Subgroups — C₁₉×#20688 #18026647

Order 4864.

1510 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2280	C₂	1	2	yes	maximal
3	C₁₉×#1820	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1819	C₂	1	4	yes	maximal
5	C₁₉×#1740	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#117	—	2	4	no	—
8	C₂×C₁₉×#117	—	2	4	no	—
9	C₂×C₁₉×#117	C₄	1	2	yes	complemented
10	C₂×C₁₉×#117	C₄	1	2	yes	complemented
11	C₂×C₁₉×#117	C₂²	1	4	yes	—
12	C₂×C₁₉×#117	C₂²	1	2	yes	—
13	C₂×C₁₉×#117	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	complemented
15	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₄	1	2	yes	complemented
17	C₁₉×#381	C₂²	1	4	yes	—
18	C₁₉×#380	—	2	4	no	—
19	C₁₉×#379	C₂²	1	4	yes	—
20	C₄×C₂²⋊C₄×C₁₉	—	2	4	no	—
21	C₁₉×#342	C₂²	1	16	yes	—
22	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	—	2	8	no	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₁₉	D₄	1	4	yes	—
30	C₂³×C₄×C₁₉	D₄	1	4	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×#117	—	2	8	no	—
34	C₁₉×#117	—	2	4	no	—
35	C₁₉×#117	—	2	4	no	—
36	C₁₉×#117	—	2	8	no	—
37	C₁₉×#117	—	2	4	no	—
38	C₁₉×#117	—	2	4	no	—
39	C₁₉×#117	—	2	16	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
46	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
47	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
49	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
50	C₂×C₄²×C₁₉	—	2	4	no	—
51	C₂×C₄²×C₁₉	—	2	4	no	—
52	C₂×C₄²×C₁₉	—	2	4	no	—
53	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
55	C₁₉×#95	—	2	8	no	—
56	C₂⁴×C₁₉	—	2	4	no	—
57	C₂⁴×C₁₉	—	4	4	no	—
58	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂⁴×C₁₉	C₂×D₄	1	4	yes	—
61	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	—	2	4	no	—
67	C₂²×C₄×C₁₉	—	2	16	no	—
68	C₂²×C₄×C₁₉	—	2	16	no	—
69	C₂²×C₄×C₁₉	—	2	4	no	—
70	C₂²×C₄×C₁₉	—	2	4	no	—
71	C₂²×C₄×C₁₉	—	2	8	no	—
72	C₂²×C₄×C₁₉	—	4	8	no	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₉	#41	1	8	yes	—
88	C₄⋊C₄×C₁₉	—	2	8	no	—
89	C₄⋊C₄×C₁₉	—	2	8	no	—
90	C₄⋊C₄×C₁₉	—	4	8	no	—
91	C₂²⋊C₄×C₁₉	—	2	8	no	—
92	C₂²⋊C₄×C₁₉	—	2	8	no	—
93	C₂²⋊C₄×C₁₉	—	2	16	no	—
94	C₂²⋊C₄×C₁₉	—	4	16	no	—
95	C₄²×C₁₉	—	2	8	no	—
96	C₄²×C₁₉	—	2	4	no	—
97	C₄²×C₁₉	—	2	4	no	—
98	C₄²×C₁₉	—	4	8	no	—
99	#20688	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	—	2	4	no	—
102	C₂³×C₁₉	—	2	2	no	—
103	C₂³×C₁₉	—	2	8	no	—
104	C₂³×C₁₉	—	2	8	no	—
105	C₂³×C₁₉	—	4	2	no	—
106	C₂³×C₁₉	—	4	4	no	—
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	—	4	16	no	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
112	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
113	C₂³×C₁₉	C₂×#41	1	4	yes	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	8	no	—
119	C₂×C₄×C₁₉	—	2	8	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	8	no	—
124	C₂×C₄×C₁₉	—	4	8	no	—
125	C₂×C₄×C₁₉	—	4	8	no	—
126	C₂×C₄×C₁₉	—	4	8	no	—
127	C₂×C₄×C₁₉	—	4	16	no	—
128	C₂×C₄×C₁₉	—	4	8	no	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#117	1	4	yes	—
135	C₂×C₄×C₁₉	#117	1	4	yes	—
136	C₂×C₄×C₁₉	#123	1	8	yes	—
137	C₂×C₄×C₁₉	#123	1	8	yes	—
138	C₂²×#117	C₂×C₁₉	1	1	yes	characteristic
139	#2280	C₂×C₁₉	1	2	yes	—
140	#1820	C₂×C₁₉	1	4	yes	complemented
141	#1819	C₂×C₁₉	1	4	yes	—
142	#1740	C₂×C₁₉	1	4	yes	complemented
143	C₂²×C₁₉	—	2	8	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	4	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	4	4	no	—
151	C₂²×C₁₉	—	4	16	no	—
152	C₂²×C₁₉	—	4	4	no	—
153	C₂²×C₁₉	#410	1	4	yes	—
154	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#117	1	2	yes	—
156	C₂²×C₁₉	C₂×#123	1	4	yes	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	4	8	no	—
160	C₄×C₁₉	—	8	8	no	—
161	C₄×C₁₉	#381	1	2	yes	—
162	C₄×C₁₉	#381	1	4	yes	—
163	C₄×C₁₉	#381	1	2	yes	—
164	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×#117	—	2	4	no	—
166	C₂×#117	—	2	4	no	—
167	C₂×#117	C₄×C₁₉	1	2	yes	complemented
168	C₂×#117	C₄×C₁₉	1	2	yes	complemented
169	C₂×#117	C₂²×C₁₉	1	4	yes	—
170	C₂×#117	C₂²×C₁₉	1	2	yes	—
171	C₂×#117	C₂²×C₁₉	1	2	yes	—
172	C₂²×C₄⋊C₄	C₄×C₁₉	1	2	yes	complemented
173	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
174	C₂²×C₄²	C₄×C₁₉	1	2	yes	complemented
175	#381	C₂²×C₁₉	1	4	yes	—
176	#380	—	2	4	no	—
177	#379	C₂²×C₁₉	1	4	yes	—
178	C₄×C₂²⋊C₄	—	2	4	no	—
