Subgroups — C₁₉×#8680 #18014639

Order 4864.

418 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1835	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1517	C₂	1	2	yes	maximal
5	C₁₉×#1286	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#131	—	2	1	no	—
7	C₁₉×#532	C₂²	1	1	yes	characteristic
8	C₁₉×#436	—	2	1	no	—
9	C₁₉×#436	C₄	1	2	yes	complemented
10	C₈×D₄×C₁₉	—	2	1	no	—
11	C₈×D₄×C₁₉	C₄	1	2	yes	complemented
12	C₁₉×#431	C₂²	1	2	yes	—
13	C₁₉×#423	C₂²	1	1	yes	characteristic
14	C₁₉×#408	—	2	1	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#382	C₂²	1	2	yes	—
17	C₁₉×#340	—	2	1	no	—
18	C₁₉×#340	—	2	1	no	—
19	C₁₉×#335	—	2	1	no	—
20	C₁₉×#334	—	2	1	no	—
21	C₁₉×#327	—	2	4	no	—
22	C₂×#41×C₁₉	—	2	1	no	—
23	C₁₉×#131	—	4	4	no	—
24	C₂×M₁₆×C₁₉	—	2	1	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₁₉×C₄⋊Q₈	Q₈	1	1	yes	characteristic
30	C₁₉×#127	Q₈	1	1	yes	characteristic
31	C₁₉×#124	—	2	1	no	—
32	C₁₉×#121	—	2	1	no	—
33	C₁₉×#121	D₄	1	2	yes	—
34	C₄×D₄×C₁₉	—	2	1	no	—
35	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₁₉×C₄⋊C₈	—	2	1	no	—
39	C₁₉×C₄⋊C₈	—	2	2	no	—
40	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₁₉×C₂²⋊C₈	—	2	2	no	—
42	C₁₉×C₂²⋊C₈	—	2	2	no	—
43	C₁₉×C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈×C₁₉	—	2	2	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₁₉×#95	—	2	2	no	—
50	C₁₉×#95	—	2	2	no	—
51	#41×C₁₉	—	4	2	no	—
52	#41×C₁₉	—	4	2	no	—
53	C₂×Q₈×C₁₉	—	2	1	no	—
54	C₂×D₄×C₁₉	—	2	2	no	—
55	C₂×D₄×C₁₉	—	2	1	no	—
56	C₂×D₄×C₁₉	—	2	1	no	—
57	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	M₁₆×C₁₉	—	4	4	no	—
64	M₁₆×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	4	4	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	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₂²⋊C₄×C₁₉	—	2	2	no	—
79	C₂²⋊C₄×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₁₉	#41	1	1	yes	characteristic
84	#8680	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	—	2	2	no	—
86	C₂³×C₁₉	#121	1	1	yes	characteristic
87	Q₈×C₁₉	—	4	2	no	—
88	D₄×C₁₉	—	4	2	no	—
89	D₄×C₁₉	—	4	2	no	—
90	D₄×C₁₉	—	4	4	no	—
91	D₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	4	1	no	—
94	C₂×C₄×C₁₉	—	4	1	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	8	2	no	—
98	C₂×C₄×C₁₉	—	8	2	no	—
99	C₂×C₄×C₁₉	—	8	2	no	—
100	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#135	1	2	yes	—
107	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₁₉	—	2	1	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	—	2	1	no	—
112	C₈×C₁₉	—	4	2	no	—
113	C₈×C₁₉	—	8	2	no	—
114	#2931	C₂×C₁₉	1	1	yes	characteristic
115	#1835	C₂×C₁₉	1	1	yes	characteristic, complemented
116	#1564	C₂×C₁₉	1	1	yes	characteristic, complemented
117	#1517	C₂×C₁₉	1	2	yes	—
118	#1286	C₂×C₁₉	1	2	yes	complemented
119	C₂²×C₁₉	—	2	1	no	—
120	C₂²×C₁₉	—	2	1	no	—
121	C₂²×C₁₉	—	4	2	no	—
122	C₂²×C₁₉	—	4	2	no	—
123	C₂²×C₁₉	—	4	2	no	—
124	C₂²×C₁₉	#380	1	1	yes	characteristic
125	C₂²×C₁₉	#463	1	2	yes	—
126	C₄×C₁₉	—	2	1	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	4	2	no	—
131	C₄×C₁₉	—	16	2	no	—
132	C₄×C₁₉	#416	1	1	yes	characteristic
