Subgroups — C₁₉×#21978 #18027937

Order 4864.

870 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1940	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1851	C₂	1	4	yes	maximal
6	C₁₉×#1842	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1831	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
9	C₂×C₁₉×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#432	—	2	4	no	—
12	C₁₉×#423	—	2	4	no	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	8	yes	—
18	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₁₉×#97	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₁₉×#376	C₄	1	1	yes	characteristic
25	C₁₉×#376	C₄	1	1	yes	characteristic
26	C₄³×C₁₉	C₂²	1	2	yes	—
27	C₁₉×#336	C₂²	1	8	yes	—
28	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	—	2	4	no	—
32	C₂×M₁₆×C₁₉	—	4	4	no	—
33	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
36	C₁₉×#117	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₉	D₄	1	4	yes	—
45	C₂×C₄²×C₁₉	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₁₉×C₄⋊C₈	—	2	4	no	—
50	C₁₉×C₄⋊C₈	—	2	4	no	—
51	C₁₉×C₄⋊C₈	—	2	2	no	—
52	C₁₉×C₄⋊C₈	—	2	2	no	—
53	C₁₉×C₄⋊C₈	—	4	8	no	—
54	C₁₉×C₂²⋊C₈	—	2	4	no	—
55	C₁₉×#97	—	2	4	no	—
56	C₁₉×#95	C₂×C₄	1	2	yes	—
57	C₁₉×#95	C₂×C₄	1	2	yes	—
58	C₁₉×#95	C₂×C₄	1	2	yes	—
59	C₁₉×#95	C₂×C₄	1	2	yes	—
60	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₁₉	—	4	8	no	—
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	8	no	—
82	C₂×C₈×C₁₉	—	2	4	no	—
83	C₂×C₈×C₁₉	—	2	4	no	—
84	C₄⋊C₄×C₁₉	—	2	8	no	—
85	C₂²⋊C₄×C₁₉	—	2	8	no	—
86	C₄²×C₁₉	—	2	4	no	—
87	C₄²×C₁₉	—	2	4	no	—
88	C₄²×C₁₉	—	2	4	no	—
89	C₄²×C₁₉	—	4	4	no	—
90	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
91	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
92	C₄²×C₁₉	M₁₆	1	8	yes	complemented
93	C₄²×C₁₉	C₂×D₄	1	2	yes	—
94	C₄²×C₁₉	C₂×D₄	1	2	yes	—
95	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
96	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
97	#21978	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	—	2	2	no	—
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	2	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	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₂×M₁₆	1	8	yes	—
123	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
128	C₈×C₁₉	—	4	2	no	—
129	C₈×C₁₉	—	4	4	no	—
130	C₈×C₁₉	—	4	2	no	—
131	C₂×#432	C₂×C₁₉	1	1	yes	characteristic
132	C₂×#423	C₂×C₁₉	1	1	yes	characteristic
133	C₄×#117	C₂×C₁₉	1	1	yes	characteristic
134	#1940	C₂×C₁₉	1	4	yes	complemented
135	#1851	C₂×C₁₉	1	4	yes	—
136	#1842	C₂×C₁₉	1	2	yes	complemented
137	#1831	C₂×C₁₉	1	2	yes	complemented
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	4	4	no	—
142	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₉	#520	1	2	yes	—
144	C₂²×C₁₉	C₂²×M₁₆	1	2	yes	—
145	C₂²×C₁₉	#568	1	2	yes	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	2	8	no	—
151	C₄×C₁₉	—	4	4	no	—
152	C₄×C₁₉	#423	1	4	yes	—
153	C₄×C₁₉	#535	1	1	yes	characteristic
154	C₄×C₁₉	#536	1	1	yes	characteristic
155	C₄×C₁₉	#544	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
157	C₂×#117	C₄×C₁₉	1	2	yes	—
158	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
159	#432	—	2	4	no	—
160	#423	—	2	4	no	—
161	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₁₉	1	8	yes	—
