Subgroups — C₁₉×#20503 #18026462

Order 4864.

1206 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×#442	C₂	1	2	yes	maximal
2	C₂×C₄×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2275	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1776	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1762	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₉×#102	C₂	1	4	yes	maximal, complemented
8	C₁₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₉	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₉	C₂²	1	4	yes	—
12	C₁₉×#442	C₄	1	16	yes	complemented
13	C₄×D₈×C₁₉	C₄	1	8	yes	complemented
14	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#106	C₂²	1	2	yes	—
16	C₂×C₁₉×#102	C₂²	1	2	yes	—
17	C₂×C₁₉×#102	C₂²	1	4	yes	—
18	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
19	C₂×C₁₉×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₁₉	—	2	2	no	—
25	C₄×C₂²⋊C₄×C₁₉	—	2	4	no	—
26	C₁₉×#376	C₂²	1	1	yes	characteristic
27	C₂×C₁₉×#95	—	2	2	no	—
28	C₁₉×#340	C₂²	1	2	yes	—
29	C₁₉×#340	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₉	—	2	4	no	—
32	C₂³×C₄×C₁₉	—	2	2	no	—
33	C₂×D₈×C₁₉	C₂×C₄	1	4	yes	—
34	C₂×D₈×C₁₉	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₁₉	—	2	8	no	—
39	C₄×D₄×C₁₉	—	2	4	no	—
40	C₄×D₄×C₁₉	C₂×C₄	1	8	yes	complemented
41	C₄×D₄×C₁₉	C₂×C₄	1	16	yes	complemented
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
46	C₂×C₄²×C₁₉	—	2	2	no	—
47	C₂×C₄²×C₁₉	—	2	4	no	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
50	C₁₉×#107	C₂×C₄	1	4	yes	complemented
51	C₁₉×#106	C₂×C₄	1	8	yes	complemented
52	C₁₉×#102	C₂×C₄	1	8	yes	complemented
53	C₁₉×#102	C₂×C₄	1	16	yes	complemented
54	C₁₉×#97	—	2	2	no	—
55	C₁₉×#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₁₉	—	2	4	no	—
57	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	complemented
58	C₁₉×#95	—	2	4	no	—
59	C₁₉×#95	—	2	2	no	—
60	C₁₉×#95	D₄	1	2	yes	—
61	C₁₉×#95	D₄	1	2	yes	—
62	C₂⁴×C₁₉	—	2	2	no	—
63	C₂×D₄×C₁₉	—	2	4	no	—
64	C₂×D₄×C₁₉	—	2	2	no	—
65	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
66	C₂×D₄×C₁₉	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	2	4	no	—
69	C₂²×C₄×C₁₉	—	2	16	no	—
70	C₂²×C₄×C₁₉	—	2	8	no	—
71	C₂²×C₄×C₁₉	—	8	4	no	—
72	C₂²×C₄×C₁₉	—	8	4	no	—
73	C₂²×C₄×C₁₉	—	8	4	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
78	D₈×C₁₉	C₄²	1	16	yes	complemented
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₁₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₉	—	2	16	no	—
92	C₄²×C₁₉	—	8	4	no	—
93	C₄²×C₁₉	—	8	2	no	—
94	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₉	#41	1	4	yes	—
97	C₄²×C₁₉	#41	1	2	yes	—
98	#20503	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₁₉	—	2	8	no	—
100	C₂³×C₁₉	—	2	4	no	—
101	C₂³×C₁₉	—	4	4	no	—
102	C₂³×C₁₉	—	4	2	no	—
103	C₂³×C₁₉	—	4	2	no	—
104	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
105	D₄×C₁₉	—	2	4	no	—
106	D₄×C₁₉	—	2	8	no	—
107	D₄×C₁₉	C₂×C₄²	1	8	yes	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	8	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	8	16	no	—
