Subgroups — C₁₉×#17739 #18023698

Order 4864.

950 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	C₂×C₁₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2920	C₂	1	2	yes	maximal
3	C₁₉×#1561	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1556	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1544	C₂	1	4	yes	maximal, complemented
6	C₁₉×#533	—	2	2	no	—
7	C₁₉×#533	—	2	2	no	—
8	C₁₉×#533	C₄	1	2	yes	—
9	C₁₉×#533	C₄	1	2	yes	—
10	C₁₉×#533	C₂²	1	2	yes	—
11	C₁₉×#533	C₂²	1	2	yes	—
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₉×#125	C₄	1	2	yes	—
14	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
15	C₂×C₄×Q₈×C₁₉	—	2	2	no	—
16	C₂×C₁₉×#117	C₄	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#432	—	2	2	no	—
19	C₁₉×#432	C₂²	1	4	yes	—
20	C₁₉×#431	—	2	2	no	—
21	C₁₉×#431	C₂²	1	4	yes	—
22	C₁₉×#423	C₂²	1	4	yes	—
23	C₁₉×#407	—	2	2	no	—
24	C₄×M₁₆×C₁₉	—	2	2	no	—
25	C₁₉×#332	C₂²	1	16	yes	complemented
26	C₂²×Q₈×C₁₉	—	2	2	no	—
27	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	D₄	1	2	yes	—
29	C₂×M₁₆×C₁₉	—	2	4	no	—
30	C₁₉×C₄⋊Q₈	—	2	2	no	—
31	C₁₉×C₄⋊Q₈	—	2	2	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
33	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
34	C₁₉×#125	—	2	4	no	—
35	C₁₉×#125	C₂×C₄	1	4	yes	—
36	C₁₉×#125	C₂×C₄	1	4	yes	—
37	C₁₉×C₂²⋊Q₈	—	2	4	no	—
38	C₁₉×C₂²⋊Q₈	—	2	4	no	—
39	C₁₉×C₂²⋊Q₈	—	2	4	no	—
40	C₁₉×C₂²⋊Q₈	—	2	4	no	—
41	C₄×Q₈×C₁₉	—	2	4	no	—
42	C₄×Q₈×C₁₉	—	2	4	no	—
43	C₄×Q₈×C₁₉	—	2	4	no	—
44	C₄×Q₈×C₁₉	—	2	4	no	—
45	C₁₉×#117	—	2	2	no	—
46	C₁₉×#117	—	2	2	no	—
47	C₁₉×#117	C₂×C₄	1	4	yes	—
48	C₁₉×#117	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₉	C₂×C₄	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	1	yes	characteristic
55	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₉	D₄	1	2	yes	—
58	C₂×C₄²×C₁₉	D₄	1	2	yes	—
59	C₂×C₄²×C₁₉	D₄	1	2	yes	—
60	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₁₉×C₄⋊C₈	—	2	4	no	—
64	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
65	C₁₉×C₂²⋊C₈	—	2	4	no	—
66	C₁₉×C₂²⋊C₈	—	2	4	no	—
67	C₁₉×#97	—	2	4	no	—
68	C₄×C₈×C₁₉	—	2	4	no	—
69	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×Q₈×C₁₉	—	2	4	no	—
71	C₂×Q₈×C₁₉	—	2	4	no	—
72	C₂×Q₈×C₁₉	—	4	8	no	—
73	C₂²×C₄×C₁₉	—	2	4	no	—
74	C₂²×C₄×C₁₉	—	2	4	no	—
75	C₂²×C₄×C₁₉	—	4	4	no	—
76	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
88	M₁₆×C₁₉	—	4	8	no	—
89	C₂×C₈×C₁₉	—	2	8	no	—
90	C₄⋊C₄×C₁₉	—	2	8	no	—
91	C₄⋊C₄×C₁₉	—	2	8	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
96	C₂²⋊C₄×C₁₉	—	2	4	no	—
97	C₂²⋊C₄×C₁₉	—	2	4	no	—
98	C₄²×C₁₉	—	2	4	no	—
99	C₄²×C₁₉	—	2	4	no	—
100	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
103	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	2	yes	—
107	C₄²×C₁₉	C₂×D₄	1	2	yes	—
108	C₄²×C₁₉	C₂×D₄	1	2	yes	—
109	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
110	#17739	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₁₉	—	2	2	no	—
112	C₂³×C₁₉	—	4	2	no	—
113	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	Q₈×C₁₉	—	4	8	no	—
117	Q₈×C₁₉	—	4	8	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	8	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	8	no	—
131	C₂×C₄×C₁₉	#101	1	4	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	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
140	C₈×C₁₉	—	4	8	no	—
141	C₂×#533	C₂×C₁₉	1	1	yes	characteristic
142	#2920	C₂×C₁₉	1	2	yes	—
143	#1561	C₂×C₁₉	1	4	yes	complemented
144	#1556	C₂×C₁₉	1	4	yes	complemented
145	#1544	C₂×C₁₉	1	4	yes	complemented
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	4	4	no	—
152	C₂²×C₁₉	C₂×#101	1	2	yes	—
153	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	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	C₂×#101	1	2	yes	—
