Subgroups — C₁₉×#9748 #18015707

Order 4864.

458 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×#487	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1988	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1981	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₁₉×#487	—	2	2	no	—
11	C₁₉×#487	C₄	1	4	yes	complemented
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#103	—	2	1	no	—
16	C₂×C₁₉×#103	—	2	1	no	—
17	C₂×C₁₉×#103	—	2	2	no	—
18	C₂×C₁₉×#97	—	2	1	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#340	—	2	2	no	—
22	C₁₉×#336	—	2	1	no	—
23	C₁₉×#335	—	2	2	no	—
24	C₁₉×#332	—	2	2	no	—
25	C₁₉×#332	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₉	—	2	1	no	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
54	C₁₉×#103	—	2	4	no	—
55	C₁₉×#103	—	2	4	no	—
56	C₁₉×#103	—	4	4	no	—
57	C₁₉×#97	—	4	2	no	—
58	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₉	D₄	1	2	yes	—
60	C₁₉×#95	—	2	1	no	—
61	C₂×Q₈×C₁₉	—	2	2	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	4	1	no	—
64	C₂×Q₈×C₁₉	—	4	1	no	—
65	C₂×Q₈×C₁₉	—	4	1	no	—
66	C₂×Q₈×C₁₉	—	4	1	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	2	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	2	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	1	no	—
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	—	4	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₂×C₈×C₁₉	—	4	2	no	—
87	C₂×C₈×C₁₉	—	4	2	no	—
88	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
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	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	—	4	1	no	—
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	#41	1	1	yes	characteristic
106	#9748	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₁₉	#121	1	1	yes	characteristic
108	Q₈×C₁₉	—	4	2	no	—
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	—	4	4	no	—
111	Q₈×C₁₉	—	8	1	no	—
112	Q₈×C₁₉	—	8	1	no	—
113	Q₈×C₁₉	—	8	1	no	—
114	Q₈×C₁₉	—	8	1	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#137	1	2	yes	—
136	C₈×C₁₉	—	2	2	no	—
137	C₈×C₁₉	—	2	2	no	—
138	C₈×C₁₉	—	4	2	no	—
139	C₈×C₁₉	—	8	1	no	—
140	C₈×C₁₉	—	8	1	no	—
141	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
142	#1988	C₂×C₁₉	1	1	yes	characteristic
143	#1981	C₂×C₁₉	1	1	yes	characteristic
144	#1770	C₂×C₁₉	1	1	yes	characteristic
145	C₂×#335	C₂×C₁₉	1	1	yes	characteristic
146	C₂×#332	C₂×C₁₉	1	1	yes	characteristic
147	C₂×#332	C₂×C₁₉	1	1	yes	characteristic
148	C₂²×C₁₉	#390	1	1	yes	characteristic
149	C₂²×C₁₉	#463	1	1	yes	characteristic
150	C₂²×C₁₉	#464	1	1	yes	characteristic
151	C₂²×C₁₉	#468	1	2	yes	—
152	C₂²×C₁₉	#473	1	2	yes	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	8	2	no	—
156	C₄×C₁₉	—	8	2	no	—
157	C₄×C₁₉	—	8	2	no	—
158	C₄×C₁₉	#416	1	1	yes	characteristic
159	C₄×C₁₉	#417	1	1	yes	characteristic
160	C₄×C₁₉	#440	1	2	yes	—
161	C₄×C₁₉	#463	1	1	yes	characteristic
162	C₄×C₁₉	#464	1	1	yes	characteristic
163	C₄×C₁₉	#496	1	1	yes	characteristic
164	C₄×C₁₉	#497	1	1	yes	characteristic
165	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
167	#487	—	2	2	no	—
168	#487	C₄×C₁₉	1	4	yes	complemented
169	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×#103	—	2	1	no	—
173	C₂×#103	—	2	1	no	—
174	C₂×#103	—	2	2	no	—
175	C₂×#97	—	2	1	no	—
176	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
177	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
178	#340	—	2	2	no	—
179	#336	—	2	1	no	—
180	#335	—	2	2	no	—
181	#332	—	2	2	no	—
182	#332	—	2	2	no	—
183	C₂×C₁₉	#1699	1	2	yes	—
184	C₂×C₁₉	#1700	1	2	yes	—
185	C₂×C₁₉	#1973	1	1	yes	characteristic
186	C₂×C₁₉	#1979	1	1	yes	characteristic
187	C₂×C₁₉	#1983	1	1	yes	characteristic
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×Q₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
202	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
216	#103	—	2	4	no	—
217	#103	—	2	4	no	—
218	#103	—	4	4	no	—
219	#97	—	4	2	no	—
220	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
221	C₄×C₈	D₄×C₁₉	1	2	yes	—
222	#95	—	2	1	no	—
223	C₁₉	#9748	1	1	yes	characteristic, direct product factor, Sylow
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
239	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
252	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
268	C₄²	#41×C₁₉	1	1	yes	characteristic
269	C₂³	C₁₉×#121	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	4	no	—
273	Q₈	—	8	1	no	—
274	Q₈	—	8	1	no	—
275	Q₈	—	8	1	no	—
276	Q₈	—	8	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
288	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
292	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
293	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
294	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
296	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
297	C₂×C₄	C₁₉×#137	1	2	yes	—
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	8	1	no	—
302	C₈	—	8	1	no	—
303	C₂²	C₁₉×#390	1	1	yes	characteristic
304	C₂²	C₁₉×#463	1	1	yes	characteristic
305	C₂²	C₁₉×#464	1	1	yes	characteristic
306	C₂²	C₁₉×#468	1	2	yes	—
307	C₂²	C₁₉×#473	1	2	yes	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₁₉×#416	1	1	yes	characteristic
314	C₄	C₁₉×#417	1	1	yes	characteristic
315	C₄	C₁₉×#440	1	2	yes	—
316	C₄	C₁₉×#463	1	1	yes	characteristic
317	C₄	C₁₉×#464	1	1	yes	characteristic
318	C₄	C₁₉×#496	1	1	yes	characteristic
319	C₄	C₁₉×#497	1	1	yes	characteristic
320	C₂	C₁₉×#1699	1	2	yes	—
321	C₂	C₁₉×#1700	1	2	yes	—
322	C₂	C₁₉×#1973	1	1	yes	characteristic
323	C₂	C₁₉×#1979	1	1	yes	characteristic
324	C₂	C₁₉×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#121
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₉×#487	C₄	semidirect
106. #9748	C₁₉	direct (N × H)
168. #487	C₄×C₁₉	semidirect
223. C₁₉	#9748	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×#487	2
C₁₉×#1988	2
C₁₉×#1981	2
C₁₉×#1770	2
C₂×C₁₉×#335	2
C₂×C₁₉×#332	2
#9748	19

Subgroups — C19×#9748 #18015707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9748 #18015707