Subgroups — C₁₉×#23658 #18029617

Order 4864.

990 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	C₂×C₁₉×#428	C₂	1	2	yes	maximal
2	C₁₉×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2084	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2083	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1736	C₂	1	2	yes	maximal
7	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
9	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#428	—	2	8	no	—
15	C₂×C₁₉×#107	C₂²	1	4	yes	—
16	C₂×C₁₉×#106	C₂²	1	4	yes	—
17	C₁₉×#389	C₂²	1	2	yes	—
18	C₁₉×#389	C₂²	1	2	yes	—
19	C₁₉×#385	—	2	2	no	—
20	C₁₉×#384	—	2	1	no	—
21	C₁₉×#384	—	2	1	no	—
22	C₁₉×#384	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
24	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₁₉×#340	C₂²	1	8	yes	—
26	C₁₉×#336	C₂²	1	4	yes	—
27	C₂³×C₄×C₁₉	—	2	2	no	—
28	C₂³×C₄×C₁₉	D₄	1	2	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	—	2	4	no	—
31	C₂×M₁₆×C₁₉	—	2	4	no	—
32	C₂×M₁₆×C₁₉	Q₈	1	8	yes	—
33	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
34	C₁₉×#117	—	2	2	no	—
35	C₁₉×#117	—	2	4	no	—
36	C₁₉×#117	D₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	—	2	4	no	—
53	C₂×C₄²×C₁₉	—	2	1	no	—
54	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
57	C₁₉×#107	—	2	8	no	—
58	C₁₉×#106	—	2	8	no	—
59	C₁₉×#95	—	2	2	no	—
60	C₁₉×#95	—	2	2	no	—
61	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	2	4	no	—
69	C₂²×C₄×C₁₉	—	4	8	no	—
70	C₂²×C₄×C₁₉	—	4	4	no	—
71	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×C₄×C₁₉	#41	1	4	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₁₉	—	2	16	no	—
80	C₂×C₈×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	—	2	4	no	—
88	C₄⋊C₄×C₁₉	—	4	8	no	—
89	C₄⋊C₄×C₁₉	—	4	4	no	—
90	C₄⋊C₄×C₁₉	—	4	4	no	—
91	C₄⋊C₄×C₁₉	—	4	8	no	—
92	C₄⋊C₄×C₁₉	—	4	4	no	—
93	C₄⋊C₄×C₁₉	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₁₉	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄×C₁₉	—	2	4	no	—
96	C₂²⋊C₄×C₁₉	—	4	2	no	—
97	C₄²×C₁₉	—	2	4	no	—
98	C₄²×C₁₉	—	2	2	no	—
99	C₄²×C₁₉	—	2	2	no	—
100	C₄²×C₁₉	—	2	4	no	—
101	C₄²×C₁₉	—	4	2	no	—
102	#23658	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	—	2	2	no	—
105	C₂³×C₁₉	#121	1	2	yes	—
106	C₂³×C₁₉	#123	1	2	yes	—
107	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	—	2	16	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	8	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	8	no	—
124	C₂×C₄×C₁₉	#121	1	2	yes	—
125	C₂×C₄×C₁₉	C₂²⋊Q₈	1	8	yes	—
126	C₂×C₄×C₁₉	#123	1	2	yes	—
127	C₂×C₄×C₁₉	#125	1	8	yes	—
128	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₁₉	#137	1	4	yes	—
131	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
135	C₈×C₁₉	—	2	8	no	—
136	C₂×#428	C₂×C₁₉	1	2	yes	—
137	#2296	C₂×C₁₉	1	1	yes	characteristic
138	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
139	#2084	C₂×C₁₉	1	4	yes	complemented
140	#2083	C₂×C₁₉	1	4	yes	complemented
141	#1736	C₂×C₁₉	1	2	yes	—
142	#1728	C₂×C₁₉	1	1	yes	characteristic
143	C₂²×C₁₉	—	2	4	no	—
144	C₂²×C₁₉	—	2	4	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	#398	1	4	yes	—
148	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#123	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂²×C₁₉	C₂×#137	1	2	yes	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	2	4	no	—
155	C₄×C₁₉	—	4	2	no	—
156	C₄×C₁₉	—	4	2	no	—
157	C₄×C₁₉	—	8	2	no	—
158	C₄×C₁₉	—	8	4	no	—
159	C₄×C₁₉	#398	1	4	yes	—
160	C₄×C₁₉	C₂×C₂²⋊Q₈	1	2	yes	—
161	C₄×C₁₉	C₂×#125	1	2	yes	—
162	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
