Subgroups — C₁₉×#20444 #18026403

Order 4864.

1018 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₂×C₁₉×#405	C₂	1	2	yes	maximal
4	C₂×C₁₉×#405	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	C₁₉×#1733	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#117	C₂²	1	2	yes	—
12	C₂×C₁₉×#117	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	2	no	—
18	C₁₉×#405	—	2	2	no	—
19	C₁₉×#405	C₄	1	8	yes	complemented
20	C₁₉×#405	C₂²	1	4	yes	—
21	C₁₉×#405	C₂²	1	4	yes	—
22	C₂×C₁₉×#97	—	2	2	no	—
23	C₂×C₁₉×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₉	—	2	2	no	—
25	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
26	C₁₉×#339	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	2	4	no	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	8	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	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₉	—	2	2	no	—
36	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₉	D₄	1	2	yes	—
38	C₂²×C₈×C₁₉	D₄	1	2	yes	—
39	C₂²×C₈×C₁₉	Q₈	1	4	yes	—
40	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×#117	—	2	2	no	—
44	C₁₉×#117	—	2	2	no	—
45	C₁₉×#117	—	2	2	no	—
46	C₁₉×#117	C₂×C₄	1	4	yes	—
47	C₁₉×#117	C₂×C₄	1	4	yes	—
48	C₁₉×#117	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
51	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
53	C₂×C₄²×C₁₉	—	2	2	no	—
54	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
55	C₁₉×#108	C₂×C₄	1	16	yes	complemented
56	C₁₉×#107	C₂×C₄	1	8	yes	—
57	C₁₉×#106	C₂×C₄	1	8	yes	—
58	C₁₉×#97	—	2	4	no	—
59	C₁₉×#97	—	2	4	no	—
60	C₁₉×#97	C₂×C₄	1	8	yes	complemented
61	C₄×C₈×C₁₉	—	2	4	no	—
62	C₄×C₈×C₁₉	—	2	4	no	—
63	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	complemented
64	C₂⁴×C₁₉	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₁₉	—	2	8	no	—
76	M₁₆×C₁₉	—	2	4	no	—
77	M₁₆×C₁₉	C₄²	1	16	yes	—
78	M₁₆×C₁₉	C₂²×C₄	1	8	yes	—
79	C₂×C₈×C₁₉	—	2	8	no	—
80	C₂×C₈×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
82	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
83	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
93	C₄⋊C₄×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
96	C₂²⋊C₄×C₁₉	—	2	4	no	—
97	C₂²⋊C₄×C₁₉	—	2	4	no	—
98	C₂²⋊C₄×C₁₉	—	2	4	no	—
99	C₄²×C₁₉	—	2	4	no	—
100	C₄²×C₁₉	—	2	4	no	—
101	C₄²×C₁₉	—	2	2	no	—
102	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
103	#20444	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₁₉	—	4	2	no	—
105	C₂³×C₁₉	#117	1	2	yes	—
106	C₂³×C₁₉	C₄×D₄	1	2	yes	—
107	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
108	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	8	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	4	8	no	—
113	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₉	#117	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
126	C₈×C₁₉	—	2	8	no	—
127	C₈×C₁₉	C₂×C₄²	1	8	yes	—
128	C₈×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
129	C₈×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×#430	C₂×C₁₉	1	1	yes	characteristic
132	#2911	C₂×C₁₉	1	1	yes	characteristic
133	C₂×#405	C₂×C₁₉	1	2	yes	—
134	C₂×#405	C₂×C₁₉	1	1	yes	characteristic
135	#1773	C₂×C₁₉	1	4	yes	complemented
136	#1772	C₂×C₁₉	1	4	yes	complemented
137	#1733	C₂×C₁₉	1	2	yes	—
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	4	2	no	—
141	C₂²×C₁₉	—	4	2	no	—
142	C₂²×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
143	C₂²×C₁₉	C₂×#117	1	1	yes	characteristic
144	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₁₉	—	4	4	no	—
147	C₄×C₁₉	—	4	8	no	—
148	C₄×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
149	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
150	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
155	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
156	C₂×#117	C₂²×C₁₉	1	2	yes	—
157	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
158	#430	C₄×C₁₉	1	8	yes	complemented
159	C₂×#108	C₂²×C₁₉	1	4	yes	—
160	#428	C₄×C₁₉	1	8	yes	complemented
161	#427	C₂²×C₁₉	1	4	yes	—
162	#405	—	2	2	no	—
163	#405	—	2	2	no	—
164	#405	C₄×C₁₉	1	8	yes	complemented
165	#405	C₂²×C₁₉	1	4	yes	—
166	#405	C₂²×C₁₉	1	4	yes	—
167	C₂×#97	—	2	2	no	—
