	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂×Q₈×C₁₉	—	2	8	no	—
70	C₂×Q₈×C₁₉	—	2	4	no	—
71	C₂×Q₈×C₁₉	—	2	16	no	—
72	C₂×Q₈×C₁₉	—	4	16	no	—
73	C₂×Q₈×C₁₉	—	4	8	no	—
74	C₂×Q₈×C₁₉	—	4	16	no	—
75	C₂×Q₈×C₁₉	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₁₉	—	2	4	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	4	no	—
79	C₂²×C₄×C₁₉	—	2	8	no	—
80	C₂²×C₄×C₁₉	—	2	4	no	—
81	C₂²×C₄×C₁₉	—	2	4	no	—
82	C₂²×C₄×C₁₉	—	4	8	no	—
83	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
84	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₁₉	—	2	4	no	—
94	C₂×C₈×C₁₉	—	2	8	no	—
95	C₄⋊C₄×C₁₉	—	2	8	no	—
96	C₄⋊C₄×C₁₉	—	2	8	no	—
97	C₄⋊C₄×C₁₉	—	2	8	no	—
98	C₄⋊C₄×C₁₉	—	2	8	no	—
99	C₄⋊C₄×C₁₉	—	2	8	no	—
100	C₂²⋊C₄×C₁₉	—	4	4	no	—
101	C₄²×C₁₉	—	4	4	no	—
102	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
103	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
104	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₉	C₂×D₄	1	2	yes	—
107	C₄²×C₁₉	C₂×D₄	1	2	yes	—
108	#21811	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₁₉	—	2	2	no	—
110	C₂³×C₁₉	—	2	2	no	—
111	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂³×C₁₉	#137	1	4	yes	—
113	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₉	—	2	16	no	—
115	Q₈×C₁₉	—	4	16	no	—
116	Q₈×C₁₉	—	4	8	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	8	no	—
125	C₂×C₄×C₁₉	—	2	8	no	—
126	C₂×C₄×C₁₉	—	4	16	no	—
127	C₂×C₄×C₁₉	—	4	8	no	—
128	C₂×C₄×C₁₉	—	4	4	no	—
129	C₂×C₄×C₁₉	—	4	4	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₉	#137	1	4	yes	—
137	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	H(5,𝔽₂)	1	4	yes	—
140	C₈×C₁₉	—	4	4	no	—
141	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
142	C₂×#432	C₂×C₁₉	1	1	yes	characteristic
143	#2283	C₂×C₁₉	1	1	yes	characteristic
144	#1981	C₂×C₁₉	1	4	yes	complemented
145	#1960	C₂×C₁₉	1	4	yes	complemented
146	#1787	C₂×C₁₉	1	4	yes	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	8	no	—
150	C₂²×C₁₉	#419	1	4	yes	—
151	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₉	#535	1	2	yes	—
153	C₂²×C₁₉	C₂×#137	1	2	yes	—
154	C₂²×C₁₉	C₂×#137	1	2	yes	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	4	8	no	—
160	C₄×C₁₉	—	8	4	no	—
161	C₄×C₁₉	#419	1	4	yes	—
162	C₄×C₁₉	#518	1	2	yes	—
163	C₄×C₁₉	#535	1	2	yes	—
164	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
165	C₂×C₄⋊Q₈	—	2	1	no	—
166	C₂×C₄⋊Q₈	—	2	1	no	—
167	C₂×C₄⋊Q₈	—	2	4	no	—
168	C₂×C₄⋊Q₈	C₄×C₁₉	1	8	yes	complemented
169	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
171	C₂²×C₄⋊C₄	—	2	1	no	—
172	C₂²×C₄⋊C₄	—	2	1	no	—
173	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
174	#432	—	2	4	no	—
175	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
176	C₂×#103	C₂²×C₁₉	1	16	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
178	C₂×#97	C₂²×C₁₉	1	2	yes	—
179	#383	—	2	1	no	—
180	#382	C₂²×C₁₉	1	2	yes	—
181	#381	—	2	1	no	—
182	#380	C₂²×C₁₉	1	2	yes	—
183	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
184	C₄×C₂²⋊C₄	—	2	1	no	—
185	#376	—	2	1	no	—
186	C₂×C₁₉	—	2	4	no	—
187	C₂×C₁₉	#2309	1	1	yes	characteristic
188	C₂×C₁₉	C₂×#419	1	1	yes	characteristic
189	C₂×C₁₉	C₂×#419	1	1	yes	characteristic
190	C₂×C₁₉	#3218	1	4	yes	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	2	2	no	—