179	#342	C₂²×C₁₉	1	16	yes	—
180	C₂×C₁₉	—	2	4	no	—
181	C₂×C₁₉	—	2	2	no	—
182	C₂×C₁₉	—	2	4	no	—
183	C₂×C₁₉	—	4	2	no	—
184	C₂×C₁₉	C₂×#381	1	1	yes	characteristic
185	C₂×C₁₉	C₂×#410	1	2	yes	—
186	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
187	C₂³×C₄	—	2	8	no	—
188	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
191	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
192	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
193	C₂³×C₄	D₄×C₁₉	1	4	yes	—
194	C₂³×C₄	D₄×C₁₉	1	4	yes	—
195	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
197	#117	—	2	8	no	—
198	#117	—	2	4	no	—
199	#117	—	2	4	no	—
200	#117	—	2	8	no	—
201	#117	—	2	4	no	—
202	#117	—	2	4	no	—
203	#117	—	2	16	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	8	no	—
210	C₂×C₂²⋊C₄	—	2	8	no	—
211	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
212	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
213	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
214	C₂×C₄²	—	2	4	no	—
215	C₂×C₄²	—	2	4	no	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
218	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
219	#95	—	2	8	no	—
220	C₁₉	#20688	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	—	2	4	no	—
222	C₂⁴	—	4	4	no	—
223	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
224	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
225	C₂⁴	C₂×D₄×C₁₉	1	4	yes	—
226	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	16	no	—
233	C₂²×C₄	—	2	16	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	4	8	no	—
238	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
239	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
240	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
242	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
247	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
250	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
252	C₂²×C₄	#41×C₁₉	1	8	yes	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	4	8	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₂²⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	2	16	no	—
259	C₂²⋊C₄	—	4	16	no	—
260	C₄²	—	2	8	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	4	8	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	8	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	4	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	16	no	—
273	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
274	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
275	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
276	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
277	C₂³	C₂×#41×C₁₉	1	4	yes	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	8	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
298	C₂×C₄	C₁₉×#117	1	4	yes	—
299	C₂×C₄	C₁₉×#117	1	4	yes	—
300	C₂×C₄	C₁₉×#123	1	8	yes	—
301	C₂×C₄	C₁₉×#123	1	8	yes	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	16	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₁₉×#410	1	4	yes	—
313	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
314	C₂²	C₂×C₁₉×#117	1	2	yes	—
315	C₂²	C₂×C₁₉×#123	1	4	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	8	no	—
319	C₄	—	8	8	no	—
320	C₄	C₁₉×#381	1	2	yes	—
321	C₄	C₁₉×#381	1	2	yes	—
322	C₄	C₁₉×#381	1	4	yes	—
323	C₂	—	2	4	no	—
324	C₂	—	2	4	no	—
325	C₂	—	2	2	no	—
326	C₂	—	4	2	no	—
327	C₂	C₂×C₁₉×#381	1	1	yes	characteristic
328	C₂	C₂×C₁₉×#410	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
prime radical	C₂⁵×C₁₉	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂⁴×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1820	C₂	semidirect
5. C₁₉×#1740	C₂	semidirect
9. C₂×C₁₉×#117	C₄	semidirect
10. C₂×C₁₉×#117	C₄	semidirect
14. C₂²×C₄⋊C₄×C₁₉	C₄	semidirect
16. C₂²×C₄²×C₁₉	C₄	semidirect
99. #20688	C₁₉	direct (N × H)
140. #1820	C₂×C₁₉	semidirect
142. #1740	C₂×C₁₉	semidirect
167. C₂×#117	C₄×C₁₉	semidirect
168. C₂×#117	C₄×C₁₉	semidirect
172. C₂²×C₄⋊C₄	C₄×C₁₉	semidirect
174. C₂²×C₄²	C₄×C₁₉	semidirect
220. C₁₉	#20688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#117	2
C₁₉×#2280	2
C₁₉×#1820	2
C₁₉×#1819	2
C₁₉×#1740	2
#20688	19

Subgroups — C19×#20688 #18026647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20688 #18026647