Subgroups — C₁₉×#23629 #18029588

Order 4864.

1010 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×#419	C₂	1	2	yes	maximal
2	C₂×C₁₉×#391	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2285	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2074	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2072	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1736	C₂	1	2	yes	maximal
7	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
10	C₂×C₁₉×#117	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#419	—	2	8	no	—
14	C₂×C₁₉×#103	C₂²	1	8	yes	—
15	C₁₉×#391	—	2	2	no	—
16	C₁₉×#391	C₂²	1	2	yes	—
17	C₁₉×#391	C₂²	1	2	yes	—
18	C₁₉×#384	C₂²	1	2	yes	—
19	C₁₉×#382	C₂²	1	2	yes	—
20	C₁₉×#380	—	2	2	no	—
21	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
22	C₂×C₁₉×#95	—	2	2	no	—
23	C₁₉×#340	C₂²	1	8	yes	—
24	C₁₉×#336	C₂²	1	2	yes	—
25	C₁₉×#336	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×Q₈×C₁₉	—	2	2	no	—
28	C₂²×Q₈×C₁₉	—	2	4	no	—
29	C₂²×Q₈×C₁₉	—	2	4	no	—
30	C₂²×Q₈×C₁₉	D₄	1	4	yes	—
31	C₂²×Q₈×C₁₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₉	—	2	1	no	—
33	C₂³×C₄×C₁₉	—	2	1	no	—
34	C₂³×C₄×C₁₉	D₄	1	2	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₉	—	2	4	no	—
37	C₂×M₁₆×C₁₉	—	2	4	no	—
38	C₂×M₁₆×C₁₉	—	2	4	no	—
39	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
40	C₁₉×#117	—	2	4	no	—
41	C₁₉×#117	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	4	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	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₄²×C₁₉	—	2	4	no	—
50	C₂×C₄²×C₁₉	—	2	2	no	—
51	C₂×C₄²×C₁₉	D₄	1	2	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₁₉×#103	—	2	16	no	—
55	C₁₉×#95	—	2	4	no	—
56	C₁₉×#95	—	2	4	no	—
57	C₁₉×#95	—	2	4	no	—
58	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	4	no	—
60	C₂×Q₈×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	—	2	16	no	—
62	C₂×Q₈×C₁₉	—	2	4	no	—
63	C₂×Q₈×C₁₉	—	2	4	no	—
64	C₂×Q₈×C₁₉	—	4	2	no	—
65	C₂×Q₈×C₁₉	—	4	2	no	—
66	C₂×Q₈×C₁₉	—	4	8	no	—
67	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₉	—	2	4	no	—
69	C₂²×C₄×C₁₉	—	2	4	no	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	—	2	4	no	—
72	C₂²×C₄×C₁₉	—	2	4	no	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	—	4	2	no	—
75	C₂²×C₄×C₁₉	—	4	2	no	—
76	C₂²×C₄×C₁₉	—	4	2	no	—
77	C₂²×C₄×C₁₉	—	4	4	no	—
78	C₂²×C₄×C₁₉	—	4	2	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₉	#41	1	2	yes	—
84	C₂²×C₄×C₁₉	#41	1	4	yes	—
85	C₂²×C₄×C₁₉	#41	1	4	yes	—
86	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₁₉	—	4	8	no	—
88	C₂×C₈×C₁₉	—	2	4	no	—
89	C₂×C₈×C₁₉	—	2	4	no	—
90	C₂×C₈×C₁₉	—	2	4	no	—
91	C₄⋊C₄×C₁₉	—	2	4	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	4	4	no	—
95	C₄⋊C₄×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	4	4	no	—
97	C₂²⋊C₄×C₁₉	—	4	4	no	—
98	C₄²×C₁₉	—	2	4	no	—
99	C₄²×C₁₉	—	2	2	no	—
100	C₄²×C₁₉	—	2	2	no	—
101	C₄²×C₁₉	—	4	4	no	—
102	#23629	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	—	2	2	no	—
105	C₂³×C₁₉	#121	1	2	yes	—
106	C₂³×C₁₉	#123	1	2	yes	—
107	C₂³×C₁₉	#124	1	2	yes	—
108	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₁₉	—	2	8	no	—
110	Q₈×C₁₉	—	4	4	no	—
111	Q₈×C₁₉	—	4	8	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	4	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	—	4	8	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	8	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	8	4	no	—
128	C₂×C₄×C₁₉	#121	1	2	yes	—
129	C₂×C₄×C₁₉	#123	1	8	yes	—
130	C₂×C₄×C₁₉	#123	1	2	yes	—
131	C₂×C₄×C₁₉	#124	1	2	yes	—
132	C₂×C₄×C₁₉	#126	1	8	yes	—
133	C₂×C₄×C₁₉	#137	1	4	yes	—
134	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
136	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
137	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
138	C₈×C₁₉	—	4	4	no	—
139	C₂×#419	C₂×C₁₉	1	2	yes	—
140	C₂×#391	C₂×C₁₉	1	1	yes	characteristic
141	#2285	C₂×C₁₉	1	1	yes	characteristic
142	#2074	C₂×C₁₉	1	4	yes	complemented
143	#2072	C₂×C₁₉	1	4	yes	complemented
144	#1736	C₂×C₁₉	1	2	yes	—
145	#1728	C₂×C₁₉	1	1	yes	characteristic
