Subgroups — C₁₉×#8387 #18014346

Order 4864.

998 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#547	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1800	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1546	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1486	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1460	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₁₉	—	2	1	no	—
7	C₁₉×#547	—	2	4	no	—
8	C₁₉×#547	—	2	4	no	—
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#123	—	2	2	no	—
11	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
12	C₂×C₁₉×#121	C₄	1	2	yes	—
13	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
16	C₁₉×#431	C₂²	1	2	yes	—
17	C₁₉×#423	C₂²	1	2	yes	—
18	C₂×C₁₉×#100	—	2	1	no	—
19	C₁₉×#343	—	2	1	no	—
20	C₁₉×#332	—	2	4	no	—
21	C₁₉×D₄⋊C₈	—	2	4	no	—
22	C₁₉×#323	—	2	2	no	—
23	C₂×#41×C₁₉	—	2	8	no	—
24	C₂×#41×C₁₉	—	4	4	no	—
25	C₂×#41×C₁₉	—	4	2	no	—
26	C₂×#41×C₁₉	—	4	2	no	—
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	2	no	—
29	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₉	—	2	2	no	—
32	C₂³×C₄×C₁₉	—	2	1	no	—
33	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₁₉	—	2	2	no	—
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	D₄	1	4	yes	—
37	C₁₉×#123	—	2	16	no	—
38	C₁₉×C₂²⋊Q₈	—	2	8	no	—
39	C₁₉×#121	—	2	4	no	—
40	C₁₉×#121	—	2	4	no	—
41	C₄×D₄×C₁₉	—	2	2	no	—
42	C₄×D₄×C₁₉	—	2	2	no	—
43	C₄×D₄×C₁₉	—	2	2	no	—
44	C₄×D₄×C₁₉	D₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
49	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	—	2	2	no	—
57	C₁₉×C₄⋊C₈	—	2	2	no	—
58	C₁₉×C₄⋊C₈	—	2	2	no	—
59	C₁₉×#100	—	4	2	no	—
60	C₁₉×C₂²⋊C₈	—	2	2	no	—
61	C₄×C₈×C₁₉	—	2	2	no	—
62	C₂⁴×C₁₉	—	2	1	no	—
63	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
64	#41×C₁₉	—	4	16	no	—
65	#41×C₁₉	—	4	16	no	—
66	C₂×Q₈×C₁₉	—	2	4	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	4	2	no	—
70	C₂×D₄×C₁₉	—	2	8	no	—
71	C₂×D₄×C₁₉	—	2	1	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	—	4	4	no	—
76	C₂×D₄×C₁₉	—	4	4	no	—
77	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×D₄×C₁₉	D₈	1	2	yes	—
79	C₂×D₄×C₁₉	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₁₉	—	2	8	no	—
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	—	2	2	no	—
83	C₂²×C₄×C₁₉	—	2	2	no	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	—	2	4	no	—
86	C₂²×C₄×C₁₉	—	4	4	no	—
87	C₂²×C₄×C₁₉	—	4	4	no	—
88	C₂²×C₄×C₁₉	—	4	4	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	—	4	1	no	—
91	C₂²×C₄×C₁₉	—	4	2	no	—
92	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	M₁₆×C₁₉	—	4	2	no	—
97	M₁₆×C₁₉	—	4	2	no	—
98	C₂×C₈×C₁₉	—	4	2	no	—
99	C₂×C₈×C₁₉	—	4	2	no	—
100	C₄⋊C₄×C₁₉	—	2	8	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	4	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
106	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
107	C₂²⋊C₄×C₁₉	—	2	8	no	—
108	C₂²⋊C₄×C₁₉	—	2	4	no	—
109	C₂²⋊C₄×C₁₉	—	2	4	no	—
110	C₄²×C₁₉	C₂×D₄	1	2	yes	—
111	C₄²×C₁₉	C₂×D₄	1	2	yes	—
112	#8387	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	—	2	2	no	—
114	C₂³×C₁₉	—	2	2	no	—
115	C₂³×C₁₉	—	2	2	no	—
116	C₂³×C₁₉	—	4	2	no	—
117	C₂³×C₁₉	—	4	2	no	—
118	C₂³×C₁₉	—	4	2	no	—
119	C₂³×C₁₉	—	4	2	no	—
120	C₂³×C₁₉	—	4	1	no	—
121	C₂³×C₁₉	—	4	1	no	—
122	C₂³×C₁₉	#102	1	4	yes	—
123	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	Q₈×C₁₉	—	4	4	no	—
125	Q₈×C₁₉	—	4	4	no	—
126	D₄×C₁₉	—	2	4	no	—
127	D₄×C₁₉	—	2	4	no	—
128	D₄×C₁₉	—	2	4	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	16	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	4	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	4	no	—
137	C₂×C₄×C₁₉	—	4	16	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	4	no	—
143	C₂×C₄×C₁₉	—	4	4	no	—
144	C₂×C₄×C₁₉	—	4	4	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#120	1	2	yes	—
149	C₂×C₄×C₁₉	#120	1	2	yes	—
150	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
152	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
153	C₈×C₁₉	—	8	2	no	—
154	C₈×C₁₉	—	8	2	no	—
155	C₂×#547	C₂×C₁₉	1	1	yes	characteristic
156	#1800	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#1546	C₂×C₁₉	1	2	yes	complemented
158	#1486	C₂×C₁₉	1	2	yes	complemented
159	#1460	C₂×C₁₉	1	1	yes	characteristic, complemented
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	4	no	—
163	C₂²×C₁₉	—	4	4	no	—
164	C₂²×C₁₉	—	4	4	no	—
165	C₂²×C₁₉	—	4	2	no	—
166	C₂²×C₁₉	—	4	2	no	—
167	C₂²×C₁₉	—	4	2	no	—
168	C₂²×C₁₉	—	4	2	no	—
169	C₂²×C₁₉	—	4	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	—	8	2	no	—
172	C₂²×C₁₉	#379	1	1	yes	characteristic
173	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
174	C₂²×C₁₉	#418	1	1	yes	characteristic
