Subgroups — C₁₉×#11622 #18017581

Order 4864.

320 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2966	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×Q₈⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₁₉×D₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₁₉×#529	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₉	—	2	1	no	—
8	C₁₉×#446	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#436	C₄	1	1	yes	characteristic
11	C₁₉×#435	—	2	1	no	—
12	C₈×D₄×C₁₉	—	2	1	no	—
13	C₈×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#433	—	2	1	no	—
15	C₁₉×#432	C₄	1	1	yes	characteristic
16	C₁₉×#405	—	2	1	no	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×C₄⋊C₁₆	—	2	1	no	—
19	C₁₉×C₄⋊C₁₆	—	2	1	no	—
20	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
22	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×#126	—	2	1	no	—
28	C₁₉×#125	C₈	1	1	yes	characteristic
29	C₁₉×#124	C₈	1	1	yes	characteristic
30	C₁₉×#123	—	2	1	no	—
31	C₁₉×C₂²⋊Q₈	C₈	1	1	yes	characteristic
32	C₁₉×#121	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₁₉	—	2	1	no	—
35	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₁₉×#117	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₉	—	2	2	no	—
39	C₂×C₁₆×C₁₉	—	2	2	no	—
40	C₂×C₁₆×C₁₉	—	4	2	no	—
41	C₁₉×C₄⋊C₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₂²⋊C₈	—	2	1	no	—
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₁₉×#97	—	2	1	no	—
49	C₁₉×#97	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
55	C₂×Q₈×C₁₉	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₁₉	—	2	1	no	—
57	C₂×D₄×C₁₉	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
62	M₁₆×C₁₉	—	4	2	no	—
63	C₂×C₈×C₁₉	—	2	1	no	—
64	C₂×C₈×C₁₉	—	2	1	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₂²⋊C₄×C₁₉	—	2	1	no	—
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
89	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₁₆×C₁₉	—	4	2	no	—
91	C₁₆×C₁₉	—	4	2	no	—
92	#11622	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
95	Q₈×C₁₉	—	2	2	no	—
96	D₄×C₁₉	—	2	2	no	—
97	D₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	4	1	no	—
106	C₂×C₄×C₁₉	—	4	1	no	—
107	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
113	C₈×C₁₉	—	2	1	no	—
114	C₈×C₁₉	—	2	1	no	—
115	C₈×C₁₉	—	2	1	no	—
116	C₈×C₁₉	—	4	1	no	—
117	C₈×C₁₉	—	4	1	no	—
118	C₈×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈×C₁₉	#137	1	1	yes	characteristic
120	#2966	C₂×C₁₉	1	1	yes	characteristic
121	#2162	C₂×C₁₉	1	1	yes	characteristic, complemented
122	C₂×C₄⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
123	Q₈⋊C₁₆	C₂×C₁₉	1	2	yes	complemented
124	D₄⋊C₁₆	C₂×C₁₉	1	2	yes	complemented
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	4	1	no	—
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	—	4	1	no	—
130	C₂²×C₁₉	#350	1	2	yes	—
131	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	—	2	1	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	4	1	no	—
137	C₄×C₁₉	—	4	1	no	—
138	C₄×C₁₉	—	4	1	no	—
139	C₄×C₁₉	#417	1	1	yes	characteristic
140	C₄×C₁₉	#421	1	1	yes	characteristic
141	C₄×C₁₉	#504	1	2	yes	—
142	#529	C₄×C₁₉	1	1	yes	characteristic
143	C₂²×C₁₆	—	2	1	no	—
144	#446	C₄×C₁₉	1	1	yes	characteristic
145	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
146	#436	C₄×C₁₉	1	1	yes	characteristic
147	#435	—	2	1	no	—
148	C₈×D₄	—	2	1	no	—
149	C₈×D₄	C₂²×C₁₉	1	1	yes	characteristic
150	#433	—	2	1	no	—
151	#432	C₄×C₁₉	1	1	yes	characteristic
152	#405	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
154	C₄⋊C₁₆	—	2	1	no	—
155	C₄⋊C₁₆	—	2	1	no	—
156	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
157	C₂²⋊C₁₆	—	2	1	no	—
158	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	—	8	1	no	—
161	C₂×C₁₉	#1477	1	1	yes	characteristic
162	C₂×C₁₉	#2113	1	1	yes	characteristic
163	C₂×C₁₉	C₂×#350	1	1	yes	characteristic
164	C₂×M₁₆	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
168	#126	—	2	1	no	—
169	#125	C₈×C₁₉	1	1	yes	characteristic
170	#124	C₈×C₁₉	1	1	yes	characteristic
171	#123	—	2	1	no	—
172	C₂²⋊Q₈	C₈×C₁₉	1	1	yes	characteristic
173	#121	C₈×C₁₉	1	1	yes	characteristic
174	C₄×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
177	#117	—	2	1	no	—
178	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	4	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	#97	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
193	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
194	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
195	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
196	C₁₉	#11622	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×Q₈	C₂×C₈×C₁₉	1	1	yes	characteristic
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×C₈×C₁₉	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
203	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
204	M₁₆	—	4	2	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
221	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
229	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
230	C₄²	M₁₆×C₁₉	1	1	yes	characteristic
231	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
232	C₁₆	—	4	2	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
253	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₈	C₁₉×#137	1	1	yes	characteristic
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₁₉×#350	1	2	yes	—
267	C₂²	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	C₁₉×#417	1	1	yes	characteristic
276	C₄	C₁₉×#421	1	1	yes	characteristic
277	C₄	C₁₉×#504	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	8	1	no	—
280	C₂	C₁₉×#1477	1	1	yes	characteristic
281	C₂	C₁₉×#2113	1	1	yes	characteristic
282	C₂	C₂×C₁₉×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×C₂²⋊C₈
prime radical	C₁₉×#121	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2162	C₂	semidirect
3. C₂×C₁₉×C₄⋊C₁₆	C₂	semidirect
4. C₁₉×Q₈⋊C₁₆	C₂	semidirect
5. C₁₉×D₄⋊C₁₆	C₂	semidirect
92. #11622	C₁₉	direct (N × H)
121. #2162	C₂×C₁₉	semidirect
122. C₂×C₄⋊C₁₆	C₂×C₁₉	semidirect
123. Q₈⋊C₁₆	C₂×C₁₉	semidirect
124. D₄⋊C₁₆	C₂×C₁₉	semidirect
196. C₁₉	#11622	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×#2966	2
C₁₉×#2162	2
C₂×C₁₉×C₄⋊C₁₆	2
C₁₉×Q₈⋊C₁₆	2
C₁₉×D₄⋊C₁₆	2
#11622	19

Subgroups — C19×#11622 #18017581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11622 #18017581