Subgroups — C₁₉×#20455 #18026414

Order 4864.

1126 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#405	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2274	C₂	1	2	yes	maximal
5	C₁₉×#1776	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₉×#107	C₂	1	4	yes	maximal, complemented
7	C₁₉×#1736	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#117	C₂²	1	2	yes	—
11	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₁₉×#427	C₄	1	8	yes	complemented
14	C₂×C₁₉×#107	C₄	1	8	yes	complemented
15	C₂×C₁₉×#107	C₂²	1	2	yes	—
16	C₂×C₁₉×#107	C₂²	1	2	yes	—
17	C₂×C₁₉×#107	C₂²	1	2	yes	—
18	C₂×C₁₉×#106	C₂²	1	2	yes	—
19	C₁₉×#405	C₄	1	8	yes	complemented
20	C₂×C₁₉×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₁₉	—	2	4	no	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
24	C₄×C₂²⋊C₄×C₁₉	—	2	4	no	—
25	C₁₉×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₉	—	2	2	no	—
27	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
37	C₁₉×#117	—	2	4	no	—
38	C₁₉×#117	C₂×C₄	1	8	yes	—
39	C₁₉×#117	C₂×C₄	1	4	yes	—
40	C₁₉×#117	Q₈	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
49	C₂×C₄²×C₁₉	—	2	2	no	—
50	C₂×C₄²×C₁₉	—	2	8	no	—
51	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
52	C₁₉×#107	C₂×C₄	1	8	yes	—
53	C₁₉×#107	C₂×C₄	1	8	yes	—
54	C₁₉×#107	C₂×C₄	1	8	yes	complemented
55	C₁₉×#106	C₂×C₄	1	8	yes	complemented
56	C₁₉×#97	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	complemented
58	C₁₉×#95	—	2	8	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	8	4	no	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₉	—	2	8	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	C₄²	1	16	yes	complemented
77	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
84	C₄⋊C₄×C₁₉	—	2	8	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	8	yes	—
88	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₁₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	8	yes	—
92	C₄²×C₁₉	—	2	8	no	—
93	C₄²×C₁₉	—	8	8	no	—
94	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
95	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
96	#20455	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	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₁₉	C₄×D₄	1	2	yes	—
101	C₂³×C₁₉	C₄×D₄	1	2	yes	—
102	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	8	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	—	4	8	no	—
109	C₂×C₄×C₁₉	—	4	4	no	—
110	C₂×C₄×C₁₉	—	8	8	no	—
111	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₉	#117	1	4	yes	—
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
125	C₈×C₁₉	—	2	4	no	—
126	C₈×C₁₉	C₂×C₄²	1	4	yes	—
127	C₈×C₁₉	C₂×C₄²	1	4	yes	—
128	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
129	C₂²×#107	C₂×C₁₉	1	1	yes	characteristic
130	C₂×#405	C₂×C₁₉	1	1	yes	characteristic
131	#2274	C₂×C₁₉	1	2	yes	—
132	#1776	C₂×C₁₉	1	4	yes	complemented
133	C₄×#107	C₂×C₁₉	1	4	yes	complemented
134	#1736	C₂×C₁₉	1	2	yes	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
140	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₁₉	#576	1	2	yes	—
144	C₂²×C₁₉	#578	1	2	yes	—
145	C₄×C₁₉	—	2	8	no	—
146	C₄×C₁₉	—	4	8	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	C₂×#117	1	1	yes	characteristic
151	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
155	C₂×#117	C₂²×C₁₉	1	2	yes	—
156	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
158	#427	C₄×C₁₉	1	8	yes	complemented
159	C₂×#107	C₄×C₁₉	1	8	yes	complemented
160	C₂×#107	C₂²×C₁₉	1	2	yes	—
161	C₂×#107	C₂²×C₁₉	1	2	yes	—
162	C₂×#107	C₂²×C₁₉	1	2	yes	—
163	C₂×#106	C₂²×C₁₉	1	2	yes	—
164	#405	C₄×C₁₉	1	8	yes	complemented
165	C₂×#97	C₂²×C₁₉	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
167	C₄×C₄⋊C₄	—	2	4	no	—
168	C₄×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
169	C₄×C₂²⋊C₄	—	2	4	no	—
170	#340	C₂²×C₁₉	1	8	yes	—
171	C₂×C₁₉	—	2	4	no	—
172	C₂×C₁₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₁₉	#2913	1	2	yes	—
174	C₂×C₁₉	#2948	1	1	yes	characteristic