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	4	2	no	—
178	C₄×C₁₉	—	4	2	no	—
179	C₄×C₁₉	—	4	4	no	—
180	C₄×C₁₉	#413	1	2	yes	—
181	C₄×C₁₉	#447	1	2	yes	—
182	C₄×C₁₉	#450	1	2	yes	—
183	C₂²×#41	—	2	1	no	—
184	#547	—	2	4	no	—
185	#547	—	2	4	no	—
186	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×#123	—	2	2	no	—
188	C₂×C₂²⋊Q₈	—	2	1	no	—
189	C₂×#121	C₄×C₁₉	1	2	yes	—
190	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
192	C₂²×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
193	#431	C₂²×C₁₉	1	2	yes	—
194	#423	C₂²×C₁₉	1	2	yes	—
195	C₂×#100	—	2	1	no	—
196	#343	—	2	1	no	—
197	#332	—	2	4	no	—
198	D₄⋊C₈	—	2	4	no	—
199	#323	—	2	2	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	1	no	—
202	C₂×C₁₉	—	4	4	no	—
203	C₂×C₁₉	—	8	2	no	—
204	C₂×C₁₉	#1785	1	1	yes	characteristic
205	C₂×C₁₉	#1786	1	1	yes	characteristic
206	C₂×C₁₉	#1796	1	1	yes	characteristic
207	C₂×#41	—	2	8	no	—
208	C₂×#41	—	4	4	no	—
209	C₂×#41	—	4	2	no	—
210	C₂×#41	—	4	2	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	—	2	2	no	—
213	C₂²×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
214	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
215	C₂³×C₄	—	2	2	no	—
216	C₂³×C₄	—	2	1	no	—
217	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
218	C₂×M₁₆	—	2	2	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	D₄×C₁₉	1	4	yes	—
221	#123	—	2	16	no	—
222	C₂²⋊Q₈	—	2	8	no	—
223	#121	—	2	4	no	—
224	#121	—	2	4	no	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	D₄×C₁₉	1	4	yes	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	4	no	—
232	C₂×C₄⋊C₄	—	2	4	no	—
233	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
235	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	2	1	no	—
238	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
239	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	2	2	no	—
243	#100	—	4	2	no	—
244	C₂²⋊C₈	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	C₁₉	#8387	1	1	yes	characteristic, direct product factor, Sylow
247	C₂⁴	—	2	1	no	—
248	C₂⁴	C₂²⋊C₄×C₁₉	1	2	yes	—
249	#41	—	4	16	no	—
250	#41	—	4	16	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	2	8	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	4	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
263	C₂×D₄	D₈×C₁₉	1	2	yes	—
264	C₂×D₄	SD₁₆×C₁₉	1	2	yes	—
265	C₂²×C₄	—	2	8	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	—	4	4	no	—
273	C₂²×C₄	—	4	4	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
278	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
281	M₁₆	—	4	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
291	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
292	C₂²⋊C₄	—	2	8	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₄²	C₂×D₄×C₁₉	1	2	yes	—
296	C₄²	C₂×D₄×C₁₉	1	2	yes	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	1	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₁₉×#102	1	4	yes	—
307	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	Q₈	—	4	4	no	—
310	D₄	—	2	4	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	16	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	16	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
331	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
332	C₂×C₄	C₁₉×#120	1	2	yes	—
333	C₂×C₄	C₁₉×#120	1	2	yes	—
334	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
335	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
336	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
337	C₈	—	8	2	no	—
338	C₈	—	8	2	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	4	4	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	C₁₉×#379	1	1	yes	characteristic
352	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
353	C₂²	C₁₉×#418	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	4	no	—
359	C₄	C₁₉×#413	1	2	yes	—
360	C₄	C₁₉×#447	1	2	yes	—
361	C₄	C₁₉×#450	1	2	yes	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	4	no	—
365	C₂	—	8	2	no	—
366	C₂	C₁₉×#1785	1	1	yes	characteristic
367	C₂	C₁₉×#1786	1	1	yes	characteristic
368	C₂	C₁₉×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂×C₁₉×#547	C₂
LCS	G ▷ C₂²×C₄ ▷ 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
2. C₁₉×#1800	C₂	semidirect
3. C₁₉×#1546	C₂	semidirect
4. C₁₉×#1486	C₂	semidirect
5. C₁₉×#1460	C₂	semidirect
112. #8387	C₁₉	direct (N × H)
156. #1800	C₂×C₁₉	semidirect
157. #1546	C₂×C₁₉	semidirect
158. #1486	C₂×C₁₉	semidirect
159. #1460	C₂×C₁₉	semidirect
246. C₁₉	#8387	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×#547	2
C₁₉×#1800	2
C₁₉×#1546	2
C₁₉×#1486	2
C₁₉×#1460	2
#8387	19

Subgroups — C19×#8387 #18014346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8387 #18014346