Subgroups — C₂×C₁₉×#1810 #18026624

Order 4864.

1566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#108	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₉×#407	C₂	1	2	yes	maximal
4	C₁₉×#1810	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#341	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
8	C₂³×C₈×C₁₉	—	2	8	no	—
9	C₂³×C₈×C₁₉	C₄	1	4	yes	—
10	C₂³×C₈×C₁₉	C₄	1	4	yes	—
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₉	C₂²	1	4	yes	—
13	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#108	—	2	8	no	—
15	C₂×C₁₉×#108	C₂²	1	8	yes	complemented
16	C₁₉×#407	—	2	4	no	—
17	C₁₉×#407	C₂²	1	8	yes	—
18	C₂×C₁₉×C₂²⋊C₈	—	2	4	no	—
19	C₁₉×#341	C₂²	1	8	yes	complemented
20	C₂⁵×C₁₉	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	2	4	no	—
22	C₂³×C₄×C₁₉	—	2	4	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	D₄	1	2	yes	—
26	C₂³×C₄×C₁₉	D₄	1	2	yes	—
27	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	—	2	4	no	—
32	C₂×M₁₆×C₁₉	—	2	8	no	—
33	C₂×M₁₆×C₁₉	—	2	8	no	—
34	C₂×M₁₆×C₁₉	C₂³	1	8	yes	—
35	C₂²×C₈×C₁₉	—	2	4	no	—
36	C₂²×C₈×C₁₉	—	2	16	no	—
37	C₂²×C₈×C₁₉	—	2	16	no	—
38	C₂²×C₈×C₁₉	—	2	16	no	—
39	C₂²×C₈×C₁₉	—	2	4	no	—
40	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₉	C₂×C₄	1	16	yes	—
43	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₁₉	D₄	1	8	yes	—
46	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
48	C₁₉×#108	—	2	16	no	—
49	C₁₉×C₂²⋊C₈	—	2	16	no	—
50	C₂⁴×C₁₉	—	2	8	no	—
51	C₂⁴×C₁₉	—	2	4	no	—
52	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
53	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
54	C₂⁴×C₁₉	C₂×Q₈	1	2	yes	—
55	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	8	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	2	8	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	#41	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₁₉	—	2	8	no	—
82	M₁₆×C₁₉	—	2	8	no	—
83	M₁₆×C₁₉	—	4	8	no	—
84	C₂×C₈×C₁₉	—	2	4	no	—
85	C₂×C₈×C₁₉	—	2	4	no	—
86	C₂×C₈×C₁₉	—	2	4	no	—
87	C₂×C₈×C₁₉	—	2	32	no	—
88	C₂×C₈×C₁₉	—	2	8	no	—
89	C₂×C₈×C₁₉	—	4	8	no	—
90	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	—
91	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	—
92	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₉	C₂×D₄	1	8	yes	—
95	C₂×#1810	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	2	32	no	—
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	—	2	4	no	—
99	C₂³×C₁₉	—	2	4	no	—
100	C₂³×C₁₉	—	2	4	no	—
101	C₂³×C₁₉	—	2	4	no	—
102	C₂³×C₁₉	—	2	8	no	—
103	C₂³×C₁₉	—	2	8	no	—
104	C₂³×C₁₉	#108	1	8	yes	complemented
105	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₉	C₂²⋊Q₈	1	4	yes	—
109	C₂³×C₁₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	8	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	8	no	—
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#117	1	2	yes	—
124	C₂×C₄×C₁₉	#117	1	2	yes	—
125	C₂×C₄×C₁₉	#121	1	4	yes	—
126	C₂×C₄×C₁₉	#121	1	4	yes	—
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
132	C₈×C₁₉	—	2	4	no	—
133	C₈×C₁₉	—	4	8	no	—
134	C₈×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
135	C₂⁴×C₈	C₂×C₁₉	1	1	yes	characteristic
136	C₂²×#108	C₂×C₁₉	1	2	yes	complemented
137	C₂×#407	C₂×C₁₉	1	2	yes	—
138	#1810	C₂×C₁₉	1	8	yes	direct product factor
139	C₂×#341	C₂×C₁₉	1	2	yes	complemented
140	C₂²×C₁₉	—	2	16	no	—
141	C₂²×C₁₉	—	2	16	no	—
142	C₂²×C₁₉	—	2	16	no	—
143	C₂²×C₁₉	—	2	8	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	4	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	4	no	—
148	C₂²×C₁₉	#380	1	2	yes	—
149	C₂²×C₁₉	#380	1	2	yes	—
150	C₂²×C₁₉	C₂×#108	1	4	yes	—
151	C₂²×C₁₉	C₂×#108	1	8	yes	complemented
152	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	2	yes	—
154	C₄×C₁₉	—	2	4	no	—
155	C₄×C₁₉	#380	1	2	yes	—
156	C₄×C₁₉	#380	1	2	yes	—
157	C₄×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₁₉	C₂×#117	1	1	yes	characteristic
159	C₄×C₁₉	C₂×#121	1	2	yes	—
160	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
162	C₂³×C₈	—	2	8	no	—
163	C₂³×C₈	C₄×C₁₉	1	4	yes	—
164	C₂³×C₈	C₄×C₁₉	1	4	yes	—
165	C₂³×C₈	C₂²×C₁₉	1	4	yes	—
166	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
168	C₂×#108	—	2	8	no	—
169	C₂×#108	C₂²×C₁₉	1	8	yes	complemented
170	#407	—	2	4	no	—
171	#407	C₂²×C₁₉	1	8	yes	—
172	C₂×C₂²⋊C₈	—	2	4	no	—
173	#341	C₂²×C₁₉	1	8	yes	complemented
174	C₂×C₁₉	—	2	2	no	—
175	C₂×C₁₉	—	2	2	no	—
176	C₂×C₁₉	—	2	8	no	—
177	C₂×C₁₉	#1810	1	4	yes	direct product factor
178	C₂×C₁₉	C₂×#380	1	1	yes	characteristic
179	C₂×C₁₉	C₂²×#108	1	2	yes	—
180	C₂⁵	Q₈×C₁₉	1	1	yes	characteristic