193	C₂²×Q₈	—	4	8	no	—
194	C₂²×Q₈	C₂×C₄×C₁₉	1	8	yes	—
195	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
196	C₂³×C₄	—	2	2	no	—
197	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
198	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
201	C₄⋊Q₈	—	2	8	no	—
202	C₄⋊Q₈	—	2	8	no	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	C₂×C₄×C₁₉	1	8	yes	—
206	C₄⋊Q₈	C₂×C₄×C₁₉	1	8	yes	—
207	C₂×C₄⋊C₄	—	2	8	no	—
208	C₂×C₄⋊C₄	—	2	8	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
217	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₁₉	1	4	yes	—
220	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
222	C₄⋊C₈	—	2	4	no	—
223	#103	—	4	16	no	—
224	C₂²⋊C₈	—	2	4	no	—
225	#97	—	2	4	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	C₁₉	#21811	1	1	yes	characteristic, direct product factor, Sylow
231	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
232	C₂×Q₈	—	2	8	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	16	no	—
235	C₂×Q₈	—	4	16	no	—
236	C₂×Q₈	—	4	8	no	—
237	C₂×Q₈	—	4	16	no	—
238	C₂×Q₈	C₂²×C₄×C₁₉	1	8	yes	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	8	no	—
246	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
247	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	8	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	2	8	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₄²	—	4	4	no	—
265	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
266	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
267	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
268	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
269	C₄²	C₂×D₄×C₁₉	1	2	yes	—
270	C₄²	C₂×D₄×C₁₉	1	2	yes	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
274	C₂³	C₁₉×#137	1	4	yes	—
275	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
276	Q₈	—	2	16	no	—
277	Q₈	—	4	16	no	—
278	Q₈	—	4	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	4	16	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
298	C₂×C₄	C₁₉×#137	1	4	yes	—
299	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
301	C₂×C₄	C₁₉×H(5,𝔽₂)	1	4	yes	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	C₁₉×#419	1	4	yes	—
307	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
308	C₂²	C₁₉×#535	1	2	yes	—
309	C₂²	C₂×C₁₉×#137	1	2	yes	—
310	C₂²	C₂×C₁₉×#137	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	8	no	—
316	C₄	—	8	4	no	—
317	C₄	C₁₉×#419	1	4	yes	—
318	C₄	C₁₉×#518	1	2	yes	—
319	C₄	C₁₉×#535	1	2	yes	—
320	C₂	—	2	4	no	—
321	C₂	C₁₉×#2309	1	1	yes	characteristic
322	C₂	C₂×C₁₉×#419	1	1	yes	characteristic
323	C₂	C₂×C₁₉×#419	1	1	yes	characteristic
324	C₂	C₁₉×#3218	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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×#1981	C₂	semidirect
5. C₁₉×#1960	C₂	semidirect
11. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
108. #21811	C₁₉	direct (N × H)
144. #1981	C₂×C₁₉	semidirect
145. #1960	C₂×C₁₉	semidirect
168. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
230. C₁₉	#21811	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×C₄⋊Q₈	2
C₂×C₁₉×#432	2
C₁₉×#2283	2
C₁₉×#1981	2
C₁₉×#1960	2
C₁₉×#1787	2
#21811	19

Subgroups — C19×#21811 #18027770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21811 #18027770