Subgroups — C₁₉×#20335 #18026294

Order 4864.

2206 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₁₉	C₂	1	2	yes	maximal
2	C₂²×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1729	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1728	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₉	—	2	8	no	—
7	C₂²×M₁₆×C₁₉	—	2	8	no	—
8	C₂²×M₁₆×C₁₉	C₄	1	16	yes	complemented
9	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
11	C₂³×C₈×C₁₉	C₂²	1	4	yes	—
12	C₂×C₁₉×#117	—	2	8	no	—
13	C₂×C₁₉×#117	C₄	1	4	yes	—
14	C₂×C₁₉×#117	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
18	C₁₉×#336	C₂²	1	16	yes	complemented
19	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₁₉	D₄	1	4	yes	—
25	C₂³×C₄×C₁₉	D₄	1	8	yes	—
26	C₂³×C₄×C₁₉	Q₈	1	4	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
29	C₂×M₁₆×C₁₉	—	2	64	no	—
30	C₂×M₁₆×C₁₉	—	2	8	no	—
31	C₂×M₁₆×C₁₉	—	2	8	no	—
32	C₂×M₁₆×C₁₉	—	2	8	no	—
33	C₂×M₁₆×C₁₉	—	2	8	no	—
34	C₂×M₁₆×C₁₉	C₂×C₄	1	16	yes	—
35	C₂×M₁₆×C₁₉	C₂×C₄	1	16	yes	complemented
36	C₂²×C₈×C₁₉	—	2	8	no	—
37	C₂²×C₈×C₁₉	—	2	8	no	—
38	C₂²×C₈×C₁₉	C₂×C₄	1	16	yes	complemented
39	C₂²×C₈×C₁₉	C₂³	1	8	yes	—
40	C₁₉×#117	—	2	16	no	—
41	C₁₉×#117	—	2	8	no	—
42	C₁₉×#117	—	2	8	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
44	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
46	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₁₉	—	2	8	no	—
48	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
50	C₂⁴×C₁₉	—	4	4	no	—
51	C₂⁴×C₁₉	C₄²	1	4	yes	—
52	C₂⁴×C₁₉	C₂²⋊C₄	1	4	yes	—
53	C₂⁴×C₁₉	C₄⋊C₄	1	4	yes	—
54	C₂⁴×C₁₉	C₂²×C₄	1	2	yes	—
55	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	2	8	no	—
57	C₂²×C₄×C₁₉	—	4	8	no	—
58	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₉	C₄²	1	8	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	32	yes	—
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	32	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₁₉	—	2	32	no	—
81	M₁₆×C₁₉	—	2	32	no	—
82	C₂×C₈×C₁₉	—	2	32	no	—
83	C₂×C₈×C₁₉	—	2	8	no	—
84	C₂×C₈×C₁₉	—	2	8	no	—
85	C₂×C₈×C₁₉	C₂²×C₄	1	16	yes	—
86	C₄⋊C₄×C₁₉	—	2	16	no	—
87	C₂²⋊C₄×C₁₉	—	2	16	no	—
88	C₄²×C₁₉	—	2	8	no	—
89	C₄²×C₁₉	—	2	8	no	—
90	#20335	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₁₉	—	2	4	no	—
92	C₂³×C₁₉	—	2	8	no	—
93	C₂³×C₁₉	—	4	8	no	—
94	C₂³×C₁₉	—	4	8	no	—
95	C₂³×C₁₉	—	4	8	no	—
96	C₂³×C₁₉	#95	1	16	yes	—
97	C₂³×C₁₉	C₂×C₄²	1	4	yes	—
98	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
102	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	—	2	8	no	—
105	C₂×C₄×C₁₉	—	2	16	no	—
106	C₂×C₄×C₁₉	—	4	16	no	—
107	C₂×C₄×C₁₉	#95	1	16	yes	—
108	C₂×C₄×C₁₉	#95	1	32	yes	—
109	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
124	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
125	C₈×C₁₉	—	2	16	no	—
126	C₂³×M₁₆	C₂×C₁₉	1	2	yes	—
127	C₂²×#117	C₂×C₁₉	1	1	yes	characteristic
128	#1729	C₂×C₁₉	1	8	yes	complemented
129	#1728	C₂×C₁₉	1	4	yes	complemented
130	C₂²×C₁₉	—	2	4	no	—
131	C₂²×C₁₉	—	2	8	no	—
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	2	8	no	—
134	C₂²×C₁₉	—	2	4	no	—
135	C₂²×C₁₉	—	2	8	no	—
136	C₂²×C₁₉	—	4	16	no	—
137	C₂²×C₁₉	C₂×#95	1	8	yes	—
138	C₂²×C₁₉	C₂×#95	1	4	yes	—
139	C₂²×C₁₉	C₂²×C₄²	1	1	yes	characteristic
140	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₉	#568	1	4	yes	—
143	C₄×C₁₉	—	4	8	no	—
144	C₄×C₁₉	C₂×#95	1	8	yes	—
145	C₄×C₁₉	C₂×#95	1	4	yes	—
146	C₄×C₁₉	C₂²×C₂²⋊C₄	1	2	yes	—
147	C₄×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
148	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
149	C₂²×M₁₆	—	2	8	no	—
150	C₂²×M₁₆	—	2	8	no	—
151	C₂²×M₁₆	C₄×C₁₉	1	16	yes	complemented
152	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
153	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
154	C₂³×C₈	C₂²×C₁₉	1	4	yes	—
155	C₂×#117	—	2	8	no	—
156	C₂×#117	C₄×C₁₉	1	4	yes	—
157	C₂×#117	C₂²×C₁₉	1	4	yes	—
158	C₂²×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
159	C₂²×C₂²⋊C₄	C₄×C₁₉	1	2	yes	—
160	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
161	#336	C₂²×C₁₉	1	16	yes	complemented
162	C₂×C₁₉	—	2	4	no	—
163	C₂×C₁₉	—	2	8	no	—
164	C₂×C₁₉	C₂²×#95	1	1	yes	characteristic
165	C₂×C₁₉	#2873	1	2	yes	—
166	C₂×C₁₉	#2879	1	2	yes	—
167	C₂×C₁₉	#2905	1	2	yes	—
168	C₂⁵	C₂×C₄×C₁₉	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