168	C₂×#97	C₂²×C₁₉	1	2	yes	—
169	C₂×C₄×C₈	—	2	2	no	—
170	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
171	#339	C₂²×C₁₉	1	8	yes	—
172	C₂×C₁₉	—	2	1	no	—
173	C₂×C₁₉	—	2	1	no	—
174	C₂×C₁₉	—	2	4	no	—
175	C₂×C₁₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₁₉	#2955	1	2	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	C₂×C₄×C₁₉	1	8	yes	—
181	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
182	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
183	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
184	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
187	C₂²×C₈	D₄×C₁₉	1	2	yes	—
188	C₂²×C₈	D₄×C₁₉	1	2	yes	—
189	C₂²×C₈	Q₈×C₁₉	1	4	yes	—
190	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	C₂×C₄×C₁₉	1	4	yes	—
197	#117	C₂×C₄×C₁₉	1	4	yes	—
198	#117	C₂³×C₁₉	1	4	yes	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
205	#108	C₂×C₄×C₁₉	1	16	yes	complemented
206	#107	C₂×C₄×C₁₉	1	8	yes	—
207	#106	C₂×C₄×C₁₉	1	8	yes	—
208	#97	—	2	4	no	—
209	#97	—	2	4	no	—
210	#97	C₂×C₄×C₁₉	1	8	yes	complemented
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	—	2	4	no	—
213	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	complemented
214	C₁₉	#20444	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	#41×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
222	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
223	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	#41×C₁₉	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
226	M₁₆	—	2	8	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	C₄²×C₁₉	1	16	yes	—
229	M₁₆	C₂²×C₄×C₁₉	1	8	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₄⋊C₄×C₁₉	1	8	yes	—
233	C₂×C₈	C₄⋊C₄×C₁₉	1	8	yes	—
234	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
237	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
238	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
242	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	C₁₉×#117	1	2	yes	—
256	C₂³	C₄×D₄×C₁₉	1	2	yes	—
257	C₂³	C₄×Q₈×C₁₉	1	2	yes	—
258	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
264	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
265	C₂×C₄	C₁₉×#117	1	2	yes	—
266	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₁₉	1	4	yes	—
268	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
269	C₂×C₄	C₄×Q₈×C₁₉	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
271	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
272	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₈	C₂×C₄²×C₁₉	1	8	yes	—
278	C₈	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
279	C₈	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
280	C₈	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
286	C₂²	C₂×C₁₉×#117	1	1	yes	characteristic
287	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
288	C₂²	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
289	C₄	—	4	4	no	—
290	C₄	—	4	8	no	—
291	C₄	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
292	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
293	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
294	C₄	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
295	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	4	no	—
298	C₂	—	2	1	no	—
299	C₂	C₂×C₄×C₄⋊C₄×C₁₉	1	1	yes	characteristic
300	C₂	C₁₉×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×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×C₈×C₁₉ ◁ C₂³×C₈×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
19. C₁₉×#405	C₄	semidirect
55. C₁₉×#108	C₂×C₄	semidirect
60. C₁₉×#97	C₂×C₄	semidirect
63. C₄×C₈×C₁₉	C₂×C₄	semidirect
103. #20444	C₁₉	direct (N × H)
135. #1773	C₂×C₁₉	semidirect
136. #1772	C₂×C₁₉	semidirect
158. #430	C₄×C₁₉	semidirect
160. #428	C₄×C₁₉	semidirect
164. #405	C₄×C₁₉	semidirect
205. #108	C₂×C₄×C₁₉	semidirect
210. #97	C₂×C₄×C₁₉	semidirect
213. C₄×C₈	C₂×C₄×C₁₉	semidirect
214. C₁₉	#20444	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
C₂×C₁₉×#405	2
C₁₉×#1773	2
C₁₉×#1772	2
C₁₉×#1733	2
#20444	19

Subgroups — C19×#20444 #18026403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20444 #18026403