Subgroups — C₁₉×#19345 #18025304

Order 4864.

568 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3255	C₂	1	2	yes	maximal
2	C₁₉×#3170	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2960	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1700	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1674	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1670	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#1561	C₂	1	2	yes	maximal
9	C₁₉×#533	C₂²	1	2	yes	—
10	C₁₉×#533	C₂²	1	1	yes	characteristic
11	C₁₉×#500	C₂²	1	1	yes	characteristic
12	C₁₉×#494	C₂²	1	1	yes	characteristic
13	C₁₉×#487	C₂²	1	2	yes	—
14	C₁₉×#475	—	2	2	no	—
15	C₁₉×#475	C₂²	1	4	yes	—
16	C₁₉×#470	—	2	4	no	—
17	C₁₉×#467	—	2	1	no	—
18	C₁₉×#467	—	2	1	no	—
19	C₁₉×#462	—	2	2	no	—
20	C₁₉×#462	C₂²	1	4	yes	—
21	C₁₉×#446	—	2	1	no	—
22	C₁₉×#446	C₂²	1	2	yes	—
23	C₈×Q₈×C₁₉	—	2	1	no	—
24	C₈×Q₈×C₁₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₁₉	—	2	1	no	—
26	C₄×Q₁₆×C₁₉	—	2	1	no	—
27	C₁₉×#424	—	2	1	no	—
28	C₁₉×#423	C₂²	1	1	yes	characteristic
29	C₁₉×#423	C₂²	1	2	yes	—
30	C₁₉×#405	—	2	1	no	—
31	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
32	C₁₉×#335	C₂²	1	2	yes	—
33	C₁₉×#335	C₂²	1	2	yes	—
34	C₁₉×#332	C₂²	1	4	yes	—
35	C₁₉×#332	C₂²	1	4	yes	—
36	C₂×Q₁₆×C₁₉	—	2	4	no	—
37	C₂×Q₁₆×C₁₉	—	2	1	no	—
38	C₂×Q₁₆×C₁₉	—	2	1	no	—
39	C₂×M₁₆×C₁₉	—	2	1	no	—
40	C₂×M₁₆×C₁₉	—	2	2	no	—
41	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
43	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₁₉×#125	—	2	1	no	—
48	C₁₉×#125	—	2	2	no	—
49	C₁₉×C₂²⋊Q₈	—	2	1	no	—
50	C₁₉×C₂²⋊Q₈	—	2	2	no	—
51	C₁₉×C₂²⋊Q₈	—	2	2	no	—
52	C₁₉×C₂²⋊Q₈	D₄	1	2	yes	—
53	C₄×Q₈×C₁₉	—	2	1	no	—
54	C₄×Q₈×C₁₉	—	2	2	no	—
55	C₄×Q₈×C₁₉	—	2	2	no	—
56	C₄×Q₈×C₁₉	C₂³	1	2	yes	—
57	C₁₉×#117	—	2	1	no	—
58	C₁₉×#117	—	2	2	no	—
59	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₁₉×#107	—	2	1	no	—
61	C₁₉×#107	—	2	1	no	—
62	C₁₉×#106	—	2	4	no	—
63	C₁₉×C₄⋊C₈	—	2	2	no	—
64	C₁₉×C₄⋊C₈	—	2	2	no	—
65	C₁₉×C₄⋊C₈	—	2	1	no	—
66	C₁₉×C₄⋊C₈	—	2	2	no	—
67	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
69	C₁₉×#103	—	2	4	no	—
70	C₁₉×#103	—	2	2	no	—
71	C₁₉×#103	—	2	2	no	—
72	C₁₉×#103	—	2	4	no	—
73	C₁₉×#97	—	2	2	no	—
74	C₄×C₈×C₁₉	—	2	2	no	—
75	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	C₂×D₄	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	#41	1	1	yes	characteristic
86	Q₁₆×C₁₉	—	4	2	no	—
87	Q₁₆×C₁₉	—	4	2	no	—
88	Q₁₆×C₁₉	—	8	4	no	—
89	M₁₆×C₁₉	—	4	2	no	—
90	M₁₆×C₁₉	—	8	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	2	4	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	2	2	no	—
108	C₄⋊C₄×C₁₉	—	4	2	no	—
109	C₄⋊C₄×C₁₉	—	4	2	no	—
110	C₄⋊C₄×C₁₉	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₁₉	—	2	2	no	—
112	C₂²⋊C₄×C₁₉	—	4	2	no	—
113	C₄²×C₁₉	—	2	2	no	—
114	C₄²×C₁₉	—	4	2	no	—
115	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	#41	1	1	yes	characteristic
118	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
119	#19345	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	#121	1	1	yes	characteristic
121	Q₈×C₁₉	—	2	2	no	—
122	Q₈×C₁₉	—	2	2	no	—
123	Q₈×C₁₉	—	4	2	no	—
124	Q₈×C₁₉	—	4	2	no	—
125	Q₈×C₁₉	—	4	2	no	—
126	Q₈×C₁₉	—	4	2	no	—
127	Q₈×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₁₉	—	2	1	no	—
144	C₈×C₁₉	—	2	1	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	—	4	2	no	—
147	C₈×C₁₉	—	4	4	no	—
148	C₈×C₁₉	#137	1	2	yes	—
149	#3255	C₂×C₁₉	1	2	yes	—
150	#3170	C₂×C₁₉	1	1	yes	characteristic
151	#2960	C₂×C₁₉	1	1	yes	characteristic
152	#1700	C₂×C₁₉	1	4	yes	complemented
153	#1674	C₂×C₁₉	1	2	yes	complemented
154	#1670	C₂×C₁₉	1	2	yes	complemented
155	#1567	C₂×C₁₉	1	1	yes	characteristic
156	#1561	C₂×C₁₉	1	2	yes	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	4	2	no	—
165	C₄×C₁₉	—	4	2	no	—
