Subgroups — C₁₉×#19001 #18024960

Order 4864.

960 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#562	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3213	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3142	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1662	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1656	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1534	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1527	C₂	1	2	yes	maximal, complemented
8	C₁₉×#562	—	2	2	no	—
9	C₁₉×#562	—	2	4	no	—
10	C₁₉×#562	C₂²	1	4	yes	—
11	C₁₉×#532	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#124	C₂²	1	2	yes	—
14	C₂×C₁₉×#123	—	2	2	no	—
15	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
16	C₂×C₁₉×#121	—	2	2	no	—
17	C₁₉×#496	C₂²	1	4	yes	complemented
18	C₁₉×#491	C₂²	1	2	yes	complemented
19	C₁₉×#490	C₂²	1	2	yes	complemented
20	C₁₉×#489	C₂²	1	4	yes	complemented
21	C₁₉×#488	—	2	2	no	—
22	C₁₉×#463	—	2	4	no	—
23	C₁₉×#448	—	2	2	no	—
24	C₁₉×#447	—	2	2	no	—
25	C₁₉×#432	C₂²	1	2	yes	—
26	C₁₉×#417	—	2	2	no	—
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#330	C₂²	1	4	yes	complemented
29	C₁₉×#329	C₂²	1	8	yes	complemented
30	C₂×#41×C₁₉	—	2	1	no	—
31	C₂²×Q₈×C₁₉	D₄	1	2	yes	—
32	C₂²×D₄×C₁₉	—	2	2	no	—
33	C₂²×D₄×C₁₉	D₄	1	2	yes	—
34	C₂³×C₄×C₁₉	—	2	2	no	—
35	C₂×SD₁₆×C₁₉	—	2	4	no	—
36	C₂×D₈×C₁₉	—	2	4	no	—
37	C₂×M₁₆×C₁₉	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
39	C₁₉×#127	C₂³	1	1	yes	characteristic
40	C₁₉×#125	—	2	2	no	—
41	C₁₉×#125	D₄	1	2	yes	—
42	C₁₉×#125	C₂³	1	2	yes	—
43	C₁₉×#124	—	2	2	no	—
44	C₁₉×#124	—	2	2	no	—
45	C₁₉×#124	—	2	1	no	—
46	C₁₉×#124	D₄	1	4	yes	—
47	C₁₉×#124	C₂³	1	4	yes	—
48	C₁₉×#123	—	2	8	no	—
49	C₁₉×#123	—	2	8	no	—
50	C₁₉×C₂²⋊Q₈	—	2	8	no	—
51	C₁₉×C₂²⋊Q₈	—	2	8	no	—
52	C₁₉×#121	—	2	8	no	—
53	C₁₉×#121	—	2	8	no	—
54	C₁₉×#121	—	2	2	no	—
55	C₄×D₄×C₁₉	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
62	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₁₉×C₄⋊C₈	—	2	2	no	—
64	C₁₉×#103	—	2	4	no	—
65	C₁₉×#103	—	2	4	no	—
66	C₁₉×#102	—	2	4	no	—
67	C₁₉×#102	—	2	4	no	—
68	C₁₉×C₂²⋊C₈	—	2	2	no	—
69	C₁₉×#97	C₂³	1	2	yes	—
70	C₁₉×#97	C₂³	1	4	yes	—
71	C₄×C₈×C₁₉	—	2	1	no	—
72	C₂⁴×C₁₉	—	2	2	no	—
73	#41×C₁₉	—	4	4	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	1	no	—
76	C₂×Q₈×C₁₉	—	8	2	no	—
77	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₁₉	—	2	8	no	—
79	C₂×D₄×C₁₉	—	2	2	no	—
80	C₂×D₄×C₁₉	—	2	2	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂×D₄×C₁₉	—	8	2	no	—
84	C₂×D₄×C₁₉	—	8	4	no	—
85	C₂×D₄×C₁₉	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₉	—	2	2	no	—
87	C₂²×C₄×C₁₉	—	2	2	no	—
88	C₂²×C₄×C₁₉	—	2	8	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	—	8	4	no	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
93	SD₁₆×C₁₉	—	8	4	no	—
94	D₈×C₁₉	—	8	4	no	—
95	M₁₆×C₁₉	—	4	2	no	—
96	C₂×C₈×C₁₉	—	2	2	no	—
97	C₂×C₈×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	2	8	no	—