133	C₄×C₁₉	#418	1	1	yes	characteristic
134	C₄×C₁₉	#478	1	2	yes	—
135	C₂×#131	—	2	1	no	—
136	#532	C₂²×C₁₉	1	1	yes	characteristic
137	#436	—	2	1	no	—
138	#436	C₄×C₁₉	1	2	yes	complemented
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	C₄×C₁₉	1	2	yes	complemented
141	#431	C₂²×C₁₉	1	2	yes	—
142	#423	C₂²×C₁₉	1	1	yes	characteristic
143	#408	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
145	#382	C₂²×C₁₉	1	2	yes	—
146	#340	—	2	1	no	—
147	#340	—	2	1	no	—
148	#335	—	2	1	no	—
149	#334	—	2	1	no	—
150	#327	—	2	4	no	—
151	C₂×C₁₉	—	2	2	no	—
152	C₂×C₁₉	—	8	2	no	—
153	C₂×C₁₉	#1804	1	1	yes	characteristic
154	C₂×C₁₉	#1812	1	1	yes	characteristic
155	C₂×C₁₉	#1814	1	1	yes	characteristic
156	C₂×#41	—	2	1	no	—
157	#131	—	4	4	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
163	C₄⋊Q₈	Q₈×C₁₉	1	1	yes	characteristic
164	#127	Q₈×C₁₉	1	1	yes	characteristic
165	#124	—	2	1	no	—
166	#121	—	2	1	no	—
167	#121	D₄×C₁₉	1	2	yes	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	C₂×C₄×C₁₉	1	2	yes	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
180	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	C₁₉	#8680	1	1	yes	characteristic, direct product factor, Sylow
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₄⋊C₄×C₁₉	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
197	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
198	M₁₆	—	4	4	no	—
199	M₁₆	—	4	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	C₄⋊C₄×C₁₉	1	2	yes	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
217	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
218	C₄²	#41×C₁₉	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	C₁₉×#121	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
236	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
237	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
238	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₄	C₁₉×#135	1	2	yes	—
241	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	8	2	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₁₉×#380	1	1	yes	characteristic
254	C₂²	C₁₉×#463	1	2	yes	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	16	2	no	—
261	C₄	C₁₉×#416	1	1	yes	characteristic
262	C₄	C₁₉×#418	1	1	yes	characteristic
263	C₄	C₁₉×#478	1	2	yes	—
264	C₂	—	2	2	no	—
265	C₂	—	8	2	no	—
266	C₂	C₁₉×#1804	1	1	yes	characteristic
267	C₂	C₁₉×#1812	1	1	yes	characteristic
268	C₂	C₁₉×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#380
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1835	C₂	semidirect
3. C₁₉×#1564	C₂	semidirect
5. C₁₉×#1286	C₂	semidirect
9. C₁₉×#436	C₄	semidirect
11. C₈×D₄×C₁₉	C₄	semidirect
84. #8680	C₁₉	direct (N × H)
115. #1835	C₂×C₁₉	semidirect
116. #1564	C₂×C₁₉	semidirect
118. #1286	C₂×C₁₉	semidirect
138. #436	C₄×C₁₉	semidirect
140. C₈×D₄	C₄×C₁₉	semidirect
185. C₁₉	#8680	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×#2931	2
C₁₉×#1835	2
C₁₉×#1564	2
C₁₉×#1517	2
C₁₉×#1286	2
#8680	19

Subgroups — C19×#8680 #18014639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8680 #18014639