Subgroups — C₁₉×#11978 #18017937

Order 4864.

312 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	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1363	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1360	C₂	1	2	yes	maximal, complemented
6	C₁₉×#529	—	2	1	no	—
7	C₂²×C₁₆×C₁₉	—	2	1	no	—
8	C₁₉×#499	C₂²	1	1	yes	characteristic
9	C₁₉×#495	C₂²	1	1	yes	characteristic
10	C₁₉×#488	—	2	1	no	—
11	C₁₉×#467	C₄	1	1	yes	characteristic
12	C₁₉×#466	C₄	1	1	yes	characteristic
13	C₁₉×#465	—	2	1	no	—
14	C₄×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
15	C₄×SD₁₆×C₁₉	—	2	1	no	—
16	C₄×D₈×C₁₉	C₄	1	1	yes	characteristic
17	C₁₉×#432	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×C₄⋊C₁₆	—	2	1	no	—
20	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₁₉×#335	—	2	1	no	—
22	C₁₉×#334	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	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	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×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₉	—	2	2	no	—
40	C₂×C₁₆×C₁₉	—	4	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×#103	—	2	1	no	—
46	C₁₉×#103	—	2	1	no	—
47	C₁₉×#102	—	2	1	no	—
48	C₁₉×#102	—	2	1	no	—
49	C₁₉×C₂²⋊C₈	—	4	1	no	—
50	C₁₉×#97	—	4	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	—	2	1	no	—
56	C₂×D₄×C₁₉	—	2	1	no	—
57	C₂×D₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
62	Q₁₆×C₁₉	—	4	1	no	—
63	SD₁₆×C₁₉	—	4	1	no	—
64	SD₁₆×C₁₉	—	4	1	no	—
65	D₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	2	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	C₂²×C₄	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	—	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	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₉	D₈	1	1	yes	characteristic
88	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₁₆×C₁₉	—	4	2	no	—
91	#11978	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₁₉	—	4	1	no	—
93	C₂³×C₁₉	#102	1	1	yes	characteristic
94	Q₈×C₁₉	—	4	1	no	—
95	Q₈×C₁₉	—	4	1	no	—
96	D₄×C₁₉	—	4	1	no	—
97	D₄×C₁₉	—	4	1	no	—
98	D₄×C₁₉	—	8	2	no	—
99	C₂×C₄×C₁₉	—	2	2	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	—	4	1	no	—
106	C₂×C₄×C₁₉	—	8	2	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	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₈×C₁₉	#137	1	1	yes	characteristic
119	#3168	C₂×C₁₉	1	1	yes	characteristic
120	C₂×C₄⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
121	#1568	C₂×C₁₉	1	1	yes	characteristic, complemented
122	#1363	C₂×C₁₉	1	2	yes	complemented
123	#1360	C₂×C₁₉	1	2	yes	complemented
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	#359	1	2	yes	—
130	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
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	#413	1	1	yes	characteristic
140	C₄×C₁₉	#417	1	1	yes	characteristic
141	#529	—	2	1	no	—
142	C₂²×C₁₆	—	2	1	no	—
143	#499	C₂²×C₁₉	1	1	yes	characteristic
144	#495	C₂²×C₁₉	1	1	yes	characteristic
145	#488	—	2	1	no	—
146	#467	C₄×C₁₉	1	1	yes	characteristic
147	#466	C₄×C₁₉	1	1	yes	characteristic
148	#465	—	2	1	no	—
149	C₄×Q₁₆	C₄×C₁₉	1	1	yes	characteristic
150	C₄×SD₁₆	—	2	1	no	—
151	C₄×D₈	C₄×C₁₉	1	1	yes	characteristic
152	#432	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
154	C₄⋊C₁₆	—	2	1	no	—
155	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
156	#335	—	2	1	no	—
157	#334	C₂²×C₁₉	1	2	yes	—
158	C₂×C₁₉	—	2	2	no	—
159	C₂×C₁₉	—	16	1	no	—
160	C₂×C₁₉	#1547	1	1	yes	characteristic
161	C₂×C₁₉	C₂×#359	1	1	yes	characteristic
162	C₂×C₁₉	#2145	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	C₂×C₄×C₁₉	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
168	#126	—	4	1	no	—
169	#125	—	2	1	no	—
170	#124	—	2	1	no	—
171	#123	—	4	1	no	—
172	C₂²⋊Q₈	—	2	1	no	—
173	#121	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	4	1	no	—
177	#117	—	4	1	no	—
178	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	4	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	#103	—	2	1	no	—
186	#103	—	2	1	no	—
187	#102	—	2	1	no	—
188	#102	—	2	1	no	—
189	C₂²⋊C₈	—	4	1	no	—
190	#97	—	4	1	no	—
191	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
192	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
193	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
195	C₁₉	#11978	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	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
201	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
202	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
203	Q₁₆	—	4	1	no	—
204	SD₁₆	—	4	1	no	—
205	SD₁₆	—	4	1	no	—
206	D₈	—	4	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
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	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	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	D₈×C₁₉	1	1	yes	characteristic
229	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
231	C₁₆	—	4	2	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₁₉×#102	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	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×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₈	C₁₉×#137	1	1	yes	characteristic
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	8	1	no	—
264	C₂²	C₁₉×#359	1	2	yes	—
265	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	8	1	no	—
272	C₄	—	8	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₁₉×#413	1	1	yes	characteristic
275	C₄	C₁₉×#417	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	16	1	no	—
278	C₂	C₁₉×#1547	1	1	yes	characteristic
279	C₂	C₂×C₁₉×#359	1	1	yes	characteristic
280	C₂	C₁₉×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#466	C₄
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. C₂×C₁₉×C₄⋊C₁₆	C₂	semidirect
3. C₁₉×#1568	C₂	semidirect
4. C₁₉×#1363	C₂	semidirect
5. C₁₉×#1360	C₂	semidirect
91. #11978	C₁₉	direct (N × H)
120. C₂×C₄⋊C₁₆	C₂×C₁₉	semidirect
121. #1568	C₂×C₁₉	semidirect
122. #1363	C₂×C₁₉	semidirect
123. #1360	C₂×C₁₉	semidirect
195. C₁₉	#11978	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
C₂×C₁₉×C₄⋊C₁₆	2
C₁₉×#1568	2
C₁₉×#1363	2
C₁₉×#1360	2
#11978	19

Subgroups — C19×#11978 #18017937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11978 #18017937