99	C₄⋊C₄×C₁₉	—	2	4	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	4	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
104	C₂²⋊C₄×C₁₉	—	2	8	no	—
105	C₂²⋊C₄×C₁₉	—	2	4	no	—
106	C₂²⋊C₄×C₁₉	—	2	4	no	—
107	C₂²⋊C₄×C₁₉	—	2	4	no	—
108	C₂²⋊C₄×C₁₉	—	2	4	no	—
109	C₂²⋊C₄×C₁₉	—	4	2	no	—
110	C₄²×C₁₉	C₂×D₄	1	2	yes	—
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
113	#19001	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₁₉	—	2	4	no	—
115	C₂³×C₁₉	—	4	4	no	—
116	C₂³×C₁₉	—	4	4	no	—
117	C₂³×C₁₉	—	4	4	no	—
118	C₂³×C₁₉	—	4	1	no	—
119	C₂³×C₁₉	#120	1	1	yes	characteristic
120	Q₈×C₁₉	—	4	4	no	—
121	Q₈×C₁₉	—	4	2	no	—
122	Q₈×C₁₉	—	8	2	no	—
123	D₄×C₁₉	—	4	4	no	—
124	D₄×C₁₉	—	4	4	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	8	2	no	—
127	D₄×C₁₉	—	8	8	no	—
128	D₄×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	4	no	—
134	C₂×C₄×C₁₉	—	4	4	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	8	2	no	—
137	C₂×C₄×C₁₉	—	8	8	no	—
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#120	1	2	yes	—
140	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
142	C₈×C₁₉	—	4	2	no	—
143	C₈×C₁₉	—	4	4	no	—
144	C₂×#562	C₂×C₁₉	1	1	yes	characteristic, complemented
145	#3213	C₂×C₁₉	1	2	yes	complemented
146	#3142	C₂×C₁₉	1	1	yes	characteristic, complemented
147	#1662	C₂×C₁₉	1	4	yes	complemented
148	#1656	C₂×C₁₉	1	4	yes	complemented
149	#1534	C₂×C₁₉	1	1	yes	characteristic, complemented
150	#1527	C₂×C₁₉	1	2	yes	complemented
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	4	4	no	—
153	C₂²×C₁₉	—	4	4	no	—
154	C₂²×C₁₉	—	4	4	no	—
155	C₂²×C₁₉	—	8	8	no	—
156	C₂²×C₁₉	—	8	1	no	—
157	C₂²×C₁₉	—	8	2	no	—
158	C₂²×C₁₉	#454	1	2	yes	—
159	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	4	4	no	—
165	C₄×C₁₉	—	4	4	no	—
166	C₄×C₁₉	—	8	1	no	—
167	#562	—	2	2	no	—
168	#562	—	2	4	no	—
169	#562	C₂²×C₁₉	1	4	yes	—
170	#532	C₂²×C₁₉	1	1	yes	characteristic
171	C₂×#125	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×#124	C₂²×C₁₉	1	2	yes	—
173	C₂×#123	—	2	2	no	—
174	C₂×C₂²⋊Q₈	—	2	2	no	—
175	C₂×#121	—	2	2	no	—
176	#496	C₂²×C₁₉	1	4	yes	complemented
177	#491	C₂²×C₁₉	1	2	yes	complemented
178	#490	C₂²×C₁₉	1	2	yes	complemented
179	#489	C₂²×C₁₉	1	4	yes	complemented
180	#488	—	2	2	no	—
181	#463	—	2	4	no	—
182	#448	—	2	2	no	—
183	#447	—	2	2	no	—
184	#432	C₂²×C₁₉	1	2	yes	—
185	#417	—	2	2	no	—
186	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
187	#330	C₂²×C₁₉	1	4	yes	complemented
188	#329	C₂²×C₁₉	1	8	yes	complemented
189	C₂×C₁₉	—	2	2	no	—
190	C₂×C₁₉	—	8	4	no	—
191	C₂×C₁₉	—	16	1	no	—
192	C₂×C₁₉	C₂×#454	1	1	yes	characteristic
193	C₂×C₁₉	#3019	1	2	yes	—
194	C₂×#41	—	2	1	no	—
195	C₂²×Q₈	D₄×C₁₉	1	2	yes	—
196	C₂²×D₄	—	2	2	no	—
197	C₂²×D₄	D₄×C₁₉	1	2	yes	—
198	C₂³×C₄	—	2	2	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
203	#127	C₂³×C₁₉	1	1	yes	characteristic
204	#125	—	2	2	no	—
205	#125	D₄×C₁₉	1	2	yes	—
206	#125	C₂³×C₁₉	1	2	yes	—