181	C₂³×C₄	—	2	4	no	—
182	C₂³×C₄	—	2	4	no	—
183	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
185	C₂³×C₄	D₄×C₁₉	1	2	yes	—
186	C₂³×C₄	D₄×C₁₉	1	2	yes	—
187	C₂³×C₄	Q₈×C₁₉	1	1	yes	characteristic
188	C₂³×C₄	Q₈×C₁₉	1	2	yes	—
189	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
190	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	—	2	8	no	—
194	C₂×M₁₆	C₂³×C₁₉	1	8	yes	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	—	2	16	no	—
197	C₂²×C₈	—	2	16	no	—
198	C₂²×C₈	—	2	16	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
201	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
202	C₂²×C₈	C₂×C₄×C₁₉	1	16	yes	—
203	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
204	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
205	C₂²×C₈	D₄×C₁₉	1	8	yes	—
206	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
207	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
208	#108	—	2	16	no	—
209	C₂²⋊C₈	—	2	16	no	—
210	C₁₉	C₂×#1810	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	—	2	4	no	—
212	C₂⁴	—	2	8	no	—
213	C₂⁴	C₄⋊C₄×C₁₉	1	2	yes	—
214	C₂⁴	C₄⋊C₄×C₁₉	1	2	yes	—
215	C₂⁴	C₂×Q₈×C₁₉	1	2	yes	—
216	C₂⁴	C₂×Q₈×C₁₉	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
228	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
229	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
230	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
235	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
236	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
239	C₂²×C₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	#41×C₁₉	1	4	yes	—
241	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
242	M₁₆	—	2	8	no	—
243	M₁₆	—	2	8	no	—
244	M₁₆	—	4	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	32	no	—
249	C₂×C₈	—	2	8	no	—
250	C₂×C₈	—	4	8	no	—
251	C₂×C₈	C₂²⋊C₄×C₁₉	1	8	yes	—
252	C₂×C₈	C₂²⋊C₄×C₁₉	1	8	yes	—
253	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
254	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
255	C₂×C₈	C₂×D₄×C₁₉	1	8	yes	—
256	C₂³	—	2	32	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	8	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₁₉×#108	1	8	yes	complemented
265	C₂³	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
266	C₂³	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
267	C₂³	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
268	C₂³	C₁₉×C₂²⋊Q₈	1	4	yes	—
269	C₂³	C₂²×Q₈×C₁₉	1	1	yes	characteristic
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
279	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
283	C₂×C₄	C₁₉×#117	1	2	yes	—
284	C₂×C₄	C₁₉×#117	1	2	yes	—
285	C₂×C₄	C₁₉×#121	1	4	yes	—
286	C₂×C₄	C₁₉×#121	1	4	yes	—
287	C₂×C₄	C₁₉×C₂²⋊Q₈	1	4	yes	—
288	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
292	C₈	—	2	4	no	—
293	C₈	—	4	8	no	—
294	C₈	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
295	C₂²	—	2	16	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	2	16	no	—
303	C₂²	C₁₉×#380	1	2	yes	—
304	C₂²	C₁₉×#380	1	2	yes	—
305	C₂²	C₂×C₁₉×#108	1	4	yes	—
306	C₂²	C₂×C₁₉×#108	1	8	yes	complemented
307	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
308	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	C₁₉×#380	1	2	yes	—
311	C₄	C₁₉×#380	1	2	yes	—
312	C₄	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
313	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
314	C₄	C₂×C₁₉×#121	1	2	yes	—
315	C₂	—	2	2	no	—
316	C₂	—	2	8	no	—
317	C₂	—	2	2	no	—
318	C₂	C₁₉×#1810	1	4	yes	direct product factor
319	C₂	C₂×C₁₉×#380	1	1	yes	characteristic
320	C₂	C₂²×C₁₉×#108	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁵×C₁₉	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₂⁴×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₉×#108	C₂	semidirect
4. C₁₉×#1810	C₂	direct (N × H)
5. C₂×C₁₉×#341	C₂	semidirect
15. C₂×C₁₉×#108	C₂²	semidirect
19. C₁₉×#341	C₂²	semidirect
95. C₂×#1810	C₁₉	direct (N × H)
104. C₂³×C₁₉	#108	semidirect
136. C₂²×#108	C₂×C₁₉	semidirect
138. #1810	C₂×C₁₉	direct (N × H)
139. C₂×#341	C₂×C₁₉	semidirect
151. C₂²×C₁₉	C₂×#108	semidirect
169. C₂×#108	C₂²×C₁₉	semidirect
173. #341	C₂²×C₁₉	semidirect
177. C₂×C₁₉	#1810	direct (N × H)
210. C₁₉	C₂×#1810	direct (N × H)
264. C₂³	C₁₉×#108	semidirect
306. C₂²	C₂×C₁₉×#108	semidirect
318. C₂	C₁₉×#1810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₁₉	2
C₂²×C₁₉×#108	2
C₂×C₁₉×#407	2
C₁₉×#1810	2
C₂×C₁₉×#341	2
C₂×#1810	19

Subgroups — C2×C19×#1810 #18026624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1810 #18026624