Subgroups — C₁₉×#23171 #18029130

Order 4864.

626 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3160	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2962	C₂	1	2	yes	maximal
3	C₁₉×#1986	C₂	1	2	yes	maximal
4	C₁₉×#1946	C₂	1	2	yes	maximal
5	C₁₉×#1922	C₂	1	4	yes	maximal
6	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1773	C₂	1	2	yes	maximal
8	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#533	C₂²	1	2	yes	—
10	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
11	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
12	C₁₉×#499	C₄	1	2	yes	complemented
13	C₁₉×#499	C₄	1	2	yes	complemented
14	C₁₉×#498	C₄	1	2	yes	complemented
15	C₁₉×#446	—	2	2	no	—
16	C₁₉×#446	—	2	4	no	—
17	C₈×Q₈×C₁₉	—	2	2	no	—
18	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
19	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
20	C₁₉×#427	C₂²	1	2	yes	—
21	C₁₉×#427	C₂²	1	2	yes	—
22	C₁₉×#424	C₂²	1	4	yes	—
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₉×#405	C₂²	1	4	yes	—
25	C₄×M₁₆×C₁₉	C₂²	1	2	yes	—
26	C₂×C₁₉×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#341	C₂²	1	2	yes	—
30	C₁₉×#339	C₂²	1	4	yes	—
31	C₁₉×#339	C₂²	1	4	yes	—
32	C₁₉×#336	C₂²	1	2	yes	—
33	C₂×M₁₆×C₁₉	C₂³	1	2	yes	—
34	C₂×M₁₆×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₁₉×#125	C₂×C₄	1	4	yes	—
41	C₁₉×C₂²⋊Q₈	—	2	2	no	—
42	C₁₉×C₂²⋊Q₈	—	2	2	no	—
43	C₄×Q₈×C₁₉	—	2	2	no	—
44	C₄×Q₈×C₁₉	—	2	2	no	—
45	C₁₉×#117	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₁₉×#108	—	2	2	no	—
50	C₁₉×#108	Q₈	1	4	yes	complemented
51	C₁₉×#107	C₂×C₄	1	2	yes	—
52	C₁₉×#107	C₂×C₄	1	2	yes	—
53	C₁₉×#107	C₂×C₄	1	2	yes	—
54	C₁₉×#107	C₂×C₄	1	2	yes	—
55	C₁₉×#106	C₂×C₄	1	4	yes	—
56	C₁₉×#106	C₂×C₄	1	4	yes	—
57	C₁₉×C₄⋊C₈	—	2	4	no	—
58	C₁₉×C₄⋊C₈	—	2	4	no	—
59	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
60	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
61	C₁₉×#97	—	2	4	no	—
62	C₁₉×#97	—	2	4	no	—
63	C₁₉×#97	—	2	2	no	—
64	C₄×C₈×C₁₉	—	2	4	no	—
65	C₄×C₈×C₁₉	—	2	2	no	—
66	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
70	C₁₉×#95	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₁₉	—	2	2	no	—
78	M₁₆×C₁₉	—	2	2	no	—
79	M₁₆×C₁₉	—	2	2	no	—
80	M₁₆×C₁₉	C₂×Q₈	1	4	yes	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₉	#41	1	2	yes	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	4	2	no	—
99	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
100	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄×C₁₉	—	2	4	no	—
102	C₄²×C₁₉	—	2	4	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	#23171	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
111	Q₈×C₁₉	—	4	4	no	—
112	Q₈×C₁₉	—	8	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
130	C₈×C₁₉	—	2	1	no	—
131	C₈×C₁₉	—	2	1	no	—
132	C₈×C₁₉	—	2	4	no	—
133	C₈×C₁₉	—	2	4	no	—
134	C₈×C₁₉	—	4	2	no	—
135	C₈×C₁₉	C₄×Q₈	1	2	yes	—
136	C₈×C₁₉	C₄×Q₈	1	2	yes	—
137	#3160	C₂×C₁₉	1	1	yes	characteristic
138	#2962	C₂×C₁₉	1	2	yes	—
139	#1986	C₂×C₁₉	1	2	yes	—
140	#1946	C₂×C₁₉	1	2	yes	—
141	#1922	C₂×C₁₉	1	4	yes	—
142	C₄×#108	C₂×C₁₉	1	1	yes	characteristic
143	#1773	C₂×C₁₉	1	2	yes	—
144	#1770	C₂×C₁₉	1	1	yes	characteristic
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₁₉	#556	1	1	yes	characteristic
149	C₂²×C₁₉	#557	1	1	yes	characteristic
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	4	4	no	—
154	C₄×C₁₉	—	4	4	no	—
155	C₄×C₁₉	—	8	2	no	—
156	C₄×C₁₉	#444	1	2	yes	—
157	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₁₉	#519	1	1	yes	characteristic
159	C₄×C₁₉	#547	1	1	yes	characteristic
160	C₄×C₁₉	D₄×Q₈	1	1	yes	characteristic
161	#533	C₂²×C₁₉	1	2	yes	—
162	#500	C₄×C₁₉	1	1	yes	characteristic, complemented
163	#500	C₄×C₁₉	1	1	yes	characteristic, complemented
164	#499	C₄×C₁₉	1	2	yes	complemented
165	#499	C₄×C₁₉	1	2	yes	complemented
166	#498	C₄×C₁₉	1	2	yes	complemented
167	#446	—	2	4	no	—
168	#446	—	2	2	no	—
169	C₈×Q₈	—	2	2	no	—
170	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
171	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
172	#427	C₂²×C₁₉	1	2	yes	—
173	#427	C₂²×C₁₉	1	2	yes	—
