Subgroups — C₁₉×#23165 #18029124

Order 4864.

706 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#499	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2370	C₂	1	2	yes	maximal
3	C₁₉×#1989	C₂	1	2	yes	maximal
4	C₁₉×#1944	C₂	1	2	yes	maximal
5	C₁₉×#1921	C₂	1	2	yes	maximal
6	C₁₉×#1918	C₂	1	2	yes	maximal
7	C₁₉×#1776	C₂	1	2	yes	maximal
8	C₄×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₉×#125	C₂²	1	2	yes	—
11	C₁₉×#499	C₄	1	8	yes	complemented
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#106	C₂²	1	2	yes	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#389	—	2	2	no	—
21	C₁₉×#389	C₂²	1	2	yes	—
22	C₁₉×#389	C₂²	1	2	yes	—
23	C₁₉×#387	—	2	2	no	—
24	C₁₉×#387	—	2	2	no	—
25	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#340	C₂²	1	2	yes	—
31	C₁₉×#340	C₂²	1	2	yes	—
32	C₁₉×#340	C₂²	1	2	yes	—
33	C₁₉×#340	C₂²	1	2	yes	—
34	C₁₉×#340	C₂²	1	2	yes	—
35	C₁₉×#336	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
39	C₁₉×#125	—	2	4	no	—
40	C₁₉×#125	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₉	—	2	2	no	—
52	C₂×C₄²×C₁₉	—	2	2	no	—
53	C₂×C₄²×C₁₉	—	2	2	no	—
54	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₁₉×#107	—	2	2	no	—
57	C₁₉×#107	C₂×C₄	1	4	yes	—
58	C₁₉×#107	C₂×C₄	1	4	yes	—
59	C₁₉×#107	Q₈	1	4	yes	—
60	C₁₉×#106	C₂×C₄	1	8	yes	—
61	C₁₉×C₄⋊C₈	—	2	2	no	—
62	C₁₉×#97	—	2	4	no	—
63	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
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	C₁₉×#95	—	2	2	no	—
69	C₁₉×#95	D₄	1	2	yes	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	2	2	no	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	1	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₁₉	#41	1	2	yes	—
88	C₄⋊C₄×C₁₉	—	2	8	no	—
89	C₄⋊C₄×C₁₉	—	2	8	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	4	no	—
98	C₄⋊C₄×C₁₉	—	8	2	no	—
99	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
100	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
101	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
102	C₄²×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	—	4	4	no	—
105	C₄²×C₁₉	—	8	2	no	—
106	C₄²×C₁₉	—	8	2	no	—
107	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₉	#41	1	2	yes	—
109	#23165	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	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	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₈×C₁₉	—	2	2	no	—
142	C₈×C₁₉	—	4	2	no	—
143	C₈×C₁₉	C₄×Q₈	1	4	yes	—
144	C₂×#499	C₂×C₁₉	1	1	yes	characteristic
145	#2370	C₂×C₁₉	1	2	yes	—
146	#1989	C₂×C₁₉	1	2	yes	—
147	#1944	C₂×C₁₉	1	2	yes	—
148	#1921	C₂×C₁₉	1	2	yes	—
149	#1918	C₂×C₁₉	1	2	yes	—
150	#1776	C₂×C₁₉	1	2	yes	—
151	C₄×#107	C₂×C₁₉	1	1	yes	characteristic
152	#1770	C₂×C₁₉	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂²×C₁₉	#547	1	1	yes	characteristic
155	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
156	C₂²×C₁₉	#575	1	1	yes	characteristic
157	C₂²×C₁₉	#575	1	1	yes	characteristic
158	C₂²×C₁₉	#576	1	1	yes	characteristic
159	C₂²×C₁₉	#578	1	1	yes	characteristic
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	4	2	no	—
163	C₄×C₁₉	—	4	2	no	—
164	C₄×C₁₉	—	4	4	no	—
165	C₄×C₁₉	—	4	4	no	—
166	C₄×C₁₉	—	4	4	no	—
167	C₄×C₁₉	—	8	2	no	—
168	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₁₉	#519	1	1	yes	characteristic
170	C₄×C₁₉	#556	1	1	yes	characteristic
171	C₄×C₁₉	#557	1	1	yes	characteristic
172	C₂×#125	C₂²×C₁₉	1	2	yes	—
173	#499	C₄×C₁₉	1	8	yes	complemented
174	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
175	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
