Subgroups — C₁₉×#21942 #18027901

Order 4864.

1234 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#407	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#380	C₂	1	2	yes	maximal
3	C₁₉×#1853	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1852	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1809	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
10	C₂²×C₂²⋊C₄×C₁₉	—	2	2	no	—
11	C₂×C₁₉×#107	C₂²	1	4	yes	complemented
12	C₂×C₁₉×#106	C₂²	1	4	yes	complemented
13	C₁₉×#407	C₄	1	8	yes	complemented
14	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₁₉×#380	—	2	2	no	—
16	C₁₉×#380	—	2	2	no	—
17	C₁₉×#380	C₂²	1	4	yes	—
18	C₁₉×#380	C₂²	1	4	yes	—
19	C₂×C₁₉×#95	—	2	2	no	—
20	C₁₉×#340	C₂²	1	8	yes	complemented
21	C₂⁵×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	—	2	2	no	—
23	C₂³×C₄×C₁₉	—	2	2	no	—
24	C₂³×C₄×C₁₉	—	4	2	no	—
25	C₂³×C₄×C₁₉	—	4	2	no	—
26	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₁₉	D₄	1	2	yes	—
28	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	Q₈	1	4	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂×M₁₆×C₁₉	—	4	2	no	—
33	C₂×M₁₆×C₁₉	—	4	2	no	—
34	C₂×M₁₆×C₁₉	C₂×C₄	1	8	yes	complemented
35	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
39	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₉	—	4	8	no	—
45	C₁₉×#107	—	4	4	no	—
46	C₁₉×#106	—	4	4	no	—
47	C₁₉×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
48	C₁₉×#95	—	2	4	no	—
49	C₁₉×#95	—	2	4	no	—
50	C₂⁴×C₁₉	—	2	4	no	—
51	C₂⁴×C₁₉	—	2	4	no	—
52	C₂⁴×C₁₉	—	4	1	no	—
53	C₂⁴×C₁₉	—	4	1	no	—
54	C₂⁴×C₁₉	C₄⋊C₄	1	4	yes	—
55	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂⁴×C₁₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	4	8	no	—
64	C₂²×C₄×C₁₉	—	4	8	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	2	no	—
68	C₂²×C₄×C₁₉	—	4	2	no	—
69	C₂²×C₄×C₁₉	—	4	4	no	—
70	C₂²×C₄×C₁₉	—	4	4	no	—
71	C₂²×C₄×C₁₉	—	4	4	no	—
72	C₂²×C₄×C₁₉	—	4	4	no	—
73	C₂²×C₄×C₁₉	—	4	4	no	—
74	C₂²×C₄×C₁₉	—	4	4	no	—
75	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
77	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₁₉	—	4	8	no	—
85	C₂×C₈×C₁₉	—	2	4	no	—
86	C₂×C₈×C₁₉	—	2	4	no	—
87	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
88	C₄⋊C₄×C₁₉	—	2	8	no	—
89	C₄⋊C₄×C₁₉	—	2	8	no	—
90	C₄⋊C₄×C₁₉	—	4	4	no	—
91	C₄⋊C₄×C₁₉	—	4	4	no	—
92	C₂²⋊C₄×C₁₉	—	4	8	no	—
93	C₂²⋊C₄×C₁₉	—	4	8	no	—
94	#21942	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	—	2	2	no	—
96	C₂³×C₁₉	—	2	2	no	—
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	—	2	2	no	—
103	C₂³×C₁₉	—	2	16	no	—
104	C₂³×C₁₉	—	4	4	no	—
105	C₂³×C₁₉	—	4	4	no	—
106	C₂³×C₁₉	—	4	8	no	—
107	C₂³×C₁₉	—	4	4	no	—
108	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
110	C₂³×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
111	C₂³×C₁₉	#137	1	4	yes	—
112	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	8	no	—
119	C₂×C₄×C₁₉	—	2	8	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	4	8	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	4	no	—
128	C₂×C₄×C₁₉	—	4	4	no	—
129	C₂×C₄×C₁₉	—	8	8	no	—
130	C₂×C₄×C₁₉	—	8	8	no	—
131	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
137	C₈×C₁₉	—	2	8	no	—
138	C₂×#407	C₂×C₁₉	1	1	yes	characteristic
139	C₂×#380	C₂×C₁₉	1	2	yes	—
140	#1853	C₂×C₁₉	1	4	yes	complemented
141	#1852	C₂×C₁₉	1	4	yes	complemented
142	#1809	C₂×C₁₉	1	4	yes	complemented
143	C₂²×C₁₉	—	2	4	no	—
144	C₂²×C₁₉	—	2	4	no	—
145	C₂²×C₁₉	—	2	4	no	—
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	—	2	4	no	—
148	C₂²×C₁₉	—	2	4	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	4	no	—
151	C₂²×C₁₉	—	4	16	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	#428	1	8	yes	—
156	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₉	#534	1	1	yes	characteristic
158	C₂²×C₁₉	#534	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₂²×C₁₉	C₂×#137	1	2	yes	—
161	C₄×C₁₉	—	2	4	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	8	4	no	—
164	C₄×C₁₉	—	8	4	no	—
165	C₄×C₁₉	#518	1	2	yes	—
166	C₄×C₁₉	#543	1	2	yes	—
167	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
168	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
169	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
170	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
171	C₂²×C₂²⋊C₄	—	2	2	no	—
172	C₂×#107	C₂²×C₁₉	1	4	yes	complemented
173	C₂×#106	C₂²×C₁₉	1	4	yes	complemented
174	#407	C₄×C₁₉	1	8	yes	complemented
