Subgroups — C₁₉×#20445 #18026404

Order 4864.

1042 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#430	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2911	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2873	C₂	1	2	yes	maximal
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1773	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1772	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₉×#339	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#430	C₄	1	8	yes	complemented
14	C₂×C₁₉×#108	C₂²	1	4	yes	—
15	C₁₉×#428	C₄	1	8	yes	complemented
16	C₁₉×#427	C₂²	1	4	yes	—
17	C₁₉×#405	—	2	4	no	—
18	C₁₉×#405	C₂²	1	8	yes	—
19	C₄×M₁₆×C₁₉	—	2	4	no	—
20	C₄×M₁₆×C₁₉	C₄	1	8	yes	complemented
21	C₂×C₁₉×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
23	C₁₉×#339	C₂²	1	4	yes	—
24	C₁₉×#339	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₉	—	2	1	no	—
26	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₂×M₁₆×C₁₉	—	2	4	no	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	8	yes	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
34	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
35	C₂×M₁₆×C₁₉	Q₈	1	4	yes	—
36	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₉	—	2	2	no	—
38	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
40	C₁₉×#117	—	2	2	no	—
41	C₁₉×#117	—	2	2	no	—
42	C₁₉×#117	C₂×C₄	1	4	yes	—
43	C₁₉×#117	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄²×C₁₉	—	2	2	no	—
47	C₂×C₄²×C₁₉	—	2	1	no	—
48	C₂×C₄²×C₁₉	—	2	2	no	—
49	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₁₉×#108	C₂×C₄	1	16	yes	complemented
53	C₁₉×#107	C₂×C₄	1	8	yes	complemented
54	C₁₉×#106	C₂×C₄	1	8	yes	complemented
55	C₁₉×#97	—	2	8	no	—
56	C₁₉×#97	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₁₉	—	2	8	no	—
58	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	complemented
59	C₂⁴×C₁₉	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	4	4	no	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₉	—	2	8	no	—
74	M₁₆×C₁₉	—	2	4	no	—
75	M₁₆×C₁₉	C₄²	1	16	yes	—
76	M₁₆×C₁₉	C₄⋊C₄	1	16	yes	—
77	M₁₆×C₁₉	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₁₉	—	2	8	no	—
79	C₂×C₈×C₁₉	—	2	4	no	—
80	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
88	C₂²⋊C₄×C₁₉	—	2	4	no	—
89	C₂²⋊C₄×C₁₉	—	2	4	no	—
90	C₄²×C₁₉	—	2	4	no	—
91	C₄²×C₁₉	—	2	4	no	—
92	C₄²×C₁₉	—	2	4	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	—	2	1	no	—
95	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
97	#20445	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₁₉	—	4	2	no	—
99	C₂³×C₁₉	#117	1	2	yes	—
100	C₂³×C₁₉	C₄×D₄	1	2	yes	—
101	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
102	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	4	8	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	8	4	no	—
110	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₁₉	#117	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
116	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
123	C₈×C₁₉	—	2	8	no	—
124	C₈×C₁₉	C₂×C₄²	1	8	yes	—
125	C₈×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
126	C₂×#430	C₂×C₁₉	1	1	yes	characteristic
127	#2911	C₂×C₁₉	1	1	yes	characteristic
128	#2873	C₂×C₁₉	1	2	yes	—
129	C₂×C₄×M₁₆	C₂×C₁₉	1	1	yes	characteristic
130	#1773	C₂×C₁₉	1	4	yes	complemented
131	#1772	C₂×C₁₉	1	4	yes	complemented
132	C₂×#339	C₂×C₁₉	1	2	yes	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	4	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
138	C₂²×C₁₉	C₂×#117	1	1	yes	characteristic
139	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₁₉	—	2	4	no	—
142	C₄×C₁₉	—	4	2	no	—
143	C₄×C₁₉	—	4	8	no	—
144	C₄×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
145	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
150	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
151	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
152	C₂×#117	C₂²×C₁₉	1	2	yes	—
153	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
154	#430	C₄×C₁₉	1	8	yes	complemented
155	C₂×#108	C₂²×C₁₉	1	4	yes	—
156	#428	C₄×C₁₉	1	8	yes	complemented
157	#427	C₂²×C₁₉	1	4	yes	—
158	#405	—	2	4	no	—
159	#405	C₂²×C₁₉	1	8	yes	—
