Subgroups — C₁₉×#22611 #18028570

Order 4864.

878 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2344	C₂	1	2	yes	maximal
3	C₁₉×#1942	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1918	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1823	C₂	1	2	yes	maximal
6	C₁₉×#1808	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₉×#107	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#117	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
12	C₂×C₁₉×#107	C₄	1	8	yes	complemented
13	C₂×C₁₉×#107	C₂²	1	2	yes	—
14	C₂×C₁₉×#107	C₂²	1	2	yes	—
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
19	C₁₉×#389	C₂²	1	4	yes	—
20	C₁₉×#387	—	2	4	no	—
21	C₁₉×#381	—	2	2	no	—
22	C₁₉×#380	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
24	C₄×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#340	C₂²	1	2	yes	—
26	C₁₉×#340	C₂²	1	4	yes	—
27	C₁₉×#340	C₂²	1	4	yes	—
28	C₁₉×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₁₉	—	2	2	no	—
30	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	—	2	4	no	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₁₉×#117	—	4	8	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₉	—	2	4	no	—
48	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
49	C₁₉×#107	—	2	4	no	—
50	C₁₉×#107	C₂×C₄	1	8	yes	—
51	C₁₉×#107	C₂×C₄	1	4	yes	—
52	C₁₉×#107	C₂×C₄	1	4	yes	—
53	C₁₉×C₂²⋊C₈	—	2	2	no	—
54	C₄×C₈×C₁₉	—	2	4	no	—
55	C₁₉×#95	—	2	4	no	—
56	C₁₉×#95	—	2	2	no	—
57	C₁₉×#95	—	2	4	no	—
58	C₁₉×#95	D₄	1	2	yes	—
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	—	4	4	no	—
67	C₂²×C₄×C₁₉	—	4	4	no	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	8	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₉	#41	1	4	yes	—
82	C₄⋊C₄×C₁₉	—	2	8	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	—	4	8	no	—
88	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₁₉	—	2	2	no	—
91	C₂²⋊C₄×C₁₉	—	4	4	no	—
92	C₂²⋊C₄×C₁₉	D₈	1	4	yes	—
93	C₄²×C₁₉	—	2	2	no	—
94	C₄²×C₁₉	—	2	2	no	—
95	C₄²×C₁₉	—	4	8	no	—
96	#22611	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	C₄×D₄	1	2	yes	—
100	C₂³×C₁₉	C₂×D₈	1	2	yes	—
101	C₂³×C₁₉	#135	1	2	yes	—
102	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	16	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
128	C₈×C₁₉	—	2	2	no	—
129	C₈×C₁₉	—	4	2	no	—
130	C₈×C₁₉	#117	1	4	yes	complemented
131	C₂²×#107	C₂×C₁₉	1	1	yes	characteristic
132	#2344	C₂×C₁₉	1	2	yes	—
133	#1942	C₂×C₁₉	1	2	yes	complemented
134	#1918	C₂×C₁₉	1	4	yes	complemented
135	#1823	C₂×C₁₉	1	2	yes	—
136	#1808	C₂×C₁₉	1	1	yes	characteristic
137	C₄×#107	C₂×C₁₉	1	2	yes	complemented
138	C₈×C₂²⋊C₄	C₂×C₁₉	1	1	yes	characteristic
139	C₂²×C₁₉	—	2	4	no	—
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	C₄×D₈	1	4	yes	—
144	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₁₉	#547	1	2	yes	—
146	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
147	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
148	C₂²×C₁₉	#578	1	2	yes	—
149	C₄×C₁₉	—	2	4	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	4	2	no	—
152	C₄×C₁₉	—	4	4	no	—
153	C₄×C₁₉	—	4	4	no	—
154	C₄×C₁₉	—	8	4	no	—
155	C₄×C₁₉	C₂×#117	1	1	yes	characteristic
156	C₄×C₁₉	#519	1	1	yes	characteristic
157	C₄×C₁₉	#552	1	2	yes	—
158	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
159	C₂×#117	—	2	2	no	—
160	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
161	C₂×#107	C₄×C₁₉	1	8	yes	complemented
162	C₂×#107	C₂²×C₁₉	1	2	yes	—
163	C₂×#107	C₂²×C₁₉	1	2	yes	—
