Subgroups — C₁₉×#11741 #18017700

Order 4864.

356 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3168	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1348	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1347	C₂	1	2	yes	maximal
6	C₁₉×#529	—	2	1	no	—
7	C₁₉×#499	C₄	1	1	yes	characteristic, complemented
8	C₁₉×#495	C₄	1	1	yes	characteristic, complemented
9	C₁₉×#488	—	2	1	no	—
10	C₁₉×#467	C₂²	1	1	yes	characteristic
11	C₁₉×#466	C₂²	1	1	yes	characteristic
12	C₁₉×#465	—	2	1	no	—
13	C₄×Q₁₆×C₁₉	C₄	1	1	yes	characteristic, complemented
14	C₄×SD₁₆×C₁₉	—	2	1	no	—
15	C₄×D₈×C₁₉	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₉×#106	—	2	1	no	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
19	C₁₉×C₄⋊C₁₆	—	2	1	no	—
20	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₄×C₁₆×C₁₉	—	2	1	no	—
22	C₁₉×#340	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×SD₁₆×C₁₉	—	2	1	no	—
25	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
28	C₁₉×#126	—	4	1	no	—
29	C₁₉×#125	—	2	1	no	—
30	C₁₉×#124	—	2	1	no	—
31	C₁₉×#123	—	4	1	no	—
32	C₁₉×C₂²⋊Q₈	—	2	1	no	—
33	C₁₉×#121	—	2	1	no	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₄×D₄×C₁₉	—	2	1	no	—
36	C₄×D₄×C₁₉	—	4	1	no	—
37	C₁₉×#117	—	4	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
39	C₂×C₄²×C₁₉	—	4	1	no	—
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₉	—	2	2	no	—
42	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
43	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
44	C₁₉×#106	—	2	4	no	—
45	C₁₉×#106	—	2	1	no	—
46	C₁₉×#103	—	2	1	no	—
47	C₁₉×#103	—	2	1	no	—
48	C₁₉×#102	—	2	1	no	—
49	C₁₉×#102	—	2	1	no	—
50	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
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	C₁₉×#95	—	4	1	no	—
55	C₂×Q₈×C₁₉	—	2	1	no	—
56	C₂×D₄×C₁₉	—	2	1	no	—
57	C₂×D₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₉	—	4	1	no	—
64	SD₁₆×C₁₉	—	4	2	no	—
65	D₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	—	2	4	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₄⋊C₄×C₁₉	—	4	1	no	—
80	C₂²⋊C₄×C₁₉	—	4	1	no	—
81	C₂²⋊C₄×C₁₉	—	4	1	no	—
82	C₂²⋊C₄×C₁₉	—	4	1	no	—
83	C₂²⋊C₄×C₁₉	—	4	1	no	—
84	C₂²⋊C₄×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	—	8	2	no	—
87	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
90	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
91	C₁₆×C₁₉	—	4	2	no	—
92	#11741	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₉	—	4	1	no	—
94	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₁₉	—	4	1	no	—
96	Q₈×C₁₉	—	4	1	no	—
97	D₄×C₁₉	—	4	1	no	—
98	D₄×C₁₉	—	4	1	no	—
99	D₄×C₁₉	—	8	2	no	—
100	C₂×C₄×C₁₉	—	2	2	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	—	4	4	no	—
105	C₂×C₄×C₁₉	—	4	1	no	—
106	C₂×C₄×C₁₉	—	4	1	no	—
107	C₂×C₄×C₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	—	8	2	no	—
109	C₂×C₄×C₁₉	—	8	2	no	—
110	C₂×C₄×C₁₉	—	8	2	no	—
111	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₈×C₁₉	—	2	1	no	—
118	C₈×C₁₉	—	2	1	no	—
119	C₈×C₁₉	—	2	1	no	—
120	C₈×C₁₉	#104	1	2	yes	—
121	#3168	C₂×C₁₉	1	1	yes	characteristic
122	#2162	C₂×C₁₉	1	1	yes	characteristic, complemented
123	C₄×#106	C₂×C₁₉	1	1	yes	characteristic, complemented
124	#1348	C₂×C₁₉	1	2	yes	complemented
125	#1347	C₂×C₁₉	1	2	yes	—
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	—	8	1	no	—
129	C₂²×C₁₉	—	8	1	no	—
130	C₂²×C₁₉	—	8	1	no	—
131	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
132	C₂²×C₁₉	#508	1	2	yes	—
133	C₄×C₁₉	—	2	1	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	8	4	no	—
138	C₄×C₁₉	—	8	1	no	—
139	C₄×C₁₉	—	8	1	no	—
140	C₄×C₁₉	—	8	1	no	—
141	C₄×C₁₉	#360	1	2	yes	—
142	C₄×C₁₉	#410	1	1	yes	characteristic
143	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
144	#529	—	2	1	no	—
145	#499	C₄×C₁₉	1	1	yes	characteristic, complemented
146	#495	C₄×C₁₉	1	1	yes	characteristic, complemented
147	#488	—	2	1	no	—
148	#467	C₂²×C₁₉	1	1	yes	characteristic
149	#466	C₂²×C₁₉	1	1	yes	characteristic
150	#465	—	2	1	no	—