207	#124	—	2	2	no	—
208	#124	—	2	2	no	—
209	#124	—	2	1	no	—
210	#124	D₄×C₁₉	1	4	yes	—
211	#124	C₂³×C₁₉	1	4	yes	—
212	#123	—	2	8	no	—
213	#123	—	2	8	no	—
214	C₂²⋊Q₈	—	2	8	no	—
215	C₂²⋊Q₈	—	2	8	no	—
216	#121	—	2	8	no	—
217	#121	—	2	8	no	—
218	#121	—	2	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
227	C₄⋊C₈	—	2	2	no	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	C₂²⋊C₈	—	2	2	no	—
233	#97	C₂³×C₁₉	1	2	yes	—
234	#97	C₂³×C₁₉	1	4	yes	—
235	C₄×C₈	—	2	1	no	—
236	C₁₉	#19001	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁴	—	2	2	no	—
238	#41	—	4	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	8	2	no	—
242	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
243	C₂×D₄	—	2	8	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	8	2	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂×D₄	C₂×D₄×C₁₉	1	4	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	8	4	no	—
256	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
258	SD₁₆	—	8	4	no	—
259	D₈	—	8	4	no	—
260	M₁₆	—	4	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
269	C₂²⋊C₄	—	2	8	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₉	1	2	yes	—
277	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
278	C₂³	—	2	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₁₉×#120	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	2	no	—
291	D₄	—	8	8	no	—
292	D₄	—	8	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	8	no	—
302	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
303	C₂×C₄	C₁₉×#120	1	2	yes	—
304	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	8	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₁₉×#454	1	2	yes	—
316	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	8	4	no	—
326	C₂	—	16	1	no	—
327	C₂	C₂×C₁₉×#454	1	1	yes	characteristic
328	C₂	C₁₉×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#562	C₂	semidirect
2. C₁₉×#3213	C₂	semidirect
3. C₁₉×#3142	C₂	semidirect
4. C₁₉×#1662	C₂	semidirect
5. C₁₉×#1656	C₂	semidirect
6. C₁₉×#1534	C₂	semidirect
7. C₁₉×#1527	C₂	semidirect
17. C₁₉×#496	C₂²	semidirect
18. C₁₉×#491	C₂²	semidirect
19. C₁₉×#490	C₂²	semidirect
20. C₁₉×#489	C₂²	semidirect
28. C₁₉×#330	C₂²	semidirect
29. C₁₉×#329	C₂²	semidirect
113. #19001	C₁₉	direct (N × H)
144. C₂×#562	C₂×C₁₉	semidirect
145. #3213	C₂×C₁₉	semidirect
146. #3142	C₂×C₁₉	semidirect
147. #1662	C₂×C₁₉	semidirect
148. #1656	C₂×C₁₉	semidirect
149. #1534	C₂×C₁₉	semidirect
150. #1527	C₂×C₁₉	semidirect
176. #496	C₂²×C₁₉	semidirect
177. #491	C₂²×C₁₉	semidirect
178. #490	C₂²×C₁₉	semidirect
179. #489	C₂²×C₁₉	semidirect
187. #330	C₂²×C₁₉	semidirect
188. #329	C₂²×C₁₉	semidirect
236. C₁₉	#19001	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×#562	2
C₁₉×#3213	2
C₁₉×#3142	2
C₁₉×#1662	2
C₁₉×#1656	2
C₁₉×#1534	2
C₁₉×#1527	2
#19001	19

Subgroups — C19×#19001 #18024960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19001 #18024960