178	C₂×#106	C₂²×C₁₉	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×#97	C₂²×C₁₉	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
182	#389	—	2	2	no	—
183	#389	C₂²×C₁₉	1	2	yes	—
184	#389	C₂²×C₁₉	1	2	yes	—
185	#387	—	2	2	no	—
186	#387	—	2	2	no	—
187	C₄×C₄⋊C₄	—	2	2	no	—
188	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
189	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
190	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
191	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
192	#340	C₂²×C₁₉	1	2	yes	—
193	#340	C₂²×C₁₉	1	2	yes	—
194	#340	C₂²×C₁₉	1	2	yes	—
195	#340	C₂²×C₁₉	1	2	yes	—
196	#340	C₂²×C₁₉	1	2	yes	—
197	#336	C₂²×C₁₉	1	2	yes	—
198	C₂×C₁₉	#2365	1	1	yes	characteristic
199	C₂×C₁₉	#2943	1	1	yes	characteristic
200	C₂×C₁₉	#2951	1	1	yes	characteristic
201	C₂×C₁₉	#3352	1	1	yes	characteristic
202	C₂×C₁₉	#3355	1	1	yes	characteristic
203	C₂×C₁₉	#3389	1	1	yes	characteristic
204	C₂×C₁₉	#3390	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
206	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
208	#125	—	2	4	no	—
209	#125	C₂×C₄×C₁₉	1	8	yes	—
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	—	2	2	no	—
214	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
224	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
225	#107	—	2	2	no	—
226	#107	C₂×C₄×C₁₉	1	4	yes	—
227	#107	C₂×C₄×C₁₉	1	4	yes	—
228	#107	Q₈×C₁₉	1	4	yes	—
229	#106	C₂×C₄×C₁₉	1	8	yes	—
230	C₄⋊C₈	—	2	2	no	—
231	#97	—	2	4	no	—
232	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	D₄×C₁₉	1	2	yes	—
239	C₁₉	#23165	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
255	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
256	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
257	C₂×C₈	#41×C₁₉	1	2	yes	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	8	2	no	—
269	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
270	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
271	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	4	no	—
275	C₄²	—	8	2	no	—
276	C₄²	—	8	2	no	—
277	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
278	C₄²	#41×C₁₉	1	2	yes	—
279	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
303	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
304	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
308	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₄×Q₈×C₁₉	1	4	yes	—
313	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
314	C₂²	C₁₉×#547	1	1	yes	characteristic
315	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
316	C₂²	C₁₉×#575	1	1	yes	characteristic
317	C₂²	C₁₉×#575	1	1	yes	characteristic
318	C₂²	C₁₉×#576	1	1	yes	characteristic
319	C₂²	C₁₉×#578	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	2	no	—
328	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
329	C₄	C₁₉×#519	1	1	yes	characteristic
330	C₄	C₁₉×#556	1	1	yes	characteristic
331	C₄	C₁₉×#557	1	1	yes	characteristic
332	C₂	C₁₉×#2365	1	1	yes	characteristic
333	C₂	C₁₉×#2943	1	1	yes	characteristic
334	C₂	C₁₉×#2951	1	1	yes	characteristic
335	C₂	C₁₉×#3352	1	1	yes	characteristic
336	C₂	C₁₉×#3355	1	1	yes	characteristic
337	C₂	C₁₉×#3389	1	1	yes	characteristic
338	C₂	C₁₉×#3390	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. C₁₉×#499	C₄	semidirect
109. #23165	C₁₉	direct (N × H)
173. #499	C₄×C₁₉	semidirect
239. C₁₉	#23165	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×#499	2
C₁₉×#2370	2
C₁₉×#1989	2
C₁₉×#1944	2
C₁₉×#1921	2
C₁₉×#1918	2
C₁₉×#1776	2
C₄×C₁₉×#107	2
C₁₉×#1770	2
#23165	19

Subgroups — C19×#23165 #18029124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23165 #18029124