151	C₄×Q₁₆	C₄×C₁₉	1	1	yes	characteristic, complemented
152	C₄×SD₁₆	—	2	1	no	—
153	C₄×D₈	C₄×C₁₉	1	1	yes	characteristic, complemented
154	C₂×#106	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
156	C₄×C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₁₆	—	2	1	no	—
158	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
159	C₄×C₁₆	—	2	1	no	—
160	#340	C₂²×C₁₉	1	2	yes	—
161	C₂×C₁₉	—	2	2	no	—
162	C₂×C₁₉	—	16	1	no	—
163	C₂×C₁₉	#1502	1	1	yes	characteristic
164	C₂×C₁₉	#2138	1	1	yes	characteristic
165	C₂×C₁₉	C₂×#360	1	1	yes	characteristic
166	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
169	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
171	#126	—	4	1	no	—
172	#125	—	2	1	no	—
173	#124	—	2	1	no	—
174	#123	—	4	1	no	—
175	C₂²⋊Q₈	—	2	1	no	—
176	#121	—	2	1	no	—
177	C₄×Q₈	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	4	1	no	—
180	#117	—	4	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄²	—	4	1	no	—
183	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
184	C₂×C₁₆	—	2	2	no	—
185	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
186	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
187	#106	—	2	4	no	—
188	#106	—	2	1	no	—
189	#103	—	2	1	no	—
190	#103	—	2	1	no	—
191	#102	—	2	1	no	—
192	#102	—	2	1	no	—
193	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
194	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
196	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
197	#95	—	4	1	no	—
198	C₁₉	#11741	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×Q₈	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
205	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
207	Q₁₆	—	4	1	no	—
208	SD₁₆	—	4	2	no	—
209	D₈	—	4	1	no	—
210	C₂×C₈	—	2	2	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	C₂²×C₄×C₁₉	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₂²⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	—	4	1	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	8	2	no	—
231	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
232	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
233	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
234	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
235	C₁₆	—	4	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
238	Q₈	—	4	1	no	—
239	Q₈	—	4	1	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	8	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
255	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
256	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
257	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
259	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	C₁₉×#104	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
270	C₂²	C₁₉×#508	1	2	yes	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	4	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₄	—	8	1	no	—
279	C₄	C₁₉×#360	1	2	yes	—
280	C₄	C₁₉×#410	1	1	yes	characteristic
281	C₄	C₂×C₁₉×#104	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	16	1	no	—
284	C₂	C₁₉×#1502	1	1	yes	characteristic
285	C₂	C₁₉×#2138	1	1	yes	characteristic
286	C₂	C₂×C₁₉×#360	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#102
prime radical	C₁₉×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×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. C₄×C₁₉×#106	C₂	semidirect
4. C₁₉×#1348	C₂	semidirect
7. C₁₉×#499	C₄	semidirect
8. C₁₉×#495	C₄	semidirect
13. C₄×Q₁₆×C₁₉	C₄	semidirect
15. C₄×D₈×C₁₉	C₄	semidirect
92. #11741	C₁₉	direct (N × H)
122. #2162	C₂×C₁₉	semidirect
123. C₄×#106	C₂×C₁₉	semidirect
124. #1348	C₂×C₁₉	semidirect
145. #499	C₄×C₁₉	semidirect
146. #495	C₄×C₁₉	semidirect
151. C₄×Q₁₆	C₄×C₁₉	semidirect
153. C₄×D₈	C₄×C₁₉	semidirect
198. C₁₉	#11741	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×#3168	2
C₁₉×#2162	2
C₄×C₁₉×#106	2
C₁₉×#1348	2
C₁₉×#1347	2
#11741	19

Subgroups — C19×#11741 #18017700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11741 #18017700