163	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	—	2	2	no	—
166	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
167	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
168	#428	—	2	8	no	—
169	C₂×#107	C₂²×C₁₉	1	4	yes	—
170	C₂×#106	C₂²×C₁₉	1	4	yes	—
171	#389	C₂²×C₁₉	1	2	yes	—
172	#389	C₂²×C₁₉	1	2	yes	—
173	#385	—	2	2	no	—
174	#384	—	2	1	no	—
175	#384	—	2	1	no	—
176	#384	C₂²×C₁₉	1	4	yes	—
177	C₄×C₄⋊C₄	—	2	1	no	—
178	C₄×C₄⋊C₄	—	2	1	no	—
179	#340	C₂²×C₁₉	1	8	yes	—
180	#336	C₂²×C₁₉	1	4	yes	—
181	C₂×C₁₉	—	2	4	no	—
182	C₂×C₁₉	C₂×#398	1	1	yes	characteristic
183	C₂×C₁₉	#3038	1	1	yes	characteristic
184	C₂×C₁₉	#3039	1	1	yes	characteristic
185	C₂×C₁₉	#3092	1	2	yes	—
186	C₂×C₁₉	#3165	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂³×C₄	D₄×C₁₉	1	2	yes	—
189	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	Q₈×C₁₉	1	8	yes	—
193	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	4	no	—
196	#117	D₄×C₁₉	1	4	yes	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	4	8	no	—
205	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
217	#107	—	2	8	no	—
218	#106	—	2	8	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	C₁₉	#23658	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	4	8	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
236	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
237	C₂²×C₄	#41×C₁₉	1	2	yes	—
238	C₂²×C₄	#41×C₁₉	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
240	M₁₆	—	2	16	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	8	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	C₂×D₄×C₁₉	1	2	yes	—
255	C₄⋊C₄	C₂×D₄×C₁₉	1	2	yes	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	4	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	C₁₉×#121	1	2	yes	—
266	C₂³	C₁₉×#123	1	2	yes	—
267	C₂³	C₁₉×C₄⋊Q₈	1	2	yes	—
268	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
269	C₂×C₄	—	2	16	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	8	no	—
284	C₂×C₄	C₁₉×#121	1	2	yes	—
285	C₂×C₄	C₁₉×C₂²⋊Q₈	1	8	yes	—
286	C₂×C₄	C₁₉×#123	1	2	yes	—
287	C₂×C₄	C₁₉×#125	1	8	yes	—
288	C₂×C₄	C₁₉×C₄⋊Q₈	1	2	yes	—
289	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
290	C₂×C₄	C₁₉×#137	1	4	yes	—
291	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
293	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
294	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	C₁₉×#398	1	4	yes	—
301	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
302	C₂²	C₂×C₁₉×#123	1	1	yes	characteristic
303	C₂²	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
304	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
305	C₂²	C₂×C₁₉×#137	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	2	4	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	4	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₉×#398	1	4	yes	—
313	C₄	C₂×C₁₉×C₂²⋊Q₈	1	2	yes	—
314	C₄	C₂×C₁₉×#125	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	C₂×C₁₉×#398	1	1	yes	characteristic
317	C₂	C₁₉×#3038	1	1	yes	characteristic
318	C₂	C₁₉×#3039	1	1	yes	characteristic
319	C₂	C₁₉×#3092	1	2	yes	—
320	C₂	C₁₉×#3165	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
4. C₁₉×#2084	C₂	semidirect
5. C₁₉×#2083	C₂	semidirect
102. #23658	C₁₉	direct (N × H)
139. #2084	C₂×C₁₉	semidirect
140. #2083	C₂×C₁₉	semidirect
221. C₁₉	#23658	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×#428	2
C₁₉×#2296	2
C₂×C₁₉×#384	2
C₁₉×#2084	2
C₁₉×#2083	2
C₁₉×#1736	2
C₁₉×#1728	2
#23658	19

Subgroups — C19×#23658 #18029617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23658 #18029617