Subgroups — C₁₉×#11855 #18017814

Order 4864.

324 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×#3168	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1835	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×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#382	—	2	1	no	—
18	C₁₉×C₄⋊C₁₆	—	2	1	no	—
19	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₄×C₁₆×C₁₉	—	2	1	no	—
21	C₁₉×#340	—	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	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₉	—	2	2	no	—
41	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
42	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
43	C₁₉×#106	—	2	1	no	—
44	C₁₉×#103	—	2	1	no	—
45	C₁₉×#103	—	2	1	no	—
46	C₁₉×#102	—	2	1	no	—
47	C₁₉×#102	—	2	1	no	—
48	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
52	C₁₉×#95	—	4	1	no	—
53	C₁₉×#95	—	4	1	no	—
54	C₂×Q₈×C₁₉	—	2	1	no	—
55	C₂×D₄×C₁₉	—	2	1	no	—
56	C₂×D₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₁₉	—	4	1	no	—
63	SD₁₆×C₁₉	—	4	2	no	—
64	D₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₂²⋊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	C₄²×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
88	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
89	C₁₆×C₁₉	—	4	2	no	—
90	#11855	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₁₉	—	4	1	no	—
92	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₁₉	—	4	1	no	—
94	Q₈×C₁₉	—	4	1	no	—
95	D₄×C₁₉	—	4	1	no	—
96	D₄×C₁₉	—	4	1	no	—
97	D₄×C₁₉	—	8	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	—	4	1	no	—
102	C₂×C₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	—	4	1	no	—
104	C₂×C₄×C₁₉	—	4	1	no	—
105	C₂×C₄×C₁₉	—	8	2	no	—
106	C₂×C₄×C₁₉	—	8	2	no	—
107	C₂×C₄×C₁₉	—	8	2	no	—
108	C₂×C₄×C₁₉	—	8	2	no	—
109	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₈×C₁₉	—	2	1	no	—
116	C₈×C₁₉	—	2	1	no	—
117	C₈×C₁₉	—	2	1	no	—
118	C₈×C₁₉	—	2	1	no	—
119	#3168	C₂×C₁₉	1	1	yes	characteristic
120	#2162	C₂×C₁₉	1	1	yes	characteristic, complemented
121	#1835	C₂×C₁₉	1	1	yes	characteristic, complemented
122	#1348	C₂×C₁₉	1	2	yes	complemented
123	#1347	C₂×C₁₉	1	2	yes	—
124	C₂²×C₁₉	—	2	1	no	—
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	—	8	1	no	—
127	C₂²×C₁₉	—	8	1	no	—
128	C₂²×C₁₉	—	8	1	no	—
129	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₉	#508	1	2	yes	—
131	C₄×C₁₉	—	2	1	no	—
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	—	8	1	no	—
137	C₄×C₁₉	—	8	1	no	—
138	C₄×C₁₉	—	8	1	no	—
139	C₄×C₁₉	—	16	2	no	—
140	C₄×C₁₉	#410	1	1	yes	characteristic
141	C₄×C₁₉	#421	1	1	yes	characteristic
142	#529	—	2	1	no	—
143	#499	C₄×C₁₉	1	1	yes	characteristic, complemented
144	#495	C₄×C₁₉	1	1	yes	characteristic, complemented
145	#488	—	2	1	no	—
146	#467	C₂²×C₁₉	1	1	yes	characteristic
147	#466	C₂²×C₁₉	1	1	yes	characteristic
148	#465	—	2	1	no	—
149	C₄×Q₁₆	C₄×C₁₉	1	1	yes	characteristic, complemented
150	C₄×SD₁₆	—	2	1	no	—
151	C₄×D₈	C₄×C₁₉	1	1	yes	characteristic, complemented
152	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
153	#382	—	2	1	no	—
154	C₄⋊C₁₆	—	2	1	no	—
155	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
156	C₄×C₁₆	—	2	1	no	—
157	#340	—	2	1	no	—
158	#340	C₂²×C₁₉	1	2	yes	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	—	16	1	no	—
161	C₂×C₁₉	#1517	1	1	yes	characteristic
162	C₂×C₁₉	#2138	1	1	yes	characteristic
163	C₂×C₁₉	#2145	1	1	yes	characteristic
164	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
165	C₂×SD₁₆	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
169	#126	—	4	1	no	—
170	#125	—	2	1	no	—
171	#124	—	2	1	no	—
172	#123	—	4	1	no	—
173	C₂²⋊Q₈	—	2	1	no	—
174	#121	—	2	1	no	—
175	C₄×Q₈	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	4	1	no	—
178	#117	—	4	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
181	C₂×C₁₆	—	2	2	no	—
182	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
183	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
184	#106	—	2	1	no	—
185	#103	—	2	1	no	—
186	#103	—	2	1	no	—
187	#102	—	2	1	no	—
188	#102	—	2	1	no	—
189	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
190	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
191	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
192	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
193	#95	—	4	1	no	—
194	#95	—	4	1	no	—
195	C₁₉	#11855	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×Q₈	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
202	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
204	Q₁₆	—	4	1	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₂²⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₂²⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
228	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
229	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
230	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
231	C₁₆	—	4	2	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
234	Q₈	—	4	1	no	—
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	8	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
251	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
252	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
253	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
255	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	8	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
266	C₂²	C₁₉×#508	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	1	no	—
273	C₄	—	8	1	no	—
274	C₄	—	8	1	no	—
275	C₄	—	16	2	no	—
276	C₄	C₁₉×#410	1	1	yes	characteristic
277	C₄	C₁₉×#421	1	1	yes	characteristic
278	C₂	—	2	2	no	—
279	C₂	—	16	1	no	—
280	C₂	C₁₉×#1517	1	1	yes	characteristic
281	C₂	C₁₉×#2138	1	1	yes	characteristic
282	C₂	C₁₉×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₁₉×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₁₉×#1835	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
90. #11855	C₁₉	direct (N × H)
120. #2162	C₂×C₁₉	semidirect
121. #1835	C₂×C₁₉	semidirect
122. #1348	C₂×C₁₉	semidirect
143. #499	C₄×C₁₉	semidirect
144. #495	C₄×C₁₉	semidirect
149. C₄×Q₁₆	C₄×C₁₉	semidirect
151. C₄×D₈	C₄×C₁₉	semidirect
195. C₁₉	#11855	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
C₁₉×#1835	2
C₁₉×#1348	2
C₁₉×#1347	2
#11855	19

Subgroups — C19×#11855 #18017814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11855 #18017814