Subgroups — C₁₉×#9170 #18015129

Order 4864.

432 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3079	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2925	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1477	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#477	—	2	1	no	—
11	C₁₉×#476	—	2	1	no	—
12	C₁₉×#475	C₄	1	2	yes	—
13	C₁₉×#474	C₄	1	2	yes	—
14	C₁₉×#433	—	2	2	no	—
15	C₁₉×#432	—	2	1	no	—
16	C₁₉×#431	—	2	1	no	—
17	C₁₉×#431	C₂²	1	1	yes	characteristic
18	C₁₉×#428	—	2	1	no	—
19	C₁₉×#423	C₂²	1	1	yes	characteristic
20	C₁₉×#423	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₉×#417	—	2	1	no	—
24	C₁₉×#408	—	2	1	no	—
25	C₁₉×#405	—	2	1	no	—
26	C₁₉×#333	—	2	1	no	—
27	C₁₉×#333	—	2	1	no	—
28	C₁₉×#332	—	2	2	no	—
29	C₁₉×Q₈⋊C₈	—	2	1	no	—
30	C₁₉×D₄⋊C₈	—	2	1	no	—
31	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×Q₈×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	—	2	1	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₂²×C₈×C₁₉	—	2	1	no	—
36	C₂²×C₈×C₁₉	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	—	2	1	no	—
40	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
43	C₄×D₄×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
47	C₁₉×#117	—	2	1	no	—
48	C₁₉×#117	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₁₉×#107	—	2	2	no	—
54	C₁₉×#106	—	2	2	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
65	C₁₉×#103	—	2	2	no	—
66	C₁₉×#102	—	2	2	no	—
67	C₁₉×C₂²⋊C₈	—	2	2	no	—
68	C₁₉×C₂²⋊C₈	—	2	2	no	—
69	C₁₉×C₂²⋊C₈	—	2	1	no	—
70	C₁₉×#97	—	2	2	no	—
71	C₄×C₈×C₁₉	—	2	2	no	—
72	C₄×C₈×C₁₉	—	2	1	no	—
73	#41×C₁₉	—	2	1	no	—
74	#41×C₁₉	—	2	1	no	—
75	#41×C₁₉	—	2	1	no	—
76	#41×C₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂×Q₈×C₁₉	—	8	2	no	—
80	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	—	2	1	no	—
84	C₂²×C₄×C₁₉	—	2	1	no	—
85	C₂²×C₄×C₁₉	—	2	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
90	M₁₆×C₁₉	—	4	1	no	—
91	M₁₆×C₁₉	—	4	1	no	—
92	M₁₆×C₁₉	—	4	1	no	—
93	M₁₆×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₂×C₈×C₁₉	—	4	1	no	—
101	C₂×C₈×C₁₉	—	4	1	no	—
102	C₂×C₈×C₁₉	—	8	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	4	1	no	—
108	C₄⋊C₄×C₁₉	—	4	1	no	—
109	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
111	C₂²⋊C₄×C₁₉	—	2	1	no	—
112	C₂²⋊C₄×C₁₉	—	2	1	no	—
113	C₂²⋊C₄×C₁₉	—	2	1	no	—
114	C₄²×C₁₉	—	2	1	no	—
115	C₄²×C₁₉	—	2	1	no	—
116	C₄²×C₁₉	—	2	1	no	—
117	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
121	#9170	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₁₉	—	2	1	no	—
123	C₂³×C₁₉	#123	1	1	yes	characteristic
124	Q₈×C₁₉	—	2	1	no	—
125	Q₈×C₁₉	—	2	1	no	—
126	Q₈×C₁₉	—	8	2	no	—
127	Q₈×C₁₉	—	8	2	no	—
128	D₄×C₁₉	—	2	1	no	—
129	D₄×C₁₉	—	2	1	no	—
130	D₄×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	8	1	no	—
144	C₂×C₄×C₁₉	—	8	2	no	—
145	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
155	C₈×C₁₉	—	4	2	no	—
156	C₈×C₁₉	—	4	1	no	—
157	C₈×C₁₉	—	4	1	no	—
158	C₈×C₁₉	—	4	2	no	—
159	C₈×C₁₉	—	8	1	no	—
160	C₈×C₁₉	—	8	1	no	—
161	#3079	C₂×C₁₉	1	1	yes	characteristic
162	#2925	C₂×C₁₉	1	1	yes	characteristic
163	#1549	C₂×C₁₉	1	1	yes	characteristic, complemented
164	C₂×#333	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1477	C₂×C₁₉	1	1	yes	characteristic
166	#1301	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#1292	C₂×C₁₉	1	1	yes	characteristic, complemented
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	4	1	no	—
171	C₂²×C₁₉	—	4	1	no	—
172	C₂²×C₁₉	—	4	1	no	—
173	C₂²×C₁₉	#386	1	1	yes	characteristic
174	C₂²×C₁₉	#485	1	1	yes	characteristic
175	C₂²×C₁₉	#485	1	1	yes	characteristic
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	4	1	no	—
182	C₄×C₁₉	—	8	2	no	—
183	C₄×C₁₉	#416	1	1	yes	characteristic
184	C₄×C₁₉	#419	1	1	yes	characteristic
185	C₄×C₁₉	#452	1	1	yes	characteristic
186	C₄×C₁₉	#452	1	1	yes	characteristic
187	C₄×C₁₉	#471	1	1	yes	characteristic
188	C₄×C₁₉	#473	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
190	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
191	#477	—	2	1	no	—
192	#476	—	2	1	no	—
193	#475	C₄×C₁₉	1	2	yes	—
194	#474	C₄×C₁₉	1	2	yes	—
195	#433	—	2	2	no	—
196	#432	—	2	1	no	—
197	#431	—	2	1	no	—
