Subgroups — C₁₉×#11648 #18017607

Order 4864.

320 conjugacy classes of subgroups.

The subgroup lattice has 282 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	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1332	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1331	C₂	1	2	yes	maximal, complemented
6	C₁₉×#529	—	2	1	no	—
7	C₁₉×#499	C₄	1	1	yes	characteristic
8	C₁₉×#495	C₄	1	1	yes	characteristic
9	C₁₉×#488	—	2	1	no	—
10	C₁₉×#467	C₄	1	1	yes	characteristic
11	C₁₉×#466	C₄	1	1	yes	characteristic
12	C₁₉×#465	—	2	1	no	—
13	C₄×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₁₉	—	2	1	no	—
15	C₄×D₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×C₄⋊C₁₆	—	2	1	no	—
19	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
20	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
21	C₁₉×#335	—	2	1	no	—
22	C₁₉×#335	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	—	4	1	no	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₁₉×#126	—	4	1	no	—
30	C₁₉×#125	—	2	1	no	—
31	C₁₉×#124	—	2	1	no	—
32	C₁₉×#123	—	4	1	no	—
33	C₁₉×C₂²⋊Q₈	—	2	1	no	—
34	C₁₉×#121	—	2	1	no	—
35	C₄×Q₈×C₁₉	—	2	1	no	—
36	C₄×D₄×C₁₉	—	2	1	no	—
37	C₄×D₄×C₁₉	—	4	1	no	—
38	C₁₉×#117	—	4	1	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×C₄⋊C₈	—	2	2	no	—
45	C₁₉×C₄⋊C₈	—	4	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	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
61	Q₁₆×C₁₉	—	2	2	no	—
62	SD₁₆×C₁₉	—	4	2	no	—
63	D₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	4	2	no	—
67	C₂×C₈×C₁₉	—	8	2	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	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	4	1	no	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₂²⋊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	C₄²×C₁₉	—	4	1	no	—
84	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₁₉	D₈	1	1	yes	characteristic
86	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₁₆×C₁₉	—	4	2	no	—
89	#11648	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₁₉	—	4	1	no	—
91	C₂³×C₁₉	#102	1	1	yes	characteristic
92	Q₈×C₁₉	—	4	1	no	—
93	Q₈×C₁₉	—	4	1	no	—
94	D₄×C₁₉	—	4	1	no	—
95	D₄×C₁₉	—	4	1	no	—
96	D₄×C₁₉	—	8	2	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	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	1	no	—
106	C₂×C₄×C₁₉	—	8	1	no	—
107	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×SD₁₆	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	—	8	2	no	—
117	C₈×C₁₉	#131	1	2	yes	—
118	#3168	C₂×C₁₉	1	1	yes	characteristic
119	#2162	C₂×C₁₉	1	1	yes	characteristic, complemented
120	C₂×#335	C₂×C₁₉	1	1	yes	characteristic, complemented
121	#1332	C₂×C₁₉	1	2	yes	complemented
122	#1331	C₂×C₁₉	1	2	yes	complemented
123	C₂²×C₁₉	—	2	1	no	—
124	C₂²×C₁₉	—	2	1	no	—
125	C₂²×C₁₉	—	8	1	no	—
126	C₂²×C₁₉	—	8	1	no	—
127	C₂²×C₁₉	—	8	1	no	—
128	C₂²×C₁₉	#359	1	2	yes	—
129	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	—	2	1	no	—
134	C₄×C₁₉	—	8	1	no	—
135	C₄×C₁₉	—	8	1	no	—
136	C₄×C₁₉	—	8	1	no	—
137	C₄×C₁₉	#408	1	1	yes	characteristic
138	C₄×C₁₉	#421	1	1	yes	characteristic
139	C₄×C₁₉	#509	1	1	yes	characteristic
140	C₄×C₁₉	#510	1	1	yes	characteristic
141	#529	—	2	1	no	—
142	#499	C₄×C₁₉	1	1	yes	characteristic
143	#495	C₄×C₁₉	1	1	yes	characteristic
144	#488	—	2	1	no	—
145	#467	C₄×C₁₉	1	1	yes	characteristic
146	#466	C₄×C₁₉	1	1	yes	characteristic
147	#465	—	2	1	no	—
148	C₄×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
149	C₄×SD₁₆	—	2	1	no	—
150	C₄×D₈	C₂²×C₁₉	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
153	C₄⋊C₁₆	—	2	1	no	—
154	C₂²⋊C₁₆	—	2	1	no	—
155	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
156	#335	—	2	1	no	—
157	#335	C₂²×C₁₉	1	2	yes	—
158	C₂×C₁₉	—	2	2	no	—
159	C₂×C₁₉	—	16	1	no	—
160	C₂×C₁₉	#1480	1	1	yes	characteristic
161	C₂×C₁₉	#2139	1	1	yes	characteristic
162	C₂×C₁₉	C₂×#359	1	1	yes	characteristic
163	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
164	C₂×SD₁₆	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
166	C₂²×C₈	—	4	1	no	—
167	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
168	C₂²×C₈	D₄×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×C₄²	D₄×C₁₉	1	1	yes	characteristic
180	C₂×C₁₆	—	2	2	no	—
181	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
182	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
183	#106	—	2	1	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	4	1	no	—
186	#103	—	2	1	no	—
187	#103	—	2	1	no	—
188	#102	—	2	1	no	—
189	#102	—	2	1	no	—
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	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
194	C₁₉	#11648	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×Q₈	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
200	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
201	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
202	Q₁₆	—	2	2	no	—
203	SD₁₆	—	4	2	no	—
204	D₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	8	2	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	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₂²⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₄²	—	4	1	no	—
225	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
226	C₄²	D₈×C₁₉	1	1	yes	characteristic
227	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
229	C₁₆	—	4	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₁₉×#102	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	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	—	8	1	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
248	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
249	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
251	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	8	2	no	—
257	C₈	C₁₉×#131	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	8	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	C₁₉×#359	1	2	yes	—
264	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	8	1	no	—
270	C₄	—	8	1	no	—
271	C₄	—	8	1	no	—
272	C₄	C₁₉×#408	1	1	yes	characteristic
273	C₄	C₁₉×#421	1	1	yes	characteristic
274	C₄	C₁₉×#509	1	1	yes	characteristic
275	C₄	C₁₉×#510	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	16	1	no	—
278	C₂	C₁₉×#1480	1	1	yes	characteristic
279	C₂	C₁₉×#2139	1	1	yes	characteristic
280	C₂	C₂×C₁₉×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#102
prime radical	C₁₉×#466	C₄
LCS	G ▷ C₂×C₈ ▷ 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. C₂×C₁₉×#335	C₂	semidirect
4. C₁₉×#1332	C₂	semidirect
5. C₁₉×#1331	C₂	semidirect
89. #11648	C₁₉	direct (N × H)
119. #2162	C₂×C₁₉	semidirect
120. C₂×#335	C₂×C₁₉	semidirect
121. #1332	C₂×C₁₉	semidirect
122. #1331	C₂×C₁₉	semidirect
194. C₁₉	#11648	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
C₂×C₁₉×#335	2
C₁₉×#1332	2
C₁₉×#1331	2
#11648	19

Subgroups — C19×#11648 #18017607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11648 #18017607