Subgroups — C₂×C₁₉×#1340 #18017643

Order 4864.

444 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#349	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1340	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₁₉	C₄	1	1	yes	characteristic, complemented
6	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
8	C₂×C₁₉×#117	—	2	1	no	—
9	C₂×C₁₉×M₃₂	—	2	1	no	—
10	C₁₉×#470	—	2	1	no	—
11	C₁₉×#470	—	2	1	no	—
12	C₁₉×#470	C₄	1	2	yes	complemented
13	C₁₉×#470	C₂²	1	2	yes	—
14	C₂×C₁₉×#106	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₉×#103	—	2	1	no	—
16	C₁₉×#349	—	2	1	no	—
17	C₁₉×#349	C₂²	1	2	yes	—
18	C₁₉×H(3,ℤ/4)	—	2	1	no	—
19	C₁₉×H(3,ℤ/4)	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₁₉	—	2	1	no	—
23	C₂×Q₁₆×C₁₉	—	4	1	no	—
24	C₂×Q₁₆×C₁₉	—	4	1	no	—
25	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
26	C₂×M₁₆×C₁₉	—	4	1	no	—
27	C₂×M₁₆×C₁₉	—	4	1	no	—
28	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₁₉×C₂²⋊Q₈	—	2	2	no	—
35	C₁₉×C₂²⋊Q₈	—	2	2	no	—
36	C₁₉×C₂²⋊Q₈	—	4	1	no	—
37	C₁₉×C₂²⋊Q₈	—	4	1	no	—
38	C₁₉×#117	—	2	2	no	—
39	C₁₉×#117	—	2	2	no	—
40	C₁₉×#117	—	4	1	no	—
41	C₁₉×#117	—	4	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
45	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
46	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
47	C₂×C₄²×C₁₉	—	2	2	no	—
48	C₁₉×M₃₂	—	2	2	no	—
49	C₁₉×M₃₂	—	4	1	no	—
50	C₂×C₁₆×C₁₉	—	4	1	no	—
51	C₁₉×#106	—	2	1	no	—
52	C₁₉×#106	C₂×C₄	1	2	yes	complemented
53	C₁₉×#103	—	2	2	no	—
54	C₁₉×#103	—	2	2	no	—
55	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂×Q₈×C₁₉	—	4	1	no	—
58	C₂×Q₈×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×Q₈×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	4	2	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₁₉	—	4	2	no	—
76	Q₁₆×C₁₉	—	4	2	no	—
77	M₁₆×C₁₉	—	4	1	no	—
78	M₁₆×C₁₉	—	4	1	no	—
79	M₁₆×C₁₉	—	4	1	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	4	2	no	—
94	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₉	—	4	2	no	—
96	C₂²⋊C₄×C₁₉	—	4	2	no	—
97	C₂²⋊C₄×C₁₉	—	4	2	no	—
98	C₂²⋊C₄×C₁₉	—	4	2	no	—
99	C₄²×C₁₉	—	2	4	no	—
100	C₄²×C₁₉	—	4	2	no	—
101	C₁₆×C₁₉	—	4	2	no	—
102	C₂×#1340	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	4	2	no	—
106	C₂³×C₁₉	#102	1	1	yes	characteristic
107	C₂³×C₁₉	#102	1	1	yes	characteristic
108	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	—	4	2	no	—
111	Q₈×C₁₉	—	8	1	no	—
112	Q₈×C₁₉	—	8	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	8	2	no	—
127	C₂×C₄×C₁₉	—	8	4	no	—
128	C₂×C₄×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	8	1	no	—
130	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
138	C₈×C₁₉	—	2	1	no	—
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₂×#470	C₂×C₁₉	1	1	yes	characteristic
143	C₂×#349	C₂×C₁₉	1	1	yes	characteristic
144	C₂×H(3,ℤ/4)	C₂×C₁₉	1	1	yes	characteristic
145	#1340	C₂×C₁₉	1	4	yes	direct product factor
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	2	no	—
152	C₂²×C₁₉	—	8	1	no	—
153	C₂²×C₁₉	#327	1	1	yes	characteristic
154	C₂²×C₁₉	#327	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	—	8	4	no	—
160	C₄×C₁₉	—	8	2	no	—
161	C₄×C₁₉	#327	1	1	yes	characteristic
162	C₄×C₁₉	#327	1	1	yes	characteristic
163	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
164	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
165	C₂²×Q₁₆	C₄×C₁₉	1	1	yes	characteristic, complemented
166	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	—	2	1	no	—
168	C₂×#117	—	2	1	no	—
169	C₂×M₃₂	—	2	1	no	—
170	#470	—	2	1	no	—
171	#470	—	2	1	no	—
