Subgroups — C₁₉×#8644 #18014603

Order 4864.

542 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1802	C₂	1	2	yes	maximal
4	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1296	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₉	—	2	1	no	—
7	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₉	—	2	2	no	—
9	C₈×D₄×C₁₉	C₄	1	4	yes	complemented
10	C₂×C₁₉×#107	—	2	1	no	—
11	C₂×C₁₉×#107	—	2	1	no	—
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#102	—	2	2	no	—
15	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#384	C₂²	1	2	yes	—
18	C₁₉×#340	—	2	2	no	—
19	C₁₉×#335	—	2	2	no	—
20	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	—	2	1	no	—
23	C₂²×C₈×C₁₉	—	2	4	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	4	2	no	—
26	C₂²×C₈×C₁₉	—	4	2	no	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₄×D₄×C₁₉	—	2	1	no	—
30	C₄×D₄×C₁₉	—	2	1	no	—
31	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₁₉	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₁₉×#107	—	4	2	no	—
40	C₁₉×#107	—	4	2	no	—
41	C₁₉×C₄⋊C₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	—	2	2	no	—
43	C₁₉×C₄⋊C₈	—	2	2	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₉×#102	—	4	4	no	—
46	C₁₉×C₂²⋊C₈	—	2	4	no	—
47	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₉	D₄	1	2	yes	—
49	C₂⁴×C₁₉	—	2	1	no	—
50	C₂×D₄×C₁₉	—	2	1	no	—
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	—	2	1	no	—
53	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₁₉	D₈	1	2	yes	—
55	C₂×D₄×C₁₉	Q₁₆	1	2	yes	—
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	—	4	8	no	—
73	C₂×C₈×C₁₉	—	4	2	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₉	D₈	1	2	yes	—
79	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄⋊C₄×C₁₉	—	8	2	no	—
86	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
87	C₂²⋊C₄×C₁₉	—	2	2	no	—
88	C₂²⋊C₄×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₁₉	#41	1	1	yes	characteristic
93	#8644	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	—	2	2	no	—
95	C₂³×C₁₉	—	2	2	no	—
96	C₂³×C₁₉	—	2	2	no	—
97	C₂³×C₁₉	—	4	1	no	—
98	C₂³×C₁₉	—	4	1	no	—
99	C₂³×C₁₉	—	4	1	no	—
100	C₂³×C₁₉	—	4	1	no	—
101	C₂³×C₁₉	#121	1	1	yes	characteristic
102	D₄×C₁₉	—	2	2	no	—
103	D₄×C₁₉	—	2	2	no	—
104	D₄×C₁₉	—	2	2	no	—
105	D₄×C₁₉	#107	1	4	yes	complemented
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	8	2	no	—
116	C₂×C₄×C₁₉	—	8	2	no	—
117	C₂×C₄×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
130	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
132	C₈×C₁₉	—	2	2	no	—
133	C₈×C₁₉	—	4	2	no	—
134	C₈×C₁₉	—	8	2	no	—
135	C₈×C₁₉	#102	1	4	yes	complemented
136	C₂×C₈×D₄	C₂×C₁₉	1	1	yes	characteristic
137	#1845	C₂×C₁₉	1	1	yes	characteristic, complemented
138	#1802	C₂×C₁₉	1	2	yes	—
139	C₂×#335	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#1296	C₂×C₁₉	1	2	yes	complemented
141	C₂²×C₁₉	—	2	4	no	—
142	C₂²×C₁₉	—	4	2	no	—
143	C₂²×C₁₉	—	4	2	no	—
144	C₂²×C₁₉	—	4	2	no	—
145	C₂²×C₁₉	—	4	2	no	—
146	C₂²×C₁₉	—	4	2	no	—
147	C₂²×C₁₉	—	4	2	no	—
148	C₂²×C₁₉	#380	1	1	yes	characteristic
149	C₂²×C₁₉	#459	1	1	yes	characteristic
150	C₂²×C₁₉	#461	1	1	yes	characteristic
151	C₂²×C₁₉	#466	1	1	yes	characteristic
152	C₂²×C₁₉	#467	1	1	yes	characteristic
153	C₂²×C₁₉	#472	1	1	yes	characteristic
154	C₂²×C₁₉	#473	1	1	yes	characteristic
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	4	2	no	—
158	C₄×C₁₉	—	16	2	no	—
159	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
160	C₄×C₁₉	#417	1	1	yes	characteristic
161	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
162	C₄×C₁₉	#427	1	1	yes	characteristic
163	C₄×C₁₉	#466	1	1	yes	characteristic
164	C₄×C₁₉	#467	1	1	yes	characteristic
165	C₄×C₁₉	#474	1	1	yes	characteristic
166	C₄×C₁₉	#476	1	1	yes	characteristic
167	C₂³×C₈	—	2	1	no	—
168	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
169	C₈×D₄	—	2	2	no	—
170	C₈×D₄	C₄×C₁₉	1	4	yes	complemented
171	C₂×#107	—	2	1	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
174	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
