Subgroups — C₁₉×#21841 #18027800

Order 4864.

1238 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#1993	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1988	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1824	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₉	—	2	1	no	—
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	C₂²	1	2	yes	—
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×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₉×#97	C₂²	1	2	yes	—
21	C₁₉×#383	—	2	1	no	—
22	C₁₉×#382	C₂²	1	2	yes	—
23	C₁₉×#381	—	2	1	no	—
24	C₁₉×#380	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
27	C₁₉×#376	—	2	1	no	—
28	C₁₉×#340	C₂²	1	16	yes	—
29	C₂²×Q₈×C₁₉	—	2	8	no	—
30	C₂²×Q₈×C₁₉	—	2	2	no	—
31	C₂²×Q₈×C₁₉	—	2	2	no	—
32	C₂²×Q₈×C₁₉	—	4	8	no	—
33	C₂³×C₄×C₁₉	—	2	2	no	—
34	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
38	C₁₉×C₄⋊Q₈	—	2	8	no	—
39	C₁₉×C₄⋊Q₈	—	2	8	no	—
40	C₁₉×C₄⋊Q₈	—	2	4	no	—
41	C₁₉×C₄⋊Q₈	—	2	4	no	—
42	C₁₉×C₄⋊Q₈	C₂×C₄	1	8	yes	—
43	C₁₉×C₄⋊Q₈	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
49	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
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×C₂²⋊C₈	—	2	4	no	—
62	C₁₉×#97	—	2	4	no	—
63	C₁₉×#95	—	2	2	no	—
64	C₁₉×#95	—	2	2	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	2	no	—
67	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂×Q₈×C₁₉	—	2	8	no	—
69	C₂×Q₈×C₁₉	—	2	8	no	—
70	C₂×Q₈×C₁₉	—	4	8	no	—
71	C₂×Q₈×C₁₉	—	4	32	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	—	2	4	no	—
76	C₂²×C₄×C₁₉	—	2	8	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	4	no	—
79	C₂²×C₄×C₁₉	—	4	8	no	—
80	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
81	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
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	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₉	—	2	4	no	—
91	C₂×C₈×C₁₉	—	2	8	no	—
92	C₄⋊C₄×C₁₉	—	2	8	no	—
93	C₄⋊C₄×C₁₉	—	2	8	no	—
94	C₄⋊C₄×C₁₉	—	2	8	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	8	no	—
97	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
98	C₂²⋊C₄×C₁₉	—	4	4	no	—
99	C₄²×C₁₉	—	4	4	no	—
100	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
101	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
102	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₉	C₂×D₄	1	2	yes	—
105	C₄²×C₁₉	C₂×D₄	1	2	yes	—
106	#21841	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₁₉	—	2	2	no	—
108	C₂³×C₁₉	—	2	2	no	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₁₉	#137	1	4	yes	—
111	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₁₉	—	4	16	no	—
113	Q₈×C₁₉	—	4	16	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	8	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	—	2	8	no	—
123	C₂×C₄×C₁₉	—	4	16	no	—
124	C₂×C₄×C₁₉	—	4	8	no	—
125	C₂×C₄×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	4	no	—
128	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	2₋¹⁺⁴	1	4	yes	—
137	C₈×C₁₉	—	4	4	no	—
138	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
139	C₂×#432	C₂×C₁₉	1	1	yes	characteristic
140	#2283	C₂×C₁₉	1	1	yes	characteristic
141	#1993	C₂×C₁₉	1	4	yes	complemented
142	#1988	C₂×C₁₉	1	4	yes	complemented
143	#1824	C₂×C₁₉	1	4	yes	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	8	no	—
147	C₂²×C₁₉	#419	1	4	yes	—
148	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₁₉	#536	1	2	yes	—
150	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
151	C₂²×C₁₉	C₂×#137	1	2	yes	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	4	8	no	—
157	C₄×C₁₉	—	8	4	no	—
158	C₄×C₁₉	#418	1	4	yes	—
159	C₄×C₁₉	#519	1	2	yes	—
160	C₄×C₁₉	#536	1	2	yes	—
161	C₂³×Q₈	—	2	1	no	—
162	C₂×C₄⋊Q₈	—	2	1	no	—
163	C₂×C₄⋊Q₈	—	2	1	no	—
164	C₂×C₄⋊Q₈	—	2	4	no	—
165	C₂×C₄⋊Q₈	C₄×C₁₉	1	8	yes	complemented
166	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
167	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
168	C₂²×C₄⋊C₄	—	2	1	no	—
169	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
170	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
171	#432	—	2	4	no	—
172	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
174	C₂×#97	C₂²×C₁₉	1	2	yes	—
175	#383	—	2	1	no	—
176	#382	C₂²×C₁₉	1	2	yes	—
177	#381	—	2	1	no	—
178	#380	C₂²×C₁₉	1	2	yes	—
179	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
180	C₄×C₂²⋊C₄	—	2	1	no	—
181	#376	—	2	1	no	—
182	#340	C₂²×C₁₉	1	16	yes	—
183	C₂×C₁₉	—	2	4	no	—
184	C₂×C₁₉	#2310	1	1	yes	characteristic
185	C₂×C₁₉	C₂×#418	1	1	yes	characteristic
186	C₂×C₁₉	C₂×#419	1	1	yes	characteristic
187	C₂×C₁₉	#3239	1	4	yes	—
188	C₂²×Q₈	—	2	8	no	—
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	4	8	no	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
194	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
195	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
197	C₄⋊Q₈	—	2	8	no	—
198	C₄⋊Q₈	—	2	8	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	C₂×C₄×C₁₉	1	8	yes	—
202	C₄⋊Q₈	C₂×C₄×C₁₉	1	8	yes	—
203	C₂×C₄⋊C₄	—	2	8	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
214	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₁₉	1	4	yes	—
217	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₂²⋊C₈	—	2	4	no	—
221	#97	—	2	4	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	C₁₉	#21841	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	—	4	32	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	8	no	—
240	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
242	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	8	no	—
252	C₄⋊C₄	—	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	8	no	—
257	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
261	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
262	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
263	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
264	C₄²	C₂×D₄×C₁₉	1	2	yes	—
265	C₄²	C₂×D₄×C₁₉	1	2	yes	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
269	C₂³	C₁₉×#137	1	4	yes	—
270	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	Q₈	—	4	16	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	16	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
292	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₁₉×2₋¹⁺⁴	1	4	yes	—
296	C₈	—	4	4	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	C₁₉×#419	1	4	yes	—
301	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
302	C₂²	C₁₉×#536	1	2	yes	—
303	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
304	C₂²	C₂×C₁₉×#137	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	8	no	—
310	C₄	—	8	4	no	—
311	C₄	C₁₉×#418	1	4	yes	—
312	C₄	C₁₉×#519	1	2	yes	—
313	C₄	C₁₉×#536	1	2	yes	—
314	C₂	—	2	4	no	—
315	C₂	C₁₉×#2310	1	1	yes	characteristic
316	C₂	C₂×C₁₉×#418	1	1	yes	characteristic
317	C₂	C₂×C₁₉×#419	1	1	yes	characteristic
318	C₂	C₁₉×#3239	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×#1993	C₂	semidirect
5. C₁₉×#1988	C₂	semidirect
11. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
106. #21841	C₁₉	direct (N × H)
141. #1993	C₂×C₁₉	semidirect
142. #1988	C₂×C₁₉	semidirect
165. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
226. C₁₉	#21841	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×#1993	2
C₁₉×#1988	2
C₁₉×#1824	2
#21841	19

Subgroups — C19×#21841 #18027800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21841 #18027800