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	—	8	2	no	—
168	C₄×C₁₉	#473	1	2	yes	—
169	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
170	C₄×C₁₉	#575	1	1	yes	characteristic
171	C₄×C₁₉	#576	1	1	yes	characteristic
172	C₄×C₁₉	#578	1	1	yes	characteristic
173	#533	C₂²×C₁₉	1	2	yes	—
174	#533	C₂²×C₁₉	1	1	yes	characteristic
175	#500	C₂²×C₁₉	1	1	yes	characteristic
176	#494	C₂²×C₁₉	1	1	yes	characteristic
177	#487	C₂²×C₁₉	1	2	yes	—
178	#475	—	2	2	no	—
179	#475	C₂²×C₁₉	1	4	yes	—
180	#470	—	2	4	no	—
181	#467	—	2	1	no	—
182	#467	—	2	1	no	—
183	#462	—	2	2	no	—
184	#462	C₂²×C₁₉	1	4	yes	—
185	#446	—	2	1	no	—
186	#446	C₂²×C₁₉	1	2	yes	—
187	C₈×Q₈	—	2	1	no	—
188	C₈×Q₈	C₂²×C₁₉	1	2	yes	—
189	C₄×Q₁₆	—	2	1	no	—
190	C₄×Q₁₆	—	2	1	no	—
191	#424	—	2	1	no	—
192	#423	C₂²×C₁₉	1	1	yes	characteristic
193	#423	C₂²×C₁₉	1	2	yes	—
194	#405	—	2	1	no	—
195	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
196	#335	C₂²×C₁₉	1	2	yes	—
197	#335	C₂²×C₁₉	1	2	yes	—
198	#332	C₂²×C₁₉	1	4	yes	—
199	#332	C₂²×C₁₉	1	4	yes	—
200	C₂×C₁₉	—	4	1	no	—
201	C₂×C₁₉	#3037	1	1	yes	characteristic
202	C₂×C₁₉	#3061	1	1	yes	characteristic
203	C₂×C₁₉	C₂×#473	1	1	yes	characteristic
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
211	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
212	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
213	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
215	#125	—	2	1	no	—
216	#125	—	2	2	no	—
217	C₂²⋊Q₈	—	2	1	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	D₄×C₁₉	1	2	yes	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	C₂³×C₁₉	1	2	yes	—
225	#117	—	2	1	no	—
226	#117	—	2	2	no	—
227	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#106	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
236	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
237	#103	—	2	4	no	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	#103	—	2	4	no	—
241	#97	—	2	2	no	—
242	C₄×C₈	—	2	2	no	—
243	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
244	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
247	C₁₉	#19345	1	1	yes	characteristic, direct product factor, Sylow
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
254	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
255	Q₁₆	—	4	2	no	—
256	Q₁₆	—	4	2	no	—
257	Q₁₆	—	8	4	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	8	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	C₂×D₄×C₁₉	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
286	C₄²	#41×C₁₉	1	1	yes	characteristic
287	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
288	C₂³	C₁₉×#121	1	1	yes	characteristic
289	Q₈	—	2	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
306	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
307	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	4	no	—
316	C₈	C₁₉×#137	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₁₉×#473	1	2	yes	—
329	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
330	C₄	C₁₉×#575	1	1	yes	characteristic
331	C₄	C₁₉×#576	1	1	yes	characteristic
332	C₄	C₁₉×#578	1	1	yes	characteristic
333	C₂	—	4	1	no	—
334	C₂	C₁₉×#3037	1	1	yes	characteristic
335	C₂	C₁₉×#3061	1	1	yes	characteristic
336	C₂	C₂×C₁₉×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	#121
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
4. C₁₉×#1700	C₂	semidirect
5. C₁₉×#1674	C₂	semidirect
6. C₁₉×#1670	C₂	semidirect
119. #19345	C₁₉	direct (N × H)
152. #1700	C₂×C₁₉	semidirect
153. #1674	C₂×C₁₉	semidirect
154. #1670	C₂×C₁₉	semidirect
247. C₁₉	#19345	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×#3255	2
C₁₉×#3170	2
C₁₉×#2960	2
C₁₉×#1700	2
C₁₉×#1674	2
C₁₉×#1670	2
C₁₉×#1567	2
C₁₉×#1561	2
#19345	19

Subgroups — C19×#19345 #18025304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19345 #18025304