Subgroups — C₁₉×#23839 #18029798

Order 4864.

766 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2084	C₂	1	2	yes	maximal
4	C₁₉×#2083	C₂	1	2	yes	maximal
5	C₁₉×#2058	C₂	1	2	yes	maximal
6	C₁₉×#2057	C₂	1	2	yes	maximal
7	C₁₉×#1776	C₂	1	4	yes	maximal
8	C₁₉×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
12	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
13	C₂×C₁₉×#107	C₂²	1	4	yes	—
14	C₂×C₁₉×#106	C₂²	1	4	yes	—
15	C₂×C₁₉×#97	C₂²	1	2	yes	—
16	C₁₉×#391	—	2	1	no	—
17	C₁₉×#391	—	2	1	no	—
18	C₁₉×#389	—	2	1	no	—
19	C₁₉×#389	—	2	1	no	—
20	C₁₉×#389	C₂²	1	2	yes	—
21	C₁₉×#389	C₂²	1	2	yes	—
22	C₁₉×#384	C₂²	1	1	yes	characteristic
23	C₁₉×#384	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₄³×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#340	C₂²	1	4	yes	—
30	C₁₉×#340	C₂²	1	4	yes	—
31	C₁₉×#336	C₂²	1	2	yes	—
32	C₁₉×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₂²×Q₈×C₁₉	—	2	1	no	—
35	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
36	C₁₉×C₄⋊Q₈	—	2	4	no	—
37	C₁₉×C₄⋊Q₈	—	2	4	no	—
38	C₄×Q₈×C₁₉	—	4	4	no	—
39	C₄×Q₈×C₁₉	—	4	4	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
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	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₉	—	2	2	no	—
53	C₂×C₄²×C₁₉	—	2	2	no	—
54	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₁₉×#107	—	2	8	no	—
62	C₁₉×#106	—	2	8	no	—
63	C₁₉×#97	Q₈	1	8	yes	complemented
64	C₁₉×#95	—	2	2	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	2	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₂×Q₈×C₁₉	—	2	4	no	—
69	C₂×Q₈×C₁₉	—	2	4	no	—
70	C₂×Q₈×C₁₉	—	4	2	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂×Q₈×C₁₉	—	4	2	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₁₉	—	2	4	no	—
86	C₂×C₈×C₁₉	—	2	4	no	—
87	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
88	C₄⋊C₄×C₁₉	—	2	4	no	—
89	C₄⋊C₄×C₁₉	—	2	4	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₄⋊C₄×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	4	2	no	—
97	C₄⋊C₄×C₁₉	—	4	4	no	—
98	C₄⋊C₄×C₁₉	—	4	4	no	—
99	C₄⋊C₄×C₁₉	—	4	2	no	—
100	C₄⋊C₄×C₁₉	#41	1	4	yes	—
101	C₄⋊C₄×C₁₉	#41	1	4	yes	—
102	C₄²×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	C₄²×C₁₉	—	2	2	no	—
106	C₄²×C₁₉	—	2	2	no	—
107	C₄²×C₁₉	—	2	2	no	—
108	C₄²×C₁₉	—	4	4	no	—
109	C₄²×C₁₉	—	4	4	no	—
110	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
111	#23839	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₉	—	4	8	no	—
114	Q₈×C₁₉	—	4	8	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	4	no	—
128	C₂×C₄×C₁₉	—	2	4	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₁₉	#137	1	2	yes	—
144	C₂×C₄×C₁₉	#137	1	2	yes	—
145	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
146	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
147	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
148	C₈×C₁₉	—	2	8	no	—
149	C₄×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
150	C₄×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
151	#2084	C₂×C₁₉	1	2	yes	—
152	#2083	C₂×C₁₉	1	2	yes	—
153	#2058	C₂×C₁₉	1	2	yes	—
154	#2057	C₂×C₁₉	1	2	yes	—
155	#1776	C₂×C₁₉	1	4	yes	—
156	#1745	C₂×C₁₉	1	1	yes	characteristic
157	C₂²×C₁₉	#525	1	1	yes	characteristic
158	C₂²×C₁₉	#525	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
163	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	4	4	no	—