175	C₂×#102	—	2	2	no	—
176	C₂×C₂²⋊C₈	—	2	1	no	—
177	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
178	#384	C₂²×C₁₉	1	2	yes	—
179	#340	—	2	2	no	—
180	#335	—	2	2	no	—
181	C₂×C₁₉	—	4	4	no	—
182	C₂×C₁₉	#1667	1	1	yes	characteristic
183	C₂×C₁₉	#1669	1	1	yes	characteristic
184	C₂×C₁₉	#1673	1	1	yes	characteristic
185	C₂×C₁₉	#1675	1	1	yes	characteristic
186	C₂×C₁₉	#1802	1	1	yes	characteristic
187	C₂×C₁₉	#1808	1	1	yes	characteristic
188	C₂×C₁₉	#1816	1	1	yes	characteristic
189	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
190	C₂³×C₄	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	4	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	—	4	2	no	—
196	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	C₂×C₄×C₁₉	1	2	yes	—
201	C₄×D₄	Q₈×C₁₉	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
208	#107	—	4	2	no	—
209	#107	—	4	2	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
214	#102	—	4	4	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
217	C₄×C₈	D₄×C₁₉	1	2	yes	—
218	C₁₉	#8644	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄⋊C₄×C₁₉	1	2	yes	—
224	C₂×D₄	D₈×C₁₉	1	2	yes	—
225	C₂×D₄	Q₁₆×C₁₉	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
236	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
247	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
248	C₂×C₈	D₈×C₁₉	1	2	yes	—
249	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	8	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₁₉	1	2	yes	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
261	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
262	C₄²	#41×C₁₉	1	1	yes	characteristic
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	C₁₉×#121	1	1	yes	characteristic
271	D₄	—	2	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	C₁₉×#107	1	4	yes	complemented
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
291	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
294	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
297	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
298	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
299	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	8	2	no	—
304	C₈	C₁₉×#102	1	4	yes	complemented
305	C₂²	—	2	4	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	C₁₉×#380	1	1	yes	characteristic
313	C₂²	C₁₉×#459	1	1	yes	characteristic
314	C₂²	C₁₉×#461	1	1	yes	characteristic
315	C₂²	C₁₉×#466	1	1	yes	characteristic
316	C₂²	C₁₉×#467	1	1	yes	characteristic
317	C₂²	C₁₉×#472	1	1	yes	characteristic
318	C₂²	C₁₉×#473	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	16	2	no	—
323	C₄	C₂×C₁₉×#102	1	1	yes	characteristic
324	C₄	C₁₉×#417	1	1	yes	characteristic
325	C₄	C₂×C₁₉×#107	1	1	yes	characteristic
326	C₄	C₁₉×#427	1	1	yes	characteristic
327	C₄	C₁₉×#466	1	1	yes	characteristic
328	C₄	C₁₉×#467	1	1	yes	characteristic
329	C₄	C₁₉×#474	1	1	yes	characteristic
330	C₄	C₁₉×#476	1	1	yes	characteristic
331	C₂	—	4	4	no	—
332	C₂	C₁₉×#1667	1	1	yes	characteristic
333	C₂	C₁₉×#1669	1	1	yes	characteristic
334	C₂	C₁₉×#1673	1	1	yes	characteristic
335	C₂	C₁₉×#1675	1	1	yes	characteristic
336	C₂	C₁₉×#1802	1	1	yes	characteristic
337	C₂	C₁₉×#1808	1	1	yes	characteristic
338	C₂	C₁₉×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	#121
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂²×D₄×C₁₉	D₄
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
2. C₁₉×#1845	C₂	semidirect
4. C₂×C₁₉×#335	C₂	semidirect
5. C₁₉×#1296	C₂	semidirect
9. C₈×D₄×C₁₉	C₄	semidirect
93. #8644	C₁₉	direct (N × H)
105. D₄×C₁₉	#107	semidirect
135. C₈×C₁₉	#102	semidirect
137. #1845	C₂×C₁₉	semidirect
139. C₂×#335	C₂×C₁₉	semidirect
140. #1296	C₂×C₁₉	semidirect
170. C₈×D₄	C₄×C₁₉	semidirect
218. C₁₉	#8644	direct (N × H)
274. D₄	C₁₉×#107	semidirect
304. C₈	C₁₉×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₉	2
C₁₉×#1845	2
C₁₉×#1802	2
C₂×C₁₉×#335	2
C₁₉×#1296	2
#8644	19

Subgroups — C19×#8644 #18014603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8644 #18014603