Subgroups — C₂×C₁₉×#1560 #18023694

Order 4864.

1322 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#423	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1560	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₉×#333	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#331	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₁₉	C₄	1	2	yes	—
7	C₂²×M₁₆×C₁₉	—	2	2	no	—
8	C₁₉×#532	—	2	4	no	—
9	C₁₉×#532	C₄	1	4	yes	—
10	C₁₉×#532	C₂²	1	4	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#124	C₄	1	2	yes	—
14	C₂×C₁₉×#121	—	2	2	no	—
15	C₂×C₄×D₄×C₁₉	—	2	2	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#423	—	2	4	no	—
18	C₁₉×#423	C₂²	1	8	yes	complemented
19	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
21	C₁₉×#333	C₂²	1	8	yes	complemented
22	C₁₉×#331	C₂²	1	8	yes	complemented
23	C₂×#41×C₁₉	—	2	4	no	—
24	C₂×#41×C₁₉	—	4	8	no	—
25	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₁₉	—	2	2	no	—
28	C₂²×D₄×C₁₉	—	2	2	no	—
29	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₉	—	2	2	no	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₉	D₄	1	2	yes	—
33	C₂×M₁₆×C₁₉	—	2	8	no	—
34	C₂×M₁₆×C₁₉	—	4	8	no	—
35	C₂²×C₈×C₁₉	—	2	4	no	—
36	C₁₉×C₄⋊Q₈	—	2	2	no	—
37	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₁₉×#127	—	2	2	no	—
40	C₁₉×#127	C₂×C₄	1	2	yes	—
41	C₁₉×#127	C₂³	1	2	yes	—
42	C₁₉×#124	—	2	4	no	—
43	C₁₉×#124	C₂×C₄	1	8	yes	—
44	C₁₉×#121	—	2	8	no	—
45	C₁₉×#121	—	2	8	no	—
46	C₄×D₄×C₁₉	—	2	8	no	—
47	C₄×D₄×C₁₉	—	2	8	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
50	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₉	D₄	1	2	yes	—
53	C₂×C₄²×C₁₉	D₄	1	4	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
56	C₁₉×C₄⋊C₈	—	2	8	no	—
57	C₁₉×C₄⋊C₈	—	2	4	no	—
58	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
59	C₂⁴×C₁₉	—	2	2	no	—
60	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	#41×C₁₉	—	4	16	no	—
62	#41×C₁₉	—	4	16	no	—
63	C₂×Q₈×C₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	—	4	4	no	—
65	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
66	C₂×D₄×C₁₉	—	2	8	no	—
67	C₂×D₄×C₁₉	—	2	4	no	—
68	C₂×D₄×C₁₉	—	2	2	no	—
69	C₂×D₄×C₁₉	—	2	4	no	—
70	C₂×D₄×C₁₉	—	4	4	no	—
71	C₂×D₄×C₁₉	—	4	8	no	—
72	C₂×D₄×C₁₉	—	4	4	no	—
73	C₂×D₄×C₁₉	—	4	4	no	—
74	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₉	—	2	4	no	—
76	C₂²×C₄×C₁₉	—	2	4	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	2	8	no	—
80	C₂²×C₄×C₁₉	—	4	8	no	—
81	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
89	M₁₆×C₁₉	—	4	16	no	—
90	C₂×C₈×C₁₉	—	2	8	no	—
91	C₂×C₈×C₁₉	—	4	8	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	4	no	—
94	C₂²⋊C₄×C₁₉	—	2	16	no	—
95	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
96	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	4	yes	—
100	C₄²×C₁₉	C₂×D₄	1	2	yes	—
101	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
102	C₂×#1560	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	—	2	4	no	—
105	C₂³×C₁₉	—	2	4	no	—
106	C₂³×C₁₉	—	4	4	no	—
107	C₂³×C₁₉	—	4	8	no	—
108	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	Q₈×C₁₉	—	4	4	no	—
111	Q₈×C₁₉	—	4	4	no	—
112	D₄×C₁₉	—	4	4	no	—
113	D₄×C₁₉	—	4	4	no	—
114	D₄×C₁₉	—	4	16	no	—
115	D₄×C₁₉	—	4	4	no	—
116	D₄×C₁₉	—	4	4	no	—
117	D₄×C₁₉	—	4	8	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	8	no	—
125	C₂×C₄×C₁₉	—	2	4	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	16	no	—
128	C₂×C₄×C₁₉	#100	1	4	yes	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₁₉	#137	1	4	yes	—
135	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
137	C₈×C₁₉	—	4	8	no	—
138	C₂×#532	C₂×C₁₉	1	1	yes	characteristic
139	C₂×#423	C₂×C₁₉	1	2	yes	complemented
140	#1560	C₂×C₁₉	1	8	yes	direct product factor
141	C₂×#333	C₂×C₁₉	1	2	yes	complemented
142	C₂×#331	C₂×C₁₉	1	2	yes	complemented
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	4	no	—
145	C₂²×C₁₉	—	4	4	no	—
146	C₂²×C₁₉	—	4	4	no	—
147	C₂²×C₁₉	—	4	8	no	—
148	C₂²×C₁₉	—	8	8	no	—
149	C₂²×C₁₉	C₂×#100	1	2	yes	—
150	C₂²×C₁₉	#417	1	4	yes	—
151	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	4	no	—
155	C₄×C₁₉	—	4	4	no	—
156	C₄×C₁₉	C₂×#100	1	4	yes	—
157	C₄×C₁₉	#417	1	4	yes	—
158	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₄×C₁₉	C₂×#137	1	2	yes	—
160	C₂²×#41	C₄×C₁₉	1	2	yes	—
161	C₂²×M₁₆	—	2	2	no	—
162	#532	—	2	4	no	—
163	#532	C₄×C₁₉	1	4	yes	—
164	#532	C₂²×C₁₉	1	4	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