171	C₄×C₁₉	—	4	4	no	—
172	C₄×C₁₉	—	8	2	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₄×C₁₉	#533	1	4	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×C₄×Q₈	—	2	1	no	—
178	C₂×C₄×Q₈	—	2	1	no	—
179	C₂×#107	C₂²×C₁₉	1	4	yes	—
180	C₂×#106	C₂²×C₁₉	1	4	yes	—
181	C₂×#97	C₂²×C₁₉	1	2	yes	—
182	#391	—	2	1	no	—
183	#391	—	2	1	no	—
184	#389	—	2	1	no	—
185	#389	—	2	1	no	—
186	#389	C₂²×C₁₉	1	2	yes	—
187	#389	C₂²×C₁₉	1	2	yes	—
188	#384	C₂²×C₁₉	1	1	yes	characteristic
189	#384	C₂²×C₁₉	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	—	2	1	no	—
192	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
193	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
194	C₄³	C₂²×C₁₉	1	1	yes	characteristic
195	#340	C₂²×C₁₉	1	4	yes	—
196	#340	C₂²×C₁₉	1	4	yes	—
197	#336	C₂²×C₁₉	1	2	yes	—
198	#336	C₂²×C₁₉	1	2	yes	—
199	C₂×C₁₉	#2399	1	1	yes	characteristic
200	C₂×C₁₉	#3109	1	1	yes	characteristic
201	C₂×C₁₉	#3110	1	1	yes	characteristic
202	C₂×C₁₉	#3112	1	1	yes	characteristic
203	C₂×C₁₉	#3113	1	1	yes	characteristic
204	C₂×C₁₉	#3165	1	1	yes	characteristic
205	C₂×C₁₉	#3165	1	1	yes	characteristic
206	C₂²×Q₈	—	2	1	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄×Q₈	—	4	4	no	—
212	C₄×Q₈	—	4	4	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
234	#107	—	2	8	no	—
235	#106	—	2	8	no	—
236	#97	Q₈×C₁₉	1	8	yes	complemented
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	C₁₉	#23839	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	#41×C₁₉	1	4	yes	—
275	C₄⋊C₄	#41×C₁₉	1	4	yes	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	4	no	—
283	C₄²	—	4	4	no	—
284	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
285	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
286	Q₈	—	4	8	no	—
287	Q₈	—	4	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	C₁₉×C₄⋊Q₈	1	2	yes	—
313	C₂×C₄	C₁₉×C₄⋊Q₈	1	2	yes	—
314	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
315	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
316	C₂×C₄	C₁₉×#137	1	2	yes	—
317	C₂×C₄	C₁₉×#137	1	2	yes	—
318	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
319	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
320	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
321	C₈	—	2	8	no	—
322	C₂²	C₁₉×#525	1	1	yes	characteristic
323	C₂²	C₁₉×#525	1	1	yes	characteristic
324	C₂²	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
325	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
326	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
328	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	4	no	—
336	C₄	—	4	4	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	C₁₉×#533	1	4	yes	—
340	C₂	C₁₉×#2399	1	1	yes	characteristic
341	C₂	C₁₉×#3109	1	1	yes	characteristic
342	C₂	C₁₉×#3110	1	1	yes	characteristic
343	C₂	C₁₉×#3112	1	1	yes	characteristic
344	C₂	C₁₉×#3113	1	1	yes	characteristic
345	C₂	C₁₉×#3165	1	1	yes	characteristic
346	C₂	C₁₉×#3165	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
63. C₁₉×#97	Q₈	semidirect
111. #23839	C₁₉	direct (N × H)
236. #97	Q₈×C₁₉	semidirect
241. C₁₉	#23839	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₉×C₄⋊Q₈	2
C₁₉×#2084	2
C₁₉×#2083	2
C₁₉×#2058	2
C₁₉×#2057	2
C₁₉×#1776	2
C₁₉×#1745	2
#23839	19

Subgroups — C19×#23839 #18029798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23839 #18029798