175	C₂×C₁₉	#2950	1	1	yes	characteristic
176	C₂×C₁₉	#2951	1	2	yes	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
182	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
183	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
184	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
185	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
188	#117	—	2	4	no	—
189	#117	C₂×C₄×C₁₉	1	8	yes	—
190	#117	C₂×C₄×C₁₉	1	4	yes	—
191	#117	Q₈×C₁₉	1	4	yes	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
195	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	8	no	—
202	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
203	#107	C₂×C₄×C₁₉	1	8	yes	—
204	#107	C₂×C₄×C₁₉	1	8	yes	—
205	#107	C₂×C₄×C₁₉	1	8	yes	complemented
206	#106	C₂×C₄×C₁₉	1	8	yes	complemented
207	#97	C₂×C₄×C₁₉	1	8	yes	complemented
208	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	complemented
209	#95	—	2	8	no	—
210	C₁₉	#20455	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
220	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
222	C₂²×C₄	#41×C₁₉	1	2	yes	—
223	C₂²×C₄	#41×C₁₉	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
225	M₁₆	—	2	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₄²×C₁₉	1	16	yes	complemented
229	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	C₄⋊C₄×C₁₉	1	8	yes	—
240	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
241	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
242	C₂²⋊C₄	—	2	8	no	—
243	C₂²⋊C₄	C₄⋊C₄×C₁₉	1	8	yes	—
244	C₄²	—	2	8	no	—
245	C₄²	—	8	8	no	—
246	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
247	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
251	C₂³	C₄×D₄×C₁₉	1	2	yes	—
252	C₂³	C₄×D₄×C₁₉	1	2	yes	—
253	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	8	8	no	—
262	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
264	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
265	C₂×C₄	C₁₉×#117	1	4	yes	—
266	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
268	C₂×C₄	C₄×Q₈×C₁₉	1	4	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×Q₈×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	C₂×C₄²×C₁₉	1	4	yes	—
278	C₈	C₂×C₄²×C₁₉	1	4	yes	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
284	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
285	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
286	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
287	C₂²	C₁₉×#576	1	2	yes	—
288	C₂²	C₁₉×#578	1	2	yes	—
289	C₄	—	2	8	no	—
290	C₄	—	4	8	no	—
291	C₄	—	4	8	no	—
292	C₄	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
293	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
294	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
295	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
296	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×C₄×C₄⋊C₄×C₁₉	1	1	yes	characteristic
299	C₂	C₁₉×#2913	1	2	yes	—
300	C₂	C₁₉×#2948	1	1	yes	characteristic
301	C₂	C₁₉×#2950	1	1	yes	characteristic
302	C₂	C₁₉×#2951	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₉×#1776	C₂	semidirect
6. C₄×C₁₉×#107	C₂	semidirect
13. C₁₉×#427	C₄	semidirect
14. C₂×C₁₉×#107	C₄	semidirect
19. C₁₉×#405	C₄	semidirect
54. C₁₉×#107	C₂×C₄	semidirect
55. C₁₉×#106	C₂×C₄	semidirect
56. C₁₉×#97	C₂×C₄	semidirect
57. C₄×C₈×C₁₉	C₂×C₄	semidirect
76. C₂×C₈×C₁₉	C₄²	semidirect
96. #20455	C₁₉	direct (N × H)
132. #1776	C₂×C₁₉	semidirect
133. C₄×#107	C₂×C₁₉	semidirect
158. #427	C₄×C₁₉	semidirect
159. C₂×#107	C₄×C₁₉	semidirect
164. #405	C₄×C₁₉	semidirect
205. #107	C₂×C₄×C₁₉	semidirect
206. #106	C₂×C₄×C₁₉	semidirect
207. #97	C₂×C₄×C₁₉	semidirect
208. C₄×C₈	C₂×C₄×C₁₉	semidirect
210. C₁₉	#20455	direct (N × H)
228. C₂×C₈	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×#427	2
C₂²×C₁₉×#107	2
C₂×C₁₉×#405	2
C₁₉×#2274	2
C₁₉×#1776	2
C₄×C₁₉×#107	2
C₁₉×#1736	2
#20455	19

Subgroups — C19×#20455 #18026414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20455 #18026414