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	4	no	—
150	C₂²×C₁₉	#397	1	4	yes	—
151	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×#123	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#137	1	2	yes	—
155	C₂²×C₁₉	#575	1	2	yes	—
156	C₄×C₁₉	—	2	4	no	—
157	C₄×C₁₉	—	4	2	no	—
158	C₄×C₁₉	—	4	4	no	—
159	C₄×C₁₉	—	8	2	no	—
160	C₄×C₁₉	—	8	4	no	—
161	C₄×C₁₉	#397	1	4	yes	—
162	C₄×C₁₉	C₂×#123	1	2	yes	—
163	C₄×C₁₉	C₂×#126	1	2	yes	—
164	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
165	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
166	C₂×#117	C₂²×C₁₉	1	2	yes	—
167	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
168	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
169	#419	—	2	8	no	—
170	C₂×#103	C₂²×C₁₉	1	8	yes	—
171	#391	—	2	2	no	—
172	#391	C₂²×C₁₉	1	2	yes	—
173	#391	C₂²×C₁₉	1	2	yes	—
174	#384	C₂²×C₁₉	1	2	yes	—
175	#382	C₂²×C₁₉	1	2	yes	—
176	#380	—	2	2	no	—
177	C₄×C₄⋊C₄	—	2	2	no	—
178	C₂×#95	—	2	2	no	—
179	#340	C₂²×C₁₉	1	8	yes	—
180	#336	C₂²×C₁₉	1	2	yes	—
181	#336	C₂²×C₁₉	1	2	yes	—
182	C₂×C₁₉	—	2	4	no	—
183	C₂×C₁₉	C₂×#397	1	1	yes	characteristic
184	C₂×C₁₉	#3037	1	2	yes	—
185	C₂×C₁₉	#3095	1	2	yes	—
186	C₂×C₁₉	#3139	1	1	yes	characteristic
187	C₂×C₁₉	#3140	1	1	yes	characteristic
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	D₄×C₁₉	1	4	yes	—
193	C₂²×Q₈	C₂³×C₁₉	1	2	yes	—
194	C₂³×C₄	—	2	1	no	—
195	C₂³×C₄	—	2	1	no	—
196	C₂³×C₄	D₄×C₁₉	1	2	yes	—
197	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	4	no	—
201	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
202	#117	—	2	4	no	—
203	#117	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	D₄×C₁₉	1	2	yes	—
214	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
216	#103	—	2	16	no	—
217	#95	—	2	4	no	—
218	#95	—	2	4	no	—
219	#95	—	2	4	no	—
220	C₁₉	#23629	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	16	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	8	no	—
230	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
246	C₂²×C₄	#41×C₁₉	1	2	yes	—
247	C₂²×C₄	#41×C₁₉	1	4	yes	—
248	C₂²×C₄	#41×C₁₉	1	4	yes	—
249	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
250	M₁₆	—	4	8	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	4	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₁₉×#121	1	2	yes	—
268	C₂³	C₁₉×#123	1	2	yes	—
269	C₂³	C₁₉×#124	1	2	yes	—
270	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
271	Q₈	—	2	8	no	—
272	Q₈	—	4	8	no	—
273	Q₈	—	4	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	C₁₉×#121	1	2	yes	—
291	C₂×C₄	C₁₉×#123	1	2	yes	—
292	C₂×C₄	C₁₉×#123	1	8	yes	—
293	C₂×C₄	C₁₉×#124	1	2	yes	—
294	C₂×C₄	C₁₉×#126	1	8	yes	—
295	C₂×C₄	C₁₉×#137	1	4	yes	—
296	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
298	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
299	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
300	C₈	—	4	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₁₉×#397	1	4	yes	—
306	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
307	C₂²	C₂×C₁₉×#123	1	1	yes	characteristic
308	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
309	C₂²	C₂×C₁₉×#137	1	2	yes	—
310	C₂²	C₁₉×#575	1	2	yes	—
311	C₄	—	2	4	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	4	no	—
314	C₄	—	8	4	no	—
315	C₄	—	8	2	no	—
316	C₄	C₁₉×#397	1	4	yes	—
317	C₄	C₂×C₁₉×#123	1	2	yes	—
318	C₄	C₂×C₁₉×#126	1	2	yes	—
319	C₂	—	2	4	no	—
320	C₂	C₂×C₁₉×#397	1	1	yes	characteristic
321	C₂	C₁₉×#3037	1	2	yes	—
322	C₂	C₁₉×#3095	1	2	yes	—
323	C₂	C₁₉×#3139	1	1	yes	characteristic
324	C₂	C₁₉×#3140	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×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×#2074	C₂	semidirect
5. C₁₉×#2072	C₂	semidirect
102. #23629	C₁₉	direct (N × H)
142. #2074	C₂×C₁₉	semidirect
143. #2072	C₂×C₁₉	semidirect
220. C₁₉	#23629	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×#419	2
C₂×C₁₉×#391	2
C₁₉×#2285	2
C₁₉×#2074	2
C₁₉×#2072	2
C₁₉×#1736	2
C₁₉×#1728	2
#23629	19

Subgroups — C19×#23629 #18029588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23629 #18029588