198	#431	C₂²×C₁₉	1	1	yes	characteristic
199	#428	—	2	1	no	—
200	#423	C₂²×C₁₉	1	1	yes	characteristic
201	#423	C₂²×C₁₉	1	1	yes	characteristic
202	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
203	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
204	#417	—	2	1	no	—
205	#408	—	2	1	no	—
206	#405	—	2	1	no	—
207	#333	—	2	1	no	—
208	#333	—	2	1	no	—
209	#332	—	2	2	no	—
210	Q₈⋊C₈	—	2	1	no	—
211	D₄⋊C₈	—	2	1	no	—
212	C₂×C₁₉	—	2	1	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	8	1	no	—
215	C₂×C₁₉	#1873	1	1	yes	characteristic
216	C₂×C₁₉	#1903	1	1	yes	characteristic
217	C₂×C₁₉	#1908	1	1	yes	characteristic
218	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
219	C₂²×Q₈	—	2	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
228	C₄×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
229	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
233	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
234	#117	—	2	1	no	—
235	#117	D₄×C₁₉	1	1	yes	characteristic
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄²	—	2	1	no	—
239	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
240	#107	—	2	2	no	—
241	#106	—	2	2	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
249	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
250	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
251	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
252	#103	—	2	2	no	—
253	#102	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	1	no	—
257	#97	—	2	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	1	no	—
260	C₁₉	#9170	1	1	yes	characteristic, direct product factor, Sylow
261	#41	—	2	1	no	—
262	#41	—	2	1	no	—
263	#41	—	2	1	no	—
264	#41	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	8	2	no	—
268	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
277	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
278	M₁₆	—	4	1	no	—
279	M₁₆	—	4	1	no	—
280	M₁₆	—	4	1	no	—
281	M₁₆	—	4	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	8	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
298	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
309	C₂³	—	2	1	no	—
310	C₂³	C₁₉×#123	1	1	yes	characteristic
311	Q₈	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	8	2	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
333	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
334	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
335	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
336	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
337	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
338	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
339	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
340	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
341	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
342	C₈	—	4	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	8	1	no	—
347	C₈	—	8	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	C₁₉×#386	1	1	yes	characteristic
354	C₂²	C₁₉×#485	1	1	yes	characteristic
355	C₂²	C₁₉×#485	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	8	2	no	—
363	C₄	C₁₉×#416	1	1	yes	characteristic
364	C₄	C₁₉×#419	1	1	yes	characteristic
365	C₄	C₁₉×#452	1	1	yes	characteristic
366	C₄	C₁₉×#452	1	1	yes	characteristic
367	C₄	C₁₉×#471	1	1	yes	characteristic
368	C₄	C₁₉×#473	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	8	1	no	—
372	C₂	C₁₉×#1873	1	1	yes	characteristic
373	C₂	C₁₉×#1903	1	1	yes	characteristic
374	C₂	C₁₉×#1908	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	D₄
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
3. C₁₉×#1549	C₂	semidirect
4. C₂×C₁₉×#333	C₂	semidirect
6. C₁₉×#1301	C₂	semidirect
7. C₁₉×#1292	C₂	semidirect
121. #9170	C₁₉	direct (N × H)
163. #1549	C₂×C₁₉	semidirect
164. C₂×#333	C₂×C₁₉	semidirect
166. #1301	C₂×C₁₉	semidirect
167. #1292	C₂×C₁₉	semidirect
260. C₁₉	#9170	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₉×#3079	2
C₁₉×#2925	2
C₁₉×#1549	2
C₂×C₁₉×#333	2
C₁₉×#1477	2
C₁₉×#1301	2
C₁₉×#1292	2
#9170	19

Subgroups — C19×#9170 #18015129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9170 #18015129