118	C₂×C₄×C₁₉	—	8	8	no	—
119	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
121	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
125	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
126	C₈×C₁₉	—	2	2	no	—
127	C₈×C₁₉	—	2	4	no	—
128	C₈×C₁₉	C₂×C₄²	1	4	yes	—
129	C₂×#442	C₂×C₁₉	1	2	yes	—
130	C₂×C₄×D₈	C₂×C₁₉	1	1	yes	characteristic
131	#2275	C₂×C₁₉	1	2	yes	complemented
132	#1776	C₂×C₁₉	1	2	yes	complemented
133	C₄×#106	C₂×C₁₉	1	1	yes	characteristic, complemented
134	#1762	C₂×C₁₉	1	2	yes	complemented
135	C₄×#102	C₂×C₁₉	1	4	yes	complemented
136	#1726	C₂×C₁₉	1	1	yes	characteristic, complemented
137	C₂²×C₁₉	—	2	8	no	—
138	C₂²×C₁₉	—	4	8	no	—
139	C₂²×C₁₉	—	4	8	no	—
140	C₂²×C₁₉	—	4	4	no	—
141	C₂²×C₁₉	—	4	4	no	—
142	C₂²×C₁₉	C₂×C₄×D₄	1	2	yes	—
143	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₁₉	#576	1	2	yes	—
145	C₂²×C₁₉	#577	1	2	yes	—
146	C₄×C₁₉	—	2	8	no	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	4	8	no	—
150	C₄×C₁₉	—	4	4	no	—
151	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
152	C₄×C₁₉	C₄×#41	1	1	yes	characteristic
153	C₄×C₁₉	C₄×#41	1	2	yes	—
154	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
155	C₂×C₄×D₄	C₂²×C₁₉	1	2	yes	—
156	C₂×C₄×D₄	C₂²×C₁₉	1	4	yes	—
157	#442	C₄×C₁₉	1	16	yes	complemented
158	C₄×D₈	C₄×C₁₉	1	8	yes	complemented
159	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
160	C₂×#106	C₂²×C₁₉	1	2	yes	—
161	C₂×#102	C₂²×C₁₉	1	2	yes	—
162	C₂×#102	C₂²×C₁₉	1	4	yes	—
163	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×#97	C₂²×C₁₉	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
167	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
169	C₄×C₂²⋊C₄	—	2	2	no	—
170	C₄×C₂²⋊C₄	—	2	4	no	—
171	#376	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×#95	—	2	2	no	—
173	#340	C₂²×C₁₉	1	2	yes	—
174	#340	C₂²×C₁₉	1	4	yes	—
175	C₂×C₁₉	—	4	8	no	—
176	C₂×C₁₉	C₄²×D₄	1	1	yes	characteristic
177	C₂×C₁₉	#2946	1	4	yes	—
178	C₂×C₁₉	#2948	1	1	yes	characteristic
179	C₂×C₁₉	#2952	1	1	yes	characteristic
180	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
181	C₂³×C₄	—	2	2	no	—
182	C₂³×C₄	—	2	4	no	—
183	C₂×D₈	C₂×C₄×C₁₉	1	4	yes	—
184	C₂×D₈	C₂×C₄×C₁₉	1	8	yes	—
185	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
188	C₄×D₄	—	2	8	no	—
189	C₄×D₄	—	2	4	no	—
190	C₄×D₄	C₂×C₄×C₁₉	1	8	yes	complemented
191	C₄×D₄	C₂×C₄×C₁₉	1	16	yes	complemented
192	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	4	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	4	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
200	#107	C₂×C₄×C₁₉	1	4	yes	complemented
201	#106	C₂×C₄×C₁₉	1	8	yes	complemented
202	#102	C₂×C₄×C₁₉	1	8	yes	complemented
203	#102	C₂×C₄×C₁₉	1	16	yes	complemented
204	#97	—	2	2	no	—
205	#97	C₂×C₄×C₁₉	1	8	yes	complemented
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	complemented
208	#95	—	2	2	no	—
209	#95	—	2	4	no	—
210	#95	D₄×C₁₉	1	2	yes	—
211	#95	D₄×C₁₉	1	2	yes	—
212	C₁₉	#20503	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁴	—	2	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