161	C₄×C₁₉	C₂×#101	1	2	yes	—
162	C₄×C₁₉	#575	1	4	yes	—
163	#533	—	2	2	no	—
164	#533	—	2	2	no	—
165	#533	C₄×C₁₉	1	2	yes	—
166	#533	C₄×C₁₉	1	2	yes	—
167	#533	C₂²×C₁₉	1	2	yes	—
168	#533	C₂²×C₁₉	1	2	yes	—
169	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
170	C₂×#125	C₄×C₁₉	1	2	yes	—
171	C₂×C₂²⋊Q₈	—	2	2	no	—
172	C₂×C₄×Q₈	—	2	2	no	—
173	C₂×#117	C₄×C₁₉	1	2	yes	—
174	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
175	#432	—	2	2	no	—
176	#432	C₂²×C₁₉	1	4	yes	—
177	#431	—	2	2	no	—
178	#431	C₂²×C₁₉	1	4	yes	—
179	#423	C₂²×C₁₉	1	4	yes	—
180	#407	—	2	2	no	—
181	C₄×M₁₆	—	2	2	no	—
182	#332	C₂²×C₁₉	1	16	yes	complemented
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	—	4	1	no	—
185	C₂×C₁₉	—	4	1	no	—
186	C₂×C₁₉	C₂²×#101	1	1	yes	characteristic
187	C₂×C₁₉	#2896	1	2	yes	—
188	C₂²×Q₈	—	2	2	no	—
189	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₁₉	1	2	yes	—
191	C₂×M₁₆	—	2	4	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	C₂×C₄×C₁₉	1	4	yes	—
195	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
196	#125	—	2	4	no	—
197	#125	C₂×C₄×C₁₉	1	4	yes	—
198	#125	C₂×C₄×C₁₉	1	4	yes	—
199	C₂²⋊Q₈	—	2	4	no	—
200	C₂²⋊Q₈	—	2	4	no	—
201	C₂²⋊Q₈	—	2	4	no	—
202	C₂²⋊Q₈	—	2	4	no	—
203	C₄×Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	C₂×C₄×C₁₉	1	4	yes	—
210	#117	C₂×C₄×C₁₉	1	4	yes	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₁₉	1	2	yes	—
220	C₂×C₄²	D₄×C₁₉	1	2	yes	—
221	C₂×C₄²	D₄×C₁₉	1	2	yes	—
222	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
227	C₂²⋊C₈	—	2	4	no	—
228	C₂²⋊C₈	—	2	4	no	—
229	#97	—	2	4	no	—
230	C₄×C₈	—	2	4	no	—
231	C₁₉	#17739	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
242	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
244	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
251	M₁₆	—	4	8	no	—
252	C₂×C₈	—	2	8	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
264	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
265	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
266	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
267	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
268	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₉	1	2	yes	—
270	C₄²	C₂×D₄×C₁₉	1	2	yes	—
271	C₄²	C₂×D₄×C₁₉	1	2	yes	—
272	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	—	4	2	no	—
275	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
277	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
278	Q₈	—	4	8	no	—
279	Q₈	—	4	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	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	8	no	—
293	C₂×C₄	C₁₉×#101	1	4	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	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
302	C₈	—	4	8	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	C₂×C₁₉×#101	1	2	yes	—
310	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	8	no	—
317	C₄	C₂×C₁₉×#101	1	2	yes	—
318	C₄	C₂×C₁₉×#101	1	2	yes	—
319	C₄	C₁₉×#575	1	4	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	4	1	no	—
322	C₂	—	4	1	no	—
323	C₂	C₂²×C₁₉×#101	1	1	yes	characteristic
324	C₂	C₁₉×#2896	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1561	C₂	semidirect
4. C₁₉×#1556	C₂	semidirect
5. C₁₉×#1544	C₂	semidirect
25. C₁₉×#332	C₂²	semidirect
110. #17739	C₁₉	direct (N × H)
143. #1561	C₂×C₁₉	semidirect
144. #1556	C₂×C₁₉	semidirect
145. #1544	C₂×C₁₉	semidirect
182. #332	C₂²×C₁₉	semidirect
231. C₁₉	#17739	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×#533	2
C₁₉×#2920	2
C₁₉×#1561	2
C₁₉×#1556	2
C₁₉×#1544	2
#17739	19

Subgroups — C19×#17739 #18023698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17739 #18023698