167	C₂×#124	C₄×C₁₉	1	2	yes	—
168	C₂×#121	—	2	2	no	—
169	C₂×C₄×D₄	—	2	2	no	—
170	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
171	#423	—	2	4	no	—
172	#423	C₂²×C₁₉	1	8	yes	complemented
173	C₂×C₄⋊C₈	—	2	2	no	—
174	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
175	#333	C₂²×C₁₉	1	8	yes	complemented
176	#331	C₂²×C₁₉	1	8	yes	complemented
177	C₂×C₁₉	—	4	2	no	—
178	C₂×C₁₉	—	8	4	no	—
179	C₂×C₁₉	#1560	1	4	yes	direct product factor
180	C₂×C₁₉	C₂²×#100	1	1	yes	characteristic
181	C₂×C₁₉	C₂×#417	1	2	yes	—
182	C₂×#41	—	2	4	no	—
183	C₂×#41	—	4	8	no	—
184	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
185	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
186	C₂²×D₄	—	2	2	no	—
187	C₂²×D₄	—	2	2	no	—
188	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
189	C₂³×C₄	—	2	2	no	—
190	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₁₉	1	2	yes	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	—	4	8	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
197	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
198	#127	—	2	2	no	—
199	#127	C₂×C₄×C₁₉	1	2	yes	—
200	#127	C₂³×C₁₉	1	2	yes	—
201	#124	—	2	4	no	—
202	#124	C₂×C₄×C₁₉	1	8	yes	—
203	#121	—	2	8	no	—
204	#121	—	2	8	no	—
205	C₄×D₄	—	2	8	no	—
206	C₄×D₄	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
210	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
211	C₂×C₄²	D₄×C₁₉	1	2	yes	—
212	C₂×C₄²	D₄×C₁₉	1	4	yes	—
213	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
215	C₄⋊C₈	—	2	8	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
218	C₁₉	C₂×#1560	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	—	2	2	no	—
220	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
221	#41	—	4	16	no	—
222	#41	—	4	16	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
226	C₂×D₄	—	2	8	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	4	8	no	—
241	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
242	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
244	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
248	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
249	M₁₆	—	4	16	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	—	4	8	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	2	16	no	—
255	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
256	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
257	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
258	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₁₉	1	4	yes	—
260	C₄²	C₂×D₄×C₁₉	1	2	yes	—
261	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	8	no	—
267	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
268	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	D₄	—	4	4	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	16	no	—
274	D₄	—	4	8	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	16	no	—
287	C₂×C₄	C₁₉×#100	1	4	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
292	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
293	C₂×C₄	C₁₉×#137	1	4	yes	—
294	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
296	C₈	—	4	8	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	8	8	no	—
303	C₂²	C₂×C₁₉×#100	1	2	yes	—
304	C₂²	C₁₉×#417	1	4	yes	—
305	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₄	C₂×C₁₉×#100	1	4	yes	—
311	C₄	C₁₉×#417	1	4	yes	—
312	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
313	C₄	C₂×C₁₉×#137	1	2	yes	—
314	C₂	—	4	2	no	—
315	C₂	—	8	4	no	—
316	C₂	C₁₉×#1560	1	4	yes	direct product factor
317	C₂	C₂²×C₁₉×#100	1	1	yes	characteristic
318	C₂	C₂×C₁₉×#417	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#423	C₂	semidirect
3. C₁₉×#1560	C₂	direct (N × H)
4. C₂×C₁₉×#333	C₂	semidirect
5. C₂×C₁₉×#331	C₂	semidirect
18. C₁₉×#423	C₂²	semidirect
21. C₁₉×#333	C₂²	semidirect
22. C₁₉×#331	C₂²	semidirect
102. C₂×#1560	C₁₉	direct (N × H)
139. C₂×#423	C₂×C₁₉	semidirect
140. #1560	C₂×C₁₉	direct (N × H)
141. C₂×#333	C₂×C₁₉	semidirect
142. C₂×#331	C₂×C₁₉	semidirect
172. #423	C₂²×C₁₉	semidirect
175. #333	C₂²×C₁₉	semidirect
176. #331	C₂²×C₁₉	semidirect
179. C₂×C₁₉	#1560	direct (N × H)
218. C₁₉	C₂×#1560	direct (N × H)
316. C₂	C₁₉×#1560	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#532	2
C₂×C₁₉×#423	2
C₁₉×#1560	2
C₂×C₁₉×#333	2
C₂×C₁₉×#331	2
C₂×#1560	19

Subgroups — C2×C19×#1560 #18023694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1560 #18023694