164	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×C₂²⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
168	#389	C₂²×C₁₉	1	4	yes	—
169	#387	—	2	4	no	—
170	#381	—	2	2	no	—
171	#380	C₂²×C₁₉	1	2	yes	—
172	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
173	C₄×C₂²⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
174	#340	C₂²×C₁₉	1	2	yes	—
175	#340	C₂²×C₁₉	1	4	yes	—
176	#340	C₂²×C₁₉	1	4	yes	—
177	#336	C₂²×C₁₉	1	2	yes	—
178	C₂×C₁₉	—	2	4	no	—
179	C₂×C₁₉	#2344	1	1	yes	characteristic
180	C₂×C₁₉	C₂×C₄×D₈	1	1	yes	characteristic
181	C₂×C₁₉	#2950	1	1	yes	characteristic
182	C₂×C₁₉	#3334	1	2	yes	—
183	C₂×C₁₉	#3336	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
188	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
191	#117	—	4	8	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	D₄×C₁₉	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	D₄×C₁₉	1	1	yes	characteristic
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
204	#107	—	2	4	no	—
205	#107	C₂×C₄×C₁₉	1	8	yes	—
206	#107	C₂×C₄×C₁₉	1	4	yes	—
207	#107	C₂×C₄×C₁₉	1	4	yes	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₄×C₈	—	2	4	no	—
210	#95	—	2	4	no	—
211	#95	—	2	2	no	—
212	#95	—	2	4	no	—
213	#95	D₄×C₁₉	1	2	yes	—
214	C₁₉	#22611	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
228	C₂²×C₄	#41×C₁₉	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
237	C₂×C₈	#41×C₁₉	1	4	yes	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	8	no	—
244	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
245	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	D₈×C₁₉	1	4	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	8	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₄×D₄×C₁₉	1	2	yes	—
255	C₂³	C₂×D₈×C₁₉	1	2	yes	—
256	C₂³	C₁₉×#135	1	2	yes	—
257	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	16	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
278	C₂×C₄	C₂×D₈×C₁₉	1	4	yes	—
279	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
281	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
282	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	C₁₉×#117	1	4	yes	complemented
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	C₄×D₈×C₁₉	1	4	yes	—
291	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
292	C₂²	C₁₉×#547	1	2	yes	—
293	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
294	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
295	C₂²	C₁₉×#578	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
303	C₄	C₁₉×#519	1	1	yes	characteristic
304	C₄	C₁₉×#552	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₁₉×#2344	1	1	yes	characteristic
307	C₂	C₂×C₄×D₈×C₁₉	1	1	yes	characteristic
308	C₂	C₁₉×#2950	1	1	yes	characteristic
309	C₂	C₁₉×#3334	1	2	yes	—
310	C₂	C₁₉×#3336	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1942	C₂	semidirect
4. C₁₉×#1918	C₂	semidirect
7. C₄×C₁₉×#107	C₂	semidirect
12. C₂×C₁₉×#107	C₄	semidirect
96. #22611	C₁₉	direct (N × H)
130. C₈×C₁₉	#117	semidirect
133. #1942	C₂×C₁₉	semidirect
134. #1918	C₂×C₁₉	semidirect
137. C₄×#107	C₂×C₁₉	semidirect
161. C₂×#107	C₄×C₁₉	semidirect
214. C₁₉	#22611	direct (N × H)
285. C₈	C₁₉×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#107	2
C₁₉×#2344	2
C₁₉×#1942	2
C₁₉×#1918	2
C₁₉×#1823	2
C₁₉×#1808	2
C₄×C₁₉×#107	2
C₈×C₂²⋊C₄×C₁₉	2
#22611	19

Subgroups — C19×#22611 #18028570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#22611 #18028570