172	#470	C₄×C₁₉	1	2	yes	complemented
173	#470	C₂²×C₁₉	1	2	yes	—
174	C₂×#106	C₄×C₁₉	1	1	yes	characteristic, complemented
175	C₂×#103	—	2	1	no	—
176	#349	—	2	1	no	—
177	#349	C₂²×C₁₉	1	2	yes	—
178	H(3,ℤ/4)	—	2	1	no	—
179	H(3,ℤ/4)	C₂²×C₁₉	1	2	yes	—
180	C₂×C₁₉	—	2	1	no	—
181	C₂×C₁₉	—	2	1	no	—
182	C₂×C₁₉	—	4	1	no	—
183	C₂×C₁₉	—	4	1	no	—
184	C₂×C₁₉	#1340	1	2	yes	direct product factor
185	C₂×C₁₉	C₂×#327	1	1	yes	characteristic
186	C₂²×Q₈	—	2	1	no	—
187	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	4	1	no	—
190	C₂×Q₁₆	—	4	1	no	—
191	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
192	C₂×M₁₆	—	4	1	no	—
193	C₂×M₁₆	—	4	1	no	—
194	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
196	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
197	C₂×M₁₆	C₂³×C₁₉	1	1	yes	characteristic
198	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
199	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	4	1	no	—
203	C₂²⋊Q₈	—	4	1	no	—
204	#117	—	2	2	no	—
205	#117	—	2	2	no	—
206	#117	—	4	1	no	—
207	#117	—	4	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₄⋊C₄	—	4	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	4	1	no	—
213	C₂×C₄²	—	2	2	no	—
214	M₃₂	—	2	2	no	—
215	M₃₂	—	4	1	no	—
216	C₂×C₁₆	—	4	1	no	—
217	#106	—	2	1	no	—
218	#106	C₂×C₄×C₁₉	1	2	yes	complemented
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	C₁₉	C₂×#1340	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
238	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
239	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
242	Q₁₆	—	4	2	no	—
243	Q₁₆	—	4	2	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	4	2	no	—
268	C₁₆	—	4	2	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₁₉×#102	1	1	yes	characteristic
273	C₂³	C₁₉×#102	1	1	yes	characteristic
274	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	1	no	—
278	Q₈	—	8	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
297	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
298	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
299	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
300	C₂×C₄	C₁₉×#104	1	1	yes	characteristic
301	C₂×C₄	C₁₉×#104	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
303	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	C₁₉×#327	1	1	yes	characteristic
316	C₂²	C₁₉×#327	1	1	yes	characteristic
317	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	4	no	—
322	C₄	—	8	2	no	—
323	C₄	C₁₉×#327	1	1	yes	characteristic
324	C₄	C₁₉×#327	1	1	yes	characteristic
325	C₄	C₂×C₁₉×#99	1	1	yes	characteristic
326	C₄	C₂×C₁₉×#104	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	C₁₉×#1340	1	2	yes	direct product factor
332	C₂	C₂×C₁₉×#327	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#327
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1340	C₂	direct (N × H)
5. C₂²×Q₁₆×C₁₉	C₄	semidirect
12. C₁₉×#470	C₄	semidirect
14. C₂×C₁₉×#106	C₄	semidirect
25. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
52. C₁₉×#106	C₂×C₄	semidirect
102. C₂×#1340	C₁₉	direct (N × H)
145. #1340	C₂×C₁₉	direct (N × H)
165. C₂²×Q₁₆	C₄×C₁₉	semidirect
172. #470	C₄×C₁₉	semidirect
174. C₂×#106	C₄×C₁₉	semidirect
184. C₂×C₁₉	#1340	direct (N × H)
191. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
218. #106	C₂×C₄×C₁₉	semidirect
221. C₁₉	C₂×#1340	direct (N × H)
331. C₂	C₁₉×#1340	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×#470	2
C₂×C₁₉×#349	2
C₂×C₁₉×H(3,ℤ/4)	2
C₁₉×#1340	2
C₂×#1340	19

Subgroups — C2×C19×#1340 #18017643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1340 #18017643