160	C₄×M₁₆	—	2	4	no	—
161	C₄×M₁₆	C₄×C₁₉	1	8	yes	complemented
162	C₂×#97	C₂²×C₁₉	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
164	#339	C₂²×C₁₉	1	4	yes	—
165	#339	C₂²×C₁₉	1	4	yes	—
166	C₂×C₁₉	—	2	1	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	—	2	4	no	—
169	C₂×C₁₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₁₉	#2956	1	2	yes	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	C₂×C₄×C₁₉	1	8	yes	—
176	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
177	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
178	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
179	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
180	C₂×M₁₆	D₄×C₁₉	1	4	yes	—
181	C₂×M₁₆	Q₈×C₁₉	1	4	yes	—
182	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×C₄×C₁₉	1	4	yes	—
189	#117	C₂³×C₁₉	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
196	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
198	#108	C₂×C₄×C₁₉	1	16	yes	complemented
199	#107	C₂×C₄×C₁₉	1	8	yes	complemented
200	#106	C₂×C₄×C₁₉	1	8	yes	complemented
201	#97	—	2	8	no	—
202	#97	C₂×C₄×C₁₉	1	8	yes	complemented
203	C₄×C₈	—	2	8	no	—
204	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	complemented
205	C₁₉	#20445	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	#41×C₁₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
217	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	#41×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
220	M₁₆	—	2	8	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	C₄²×C₁₉	1	16	yes	—
223	M₁₆	C₄⋊C₄×C₁₉	1	16	yes	—
224	M₁₆	C₂²×C₄×C₁₉	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
228	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
229	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
231	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
232	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
243	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	C₁₉×#117	1	2	yes	—
246	C₂³	C₄×D₄×C₁₉	1	2	yes	—
247	C₂³	C₄×Q₈×C₁₉	1	2	yes	—
248	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
258	C₂×C₄	C₁₉×#117	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₉	1	4	yes	—
261	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
262	C₂×C₄	C₄×Q₈×C₁₉	1	4	yes	—
263	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
264	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
265	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
269	C₈	—	2	8	no	—
270	C₈	C₂×C₄²×C₁₉	1	8	yes	—
271	C₈	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
277	C₂²	C₂×C₁₉×#117	1	1	yes	characteristic
278	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
279	C₂²	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	8	no	—
283	C₄	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
284	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
285	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
286	C₄	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
287	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	4	no	—
290	C₂	—	2	1	no	—
291	C₂	C₂×C₄×C₄⋊C₄×C₁₉	1	1	yes	characteristic
292	C₂	C₁₉×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×C₄×D₄
prime radical	C₂⁴×C₁₉	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₉×#1773	C₂	semidirect
6. C₁₉×#1772	C₂	semidirect
13. C₁₉×#430	C₄	semidirect
15. C₁₉×#428	C₄	semidirect
20. C₄×M₁₆×C₁₉	C₄	semidirect
52. C₁₉×#108	C₂×C₄	semidirect
53. C₁₉×#107	C₂×C₄	semidirect
54. C₁₉×#106	C₂×C₄	semidirect
56. C₁₉×#97	C₂×C₄	semidirect
58. C₄×C₈×C₁₉	C₂×C₄	semidirect
97. #20445	C₁₉	direct (N × H)
130. #1773	C₂×C₁₉	semidirect
131. #1772	C₂×C₁₉	semidirect
154. #430	C₄×C₁₉	semidirect
156. #428	C₄×C₁₉	semidirect
161. C₄×M₁₆	C₄×C₁₉	semidirect
198. #108	C₂×C₄×C₁₉	semidirect
199. #107	C₂×C₄×C₁₉	semidirect
200. #106	C₂×C₄×C₁₉	semidirect
202. #97	C₂×C₄×C₁₉	semidirect
204. C₄×C₈	C₂×C₄×C₁₉	semidirect
205. C₁₉	#20445	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×#430	2
C₁₉×#2911	2
C₁₉×#2873	2
C₂×C₄×M₁₆×C₁₉	2
C₁₉×#1773	2
C₁₉×#1772	2
C₂×C₁₉×#339	2
#20445	19

Subgroups — C19×#20445 #18026404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20445 #18026404