217	C₂×D₄	C₂²×C₄×C₁₉	1	8	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	16	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	—	8	4	no	—
224	C₂²×C₄	—	8	4	no	—
225	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
229	D₈	C₄²×C₁₉	1	16	yes	complemented
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
240	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₄²	—	8	4	no	—
244	C₄²	—	8	2	no	—
245	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
246	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
247	C₄²	#41×C₁₉	1	4	yes	—
248	C₄²	#41×C₁₉	1	2	yes	—
249	C₂³	—	2	8	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	8	no	—
257	D₄	C₂×C₄²×C₁₉	1	8	yes	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	16	no	—
268	C₂×C₄	—	8	8	no	—
269	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
270	C₂×C₄	C₄×D₄×C₁₉	1	8	yes	—
271	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
272	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
273	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
275	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	4	no	—
278	C₈	C₂×C₄²×C₁₉	1	4	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	C₂×C₄×D₄×C₁₉	1	2	yes	—
285	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
286	C₂²	C₁₉×#576	1	2	yes	—
287	C₂²	C₁₉×#577	1	2	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	8	no	—
292	C₄	—	4	4	no	—
293	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
294	C₄	C₄×#41×C₁₉	1	1	yes	characteristic
295	C₄	C₄×#41×C₁₉	1	2	yes	—
296	C₂	—	4	8	no	—
297	C₂	C₄²×D₄×C₁₉	1	1	yes	characteristic
298	C₂	C₁₉×#2946	1	4	yes	—
299	C₂	C₁₉×#2948	1	1	yes	characteristic
300	C₂	C₁₉×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2275	C₂	semidirect
4. C₁₉×#1776	C₂	semidirect
5. C₄×C₁₉×#106	C₂	semidirect
6. C₁₉×#1762	C₂	semidirect
7. C₄×C₁₉×#102	C₂	semidirect
8. C₁₉×#1726	C₂	semidirect
12. C₁₉×#442	C₄	semidirect
13. C₄×D₈×C₁₉	C₄	semidirect
40. C₄×D₄×C₁₉	C₂×C₄	semidirect
41. C₄×D₄×C₁₉	C₂×C₄	semidirect
50. C₁₉×#107	C₂×C₄	semidirect
51. C₁₉×#106	C₂×C₄	semidirect
52. C₁₉×#102	C₂×C₄	semidirect
53. C₁₉×#102	C₂×C₄	semidirect
55. C₁₉×#97	C₂×C₄	semidirect
57. C₄×C₈×C₁₉	C₂×C₄	semidirect
78. D₈×C₁₉	C₄²	semidirect
98. #20503	C₁₉	direct (N × H)
131. #2275	C₂×C₁₉	semidirect
132. #1776	C₂×C₁₉	semidirect
133. C₄×#106	C₂×C₁₉	semidirect
134. #1762	C₂×C₁₉	semidirect
135. C₄×#102	C₂×C₁₉	semidirect
136. #1726	C₂×C₁₉	semidirect
157. #442	C₄×C₁₉	semidirect
158. C₄×D₈	C₄×C₁₉	semidirect
190. C₄×D₄	C₂×C₄×C₁₉	semidirect
191. C₄×D₄	C₂×C₄×C₁₉	semidirect
200. #107	C₂×C₄×C₁₉	semidirect
201. #106	C₂×C₄×C₁₉	semidirect
202. #102	C₂×C₄×C₁₉	semidirect
203. #102	C₂×C₄×C₁₉	semidirect
205. #97	C₂×C₄×C₁₉	semidirect
207. C₄×C₈	C₂×C₄×C₁₉	semidirect
212. C₁₉	#20503	direct (N × H)
229. D₈	C₄²×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#442	2
C₂×C₄×D₈×C₁₉	2
C₁₉×#2275	2
C₁₉×#1776	2
C₄×C₁₉×#106	2
C₁₉×#1762	2
C₄×C₁₉×#102	2
C₁₉×#1726	2
#20503	19

Subgroups — C19×#20503 #18026462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20503 #18026462