175	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
176	#380	—	2	2	no	—
177	#380	—	2	2	no	—
178	#380	C₂²×C₁₉	1	4	yes	—
179	#380	C₂²×C₁₉	1	4	yes	—
180	C₂×#95	—	2	2	no	—
181	#340	C₂²×C₁₉	1	8	yes	complemented
182	C₂×C₁₉	—	2	8	no	—
183	C₂×C₁₉	—	4	2	no	—
184	C₂×C₁₉	#2315	1	1	yes	characteristic
185	C₂×C₁₉	C₂×#428	1	2	yes	—
186	C₂×C₁₉	#3192	1	1	yes	characteristic
187	C₂×C₁₉	#3193	1	2	yes	—
188	C₂×C₁₉	#3195	1	1	yes	characteristic
189	C₂⁵	D₄×C₁₉	1	1	yes	characteristic
190	C₂³×C₄	—	2	2	no	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	—	4	2	no	—
193	C₂³×C₄	—	4	2	no	—
194	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
195	C₂³×C₄	D₄×C₁₉	1	2	yes	—
196	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
197	C₂³×C₄	Q₈×C₁₉	1	4	yes	—
198	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
199	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
200	C₂×M₁₆	—	4	2	no	—
201	C₂×M₁₆	—	4	2	no	—
202	C₂×M₁₆	C₂×C₄×C₁₉	1	8	yes	complemented
203	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	4	4	no	—
207	C₂×C₄⋊C₄	—	4	4	no	—
208	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	—	2	4	no	—
212	C₂×C₂²⋊C₄	—	4	8	no	—
213	#107	—	4	4	no	—
214	#106	—	4	4	no	—
215	C₂²⋊C₈	C₂×C₄×C₁₉	1	16	yes	complemented
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₁₉	#21942	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	—	2	4	no	—
220	C₂⁴	—	2	4	no	—
221	C₂⁴	—	4	1	no	—
222	C₂⁴	—	4	1	no	—
223	C₂⁴	C₄⋊C₄×C₁₉	1	4	yes	—
224	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₂⁴	C₂×D₄×C₁₉	1	2	yes	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	8	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
245	C₂²×C₄	C₄⋊C₄×C₁₉	1	8	yes	—
246	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
247	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
250	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
251	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
252	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
253	M₁₆	—	4	8	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	4	8	no	—
262	C₂²⋊C₄	—	4	8	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	16	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	8	no	—
275	C₂³	—	4	4	no	—
276	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
277	C₂³	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
278	C₂³	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
279	C₂³	C₁₉×#137	1	4	yes	—
280	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	8	no	—
298	C₂×C₄	—	8	8	no	—
299	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
300	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
301	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
303	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
304	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
305	C₈	—	2	8	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	16	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	C₁₉×#428	1	8	yes	—
319	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
320	C₂²	C₁₉×#534	1	1	yes	characteristic
321	C₂²	C₁₉×#534	1	1	yes	characteristic
322	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
323	C₂²	C₂×C₁₉×#137	1	2	yes	—
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	8	4	no	—
327	C₄	—	8	4	no	—
328	C₄	C₁₉×#518	1	2	yes	—
329	C₄	C₁₉×#543	1	2	yes	—
330	C₂	—	2	8	no	—
331	C₂	—	4	2	no	—
332	C₂	C₁₉×#2315	1	1	yes	characteristic
333	C₂	C₂×C₁₉×#428	1	2	yes	—
334	C₂	C₁₉×#3192	1	1	yes	characteristic
335	C₂	C₁₉×#3193	1	2	yes	—
336	C₂	C₁₉×#3195	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1853	C₂	semidirect
4. C₁₉×#1852	C₂	semidirect
5. C₁₉×#1809	C₂	semidirect
11. C₂×C₁₉×#107	C₂²	semidirect
12. C₂×C₁₉×#106	C₂²	semidirect
13. C₁₉×#407	C₄	semidirect
20. C₁₉×#340	C₂²	semidirect
34. C₂×M₁₆×C₁₉	C₂×C₄	semidirect
47. C₁₉×C₂²⋊C₈	C₂×C₄	semidirect
94. #21942	C₁₉	direct (N × H)
140. #1853	C₂×C₁₉	semidirect
141. #1852	C₂×C₁₉	semidirect
142. #1809	C₂×C₁₉	semidirect
172. C₂×#107	C₂²×C₁₉	semidirect
173. C₂×#106	C₂²×C₁₉	semidirect
174. #407	C₄×C₁₉	semidirect
181. #340	C₂²×C₁₉	semidirect
202. C₂×M₁₆	C₂×C₄×C₁₉	semidirect
215. C₂²⋊C₈	C₂×C₄×C₁₉	semidirect
218. C₁₉	#21942	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×#407	2
C₂×C₁₉×#380	2
C₁₉×#1853	2
C₁₉×#1852	2
C₁₉×#1809	2
#21942	19

Subgroups — C19×#21942 #18027901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21942 #18027901