166	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
167	C₂×#97	C₂²×C₁₉	1	2	yes	—
168	C₄×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
170	C₄×C₂²⋊C₄	C₄×C₁₉	1	2	yes	—
171	C₄×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
172	#376	C₄×C₁₉	1	1	yes	characteristic
173	#376	C₄×C₁₉	1	1	yes	characteristic
174	C₄³	C₂²×C₁₉	1	2	yes	—
175	#336	C₂²×C₁₉	1	8	yes	—
176	C₂×C₁₉	—	4	2	no	—
177	C₂×C₁₉	#2317	1	1	yes	characteristic
178	C₂×C₁₉	C₂×#423	1	1	yes	characteristic
179	C₂×C₁₉	#2905	1	1	yes	characteristic
180	C₂×C₁₉	#2989	1	4	yes	—
181	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
183	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	4	4	no	—
186	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
187	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
189	#117	—	2	8	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
192	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
193	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
194	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
197	C₂×C₄²	D₄×C₁₉	1	4	yes	—
198	C₂×C₄²	Q₈×C₁₉	1	4	yes	—
199	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
200	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	4	8	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	#97	—	2	4	no	—
209	#95	C₂×C₄×C₁₉	1	2	yes	—
210	#95	C₂×C₄×C₁₉	1	2	yes	—
211	#95	C₂×C₄×C₁₉	1	2	yes	—
212	#95	C₂×C₄×C₁₉	1	2	yes	—
213	C₁₉	#21978	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
220	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
229	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
231	M₁₆	—	4	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₄⋊C₄×C₁₉	1	4	yes	—
245	C₄²	C₄⋊C₄×C₁₉	1	4	yes	—
246	C₄²	M₁₆×C₁₉	1	8	yes	complemented
247	C₄²	C₂×D₄×C₁₉	1	2	yes	—
248	C₄²	C₂×D₄×C₁₉	1	2	yes	—
249	C₄²	C₂×Q₈×C₁₉	1	2	yes	—
250	C₄²	C₂×Q₈×C₁₉	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
255	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
275	C₂×C₄	C₂×M₁₆×C₁₉	1	8	yes	—
276	C₂×C₄	C₂×M₁₆×C₁₉	1	4	yes	—
277	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
279	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
280	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
281	C₈	—	4	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
289	C₂²	C₁₉×#520	1	2	yes	—
290	C₂²	C₂²×M₁₆×C₁₉	1	2	yes	—
291	C₂²	C₁₉×#568	1	2	yes	—
292	C₄	—	2	8	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	C₁₉×#423	1	4	yes	—
299	C₄	C₁₉×#535	1	1	yes	characteristic
300	C₄	C₁₉×#536	1	1	yes	characteristic
301	C₄	C₁₉×#544	1	2	yes	—
302	C₂	—	4	2	no	—
303	C₂	C₁₉×#2317	1	1	yes	characteristic
304	C₂	C₂×C₁₉×#423	1	1	yes	characteristic
305	C₂	C₁₉×#2905	1	1	yes	characteristic
306	C₂	C₁₉×#2989	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1940	C₂	semidirect
6. C₁₉×#1842	C₂	semidirect
7. C₁₉×#1831	C₂	semidirect
92. C₄²×C₁₉	M₁₆	semidirect
97. #21978	C₁₉	direct (N × H)
134. #1940	C₂×C₁₉	semidirect
136. #1842	C₂×C₁₉	semidirect
137. #1831	C₂×C₁₉	semidirect
213. C₁₉	#21978	direct (N × H)
246. C₄²	M₁₆×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×#432	2
C₂×C₁₉×#423	2
C₄×C₁₉×#117	2
C₁₉×#1940	2
C₁₉×#1851	2
C₁₉×#1842	2
C₁₉×#1831	2
#21978	19

Subgroups — C19×#21978 #18027937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21978 #18027937