170	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
172	C₂³×C₄	C₂×C₄×C₁₉	1	8	yes	—
173	C₂³×C₄	D₄×C₁₉	1	4	yes	—
174	C₂³×C₄	D₄×C₁₉	1	8	yes	—
175	C₂³×C₄	Q₈×C₁₉	1	4	yes	—
176	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
178	C₂×M₁₆	—	2	64	no	—
179	C₂×M₁₆	—	2	8	no	—
180	C₂×M₁₆	—	2	8	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂×M₁₆	—	2	8	no	—
183	C₂×M₁₆	C₂×C₄×C₁₉	1	16	yes	—
184	C₂×M₁₆	C₂×C₄×C₁₉	1	16	yes	complemented
185	C₂²×C₈	—	2	8	no	—
186	C₂²×C₈	—	2	8	no	—
187	C₂²×C₈	C₂×C₄×C₁₉	1	16	yes	complemented
188	C₂²×C₈	C₂³×C₁₉	1	8	yes	—
189	#117	—	2	16	no	—
190	#117	—	2	8	no	—
191	#117	—	2	8	no	—
192	C₂×C₄⋊C₄	—	2	8	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
194	C₂×C₂²⋊C₄	—	2	8	no	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
196	C₂×C₄²	—	2	8	no	—
197	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
198	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
199	C₁₉	#20335	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	—	4	4	no	—
201	C₂⁴	C₄²×C₁₉	1	4	yes	—
202	C₂⁴	C₂²⋊C₄×C₁₉	1	4	yes	—
203	C₂⁴	C₄⋊C₄×C₁₉	1	4	yes	—
204	C₂⁴	C₂²×C₄×C₁₉	1	2	yes	—
205	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	C₄²×C₁₉	1	4	yes	—
209	C₂²×C₄	C₄²×C₁₉	1	8	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₁₉	1	32	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
213	C₂²×C₄	C₄⋊C₄×C₁₉	1	32	yes	—
214	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
215	C₂²×C₄	C₄⋊C₄×C₁₉	1	8	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
222	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
228	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
230	M₁₆	—	2	32	no	—
231	M₁₆	—	2	32	no	—
232	C₂×C₈	—	2	32	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	C₂²×C₄×C₁₉	1	16	yes	—
236	C₄⋊C₄	—	2	16	no	—
237	C₂²⋊C₄	—	2	16	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	8	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	4	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	8	no	—
245	C₂³	C₁₉×#95	1	16	yes	—
246	C₂³	C₂×C₄²×C₁₉	1	4	yes	—
247	C₂³	C₂×C₄²×C₁₉	1	2	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
249	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
250	C₂³	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
251	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
252	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	16	no	—
255	C₂×C₄	—	4	16	no	—
256	C₂×C₄	C₁₉×#95	1	16	yes	—
257	C₂×C₄	C₁₉×#95	1	32	yes	—
258	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
259	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
267	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
270	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
271	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
273	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
274	C₈	—	2	16	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	4	16	no	—
282	C₂²	C₂×C₁₉×#95	1	8	yes	—
283	C₂²	C₂×C₁₉×#95	1	4	yes	—
284	C₂²	C₂²×C₄²×C₁₉	1	1	yes	characteristic
285	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
286	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
287	C₂²	C₁₉×#568	1	4	yes	—
288	C₄	—	4	8	no	—
289	C₄	C₂×C₁₉×#95	1	4	yes	—
290	C₄	C₂×C₁₉×#95	1	8	yes	—
291	C₄	C₂²×C₂²⋊C₄×C₁₉	1	2	yes	—
292	C₄	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
293	C₂	—	2	4	no	—
294	C₂	—	2	8	no	—
295	C₂	C₂²×C₁₉×#95	1	1	yes	characteristic
296	C₂	C₁₉×#2873	1	2	yes	—
297	C₂	C₁₉×#2879	1	2	yes	—
298	C₂	C₁₉×#2905	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂³×C₄
prime radical	C₂⁵×C₁₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1729	C₂	semidirect
4. C₁₉×#1728	C₂	semidirect
8. C₂²×M₁₆×C₁₉	C₄	semidirect
18. C₁₉×#336	C₂²	semidirect
35. C₂×M₁₆×C₁₉	C₂×C₄	semidirect
38. C₂²×C₈×C₁₉	C₂×C₄	semidirect
90. #20335	C₁₉	direct (N × H)
128. #1729	C₂×C₁₉	semidirect
129. #1728	C₂×C₁₉	semidirect
151. C₂²×M₁₆	C₄×C₁₉	semidirect
161. #336	C₂²×C₁₉	semidirect
184. C₂×M₁₆	C₂×C₄×C₁₉	semidirect
187. C₂²×C₈	C₂×C₄×C₁₉	semidirect
199. C₁₉	#20335	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₁₉	2
C₂²×C₁₉×#117	2
C₁₉×#1729	2
C₁₉×#1728	2
#20335	19

Subgroups — C19×#20335 #18026294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20335 #18026294