174	#424	C₂²×C₁₉	1	4	yes	—
175	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
176	#405	C₂²×C₁₉	1	4	yes	—
177	C₄×M₁₆	C₂²×C₁₉	1	2	yes	—
178	C₂×#97	C₂²×C₁₉	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
180	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
181	#341	C₂²×C₁₉	1	2	yes	—
182	#339	C₂²×C₁₉	1	4	yes	—
183	#339	C₂²×C₁₉	1	4	yes	—
184	#336	C₂²×C₁₉	1	2	yes	—
185	C₂×C₁₉	—	2	1	no	—
186	C₂×C₁₉	—	2	1	no	—
187	C₂×C₁₉	#2365	1	1	yes	characteristic
188	C₂×C₁₉	C₂×#444	1	1	yes	characteristic
189	C₂×C₁₉	#2957	1	1	yes	characteristic
190	C₂×M₁₆	C₂³×C₁₉	1	2	yes	—
191	C₂×M₁₆	C₂³×C₁₉	1	2	yes	—
192	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
194	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
195	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
196	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
197	#125	C₂×C₄×C₁₉	1	4	yes	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₂²⋊Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	#117	C₂³×C₁₉	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
206	#108	—	2	2	no	—
207	#108	Q₈×C₁₉	1	4	yes	complemented
208	#107	C₂×C₄×C₁₉	1	2	yes	—
209	#107	C₂×C₄×C₁₉	1	2	yes	—
210	#107	C₂×C₄×C₁₉	1	2	yes	—
211	#107	C₂×C₄×C₁₉	1	2	yes	—
212	#106	C₂×C₄×C₁₉	1	4	yes	—
213	#106	C₂×C₄×C₁₉	1	4	yes	—
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
217	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
218	#97	—	2	4	no	—
219	#97	—	2	4	no	—
220	#97	—	2	2	no	—
221	C₄×C₈	—	2	4	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
224	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
225	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
226	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
227	#95	—	2	1	no	—
228	C₁₉	#23171	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
233	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	C₂×Q₈×C₁₉	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
250	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
251	C₂×C₈	#41×C₁₉	1	2	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
258	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
264	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
268	Q₈	—	4	4	no	—
269	Q₈	—	8	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
278	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
280	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
281	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
285	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
286	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	2	4	no	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	C₄×Q₈×C₁₉	1	2	yes	—
293	C₈	C₄×Q₈×C₁₉	1	2	yes	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
297	C₂²	C₁₉×#556	1	1	yes	characteristic
298	C₂²	C₁₉×#557	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	8	2	no	—
305	C₄	C₁₉×#444	1	2	yes	—
306	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
307	C₄	C₁₉×#519	1	1	yes	characteristic
308	C₄	C₁₉×#547	1	1	yes	characteristic
309	C₄	D₄×Q₈×C₁₉	1	1	yes	characteristic
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₁₉×#2365	1	1	yes	characteristic
313	C₂	C₂×C₁₉×#444	1	1	yes	characteristic
314	C₂	C₁₉×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×C₄×D₄
prime radical	C₂³×C₁₉	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₁₉×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₁₉×#500	C₄	semidirect
11. C₁₉×#500	C₄	semidirect
12. C₁₉×#499	C₄	semidirect
13. C₁₉×#499	C₄	semidirect
14. C₁₉×#498	C₄	semidirect
50. C₁₉×#108	Q₈	semidirect
109. #23171	C₁₉	direct (N × H)
162. #500	C₄×C₁₉	semidirect
163. #500	C₄×C₁₉	semidirect
164. #499	C₄×C₁₉	semidirect
165. #499	C₄×C₁₉	semidirect
166. #498	C₄×C₁₉	semidirect
207. #108	Q₈×C₁₉	semidirect
228. C₁₉	#23171	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3160	2
C₁₉×#2962	2
C₁₉×#1986	2
C₁₉×#1946	2
C₁₉×#1922	2
C₄×C₁₉×#108	2
C₁₉×#1773	2
C₁₉×#1770	2
#23171	19

Subgroups — C19×#23171 #18029130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23171 #18029130