Subgroups — C₁₉×#12870 #18018829

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2112	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1807	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1386	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
6	C₂×C₁₉×#117	—	2	1	no	—
7	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
8	C₁₉×#427	—	2	1	no	—
9	C₁₉×#427	—	2	1	no	—
10	C₁₉×#427	C₄	1	2	yes	complemented
11	C₁₉×#427	C₂²	1	2	yes	—
12	C₂×C₁₉×#107	C₄	1	1	yes	characteristic, complemented
13	C₂×C₁₉×#107	C₄	1	1	yes	characteristic, complemented
14	C₂×C₁₉×#106	—	2	1	no	—
15	C₂×C₁₉×#106	C₂²	1	2	yes	—
16	C₁₉×#380	—	2	1	no	—
17	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
18	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
19	C₁₉×#340	—	2	1	no	—
20	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	—	2	4	no	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	—	2	2	no	—
30	C₁₉×#117	—	4	1	no	—
31	C₁₉×#117	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
37	C₂×C₄²×C₁₉	—	4	1	no	—
38	C₁₉×M₃₂	—	2	4	no	—
39	C₂×C₁₆×C₁₉	—	2	2	no	—
40	C₁₉×#107	—	2	1	no	—
41	C₁₉×#107	—	2	1	no	—
42	C₁₉×#107	C₂×C₄	1	2	yes	—
43	C₁₉×#107	C₂×C₄	1	2	yes	—
44	C₁₉×#106	—	2	2	no	—
45	C₁₉×#106	—	2	2	no	—
46	C₁₉×#106	—	2	4	no	—
47	C₁₉×#95	—	4	1	no	—
48	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	4	1	no	—
52	C₂²×C₄×C₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	#41	1	4	yes	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	2	no	—
72	C₄⋊C₄×C₁₉	—	4	1	no	—
73	C₄⋊C₄×C₁₉	—	4	1	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₂²⋊C₄×C₁₉	—	4	2	no	—
77	C₂²⋊C₄×C₁₉	—	4	2	no	—
78	C₂²⋊C₄×C₁₉	—	8	2	no	—
79	C₄²×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	—	4	2	no	—
81	C₁₆×C₁₉	—	4	2	no	—
82	#12870	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₉	—	2	2	no	—
84	C₂³×C₁₉	—	2	2	no	—
85	C₂³×C₁₉	—	4	1	no	—
86	C₂³×C₁₉	#102	1	1	yes	characteristic
87	C₂³×C₁₉	#102	1	1	yes	characteristic
88	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	8	2	no	—
99	C₂×C₄×C₁₉	—	8	2	no	—
100	C₂×C₄×C₁₉	—	8	2	no	—
101	C₂×C₄×C₁₉	—	8	2	no	—
102	C₂×C₄×C₁₉	—	8	2	no	—
103	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#123	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#137	1	2	yes	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	#117	1	2	yes	—
112	C₈×C₁₉	#123	1	2	yes	—
113	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
114	#2112	C₂×C₁₉	1	1	yes	characteristic
115	#1807	C₂×C₁₉	1	1	yes	characteristic
116	#1386	C₂×C₁₉	1	4	yes	complemented
117	C₂²×C₁₉	—	2	1	no	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	—	2	1	no	—
120	C₂²×C₁₉	—	2	1	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	—	2	1	no	—
124	C₂²×C₁₉	—	4	4	no	—
125	C₂²×C₁₉	#360	1	2	yes	—
126	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
127	C₂²×C₁₉	#480	1	2	yes	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	8	2	no	—
131	C₄×C₁₉	—	8	2	no	—
132	C₄×C₁₉	—	16	2	no	—
133	C₄×C₁₉	#381	1	1	yes	characteristic
134	C₄×C₁₉	#419	1	1	yes	characteristic
135	C₄×C₁₉	#483	1	2	yes	—
136	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
137	C₂×#117	—	2	1	no	—
138	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
139	#427	—	2	1	no	—
140	#427	—	2	1	no	—
141	#427	C₄×C₁₉	1	2	yes	complemented
142	#427	C₂²×C₁₉	1	2	yes	—
143	C₂×#107	C₄×C₁₉	1	1	yes	characteristic, complemented
144	C₂×#107	C₄×C₁₉	1	1	yes	characteristic, complemented
145	C₂×#106	—	2	1	no	—
146	C₂×#106	C₂²×C₁₉	1	2	yes	—
147	#380	—	2	1	no	—
148	C₂²⋊C₁₆	—	2	1	no	—
149	C₂²⋊C₁₆	—	2	1	no	—
150	#340	—	2	1	no	—
151	C₂×C₁₉	—	2	2	no	—
152	C₂×C₁₉	—	2	2	no	—
153	C₂×C₁₉	—	4	1	no	—
154	C₂×C₁₉	#1823	1	1	yes	characteristic
155	C₂×C₁₉	C₂×#360	1	1	yes	characteristic
156	C₂×C₁₉	#2145	1	1	yes	characteristic
157	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
161	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	#117	—	4	1	no	—
168	#117	—	4	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	4	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₄²	—	4	1	no	—
175	M₃₂	—	2	4	no	—
176	C₂×C₁₆	—	2	2	no	—
177	#107	—	2	1	no	—
178	#107	—	2	1	no	—
179	#107	C₂×C₄×C₁₉	1	2	yes	—
180	#107	C₂×C₄×C₁₉	1	2	yes	—
181	#106	—	2	2	no	—
182	#106	—	2	2	no	—
183	#106	—	2	4	no	—
184	#95	—	4	1	no	—
185	C₁₉	#12870	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
194	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
195	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
196	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
205	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
207	C₂×C₈	#41×C₁₉	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	8	2	no	—
217	C₄²	—	4	2	no	—
218	C₄²	—	4	2	no	—
219	C₁₆	—	4	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₁₉×#102	1	1	yes	characteristic
224	C₂³	C₁₉×#102	1	1	yes	characteristic
225	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
241	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
243	C₂×C₄	C₁₉×#123	1	2	yes	—
244	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
245	C₂×C₄	C₁₉×#137	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	C₁₉×#117	1	2	yes	—
249	C₈	C₁₉×#123	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₁₉×#360	1	2	yes	—
259	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
260	C₂²	C₁₉×#480	1	2	yes	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	16	2	no	—
266	C₄	C₁₉×#381	1	1	yes	characteristic
267	C₄	C₁₉×#419	1	1	yes	characteristic
268	C₄	C₁₉×#483	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	—	4	1	no	—
272	C₂	C₁₉×#1823	1	1	yes	characteristic
273	C₂	C₂×C₁₉×#360	1	1	yes	characteristic
274	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	C₂⁴×C₁₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1386	C₂	semidirect
10. C₁₉×#427	C₄	semidirect
12. C₂×C₁₉×#107	C₄	semidirect
13. C₂×C₁₉×#107	C₄	semidirect
82. #12870	C₁₉	direct (N × H)
116. #1386	C₂×C₁₉	semidirect
141. #427	C₄×C₁₉	semidirect
143. C₂×#107	C₄×C₁₉	semidirect
144. C₂×#107	C₄×C₁₉	semidirect
185. C₁₉	#12870	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₉×#427	2
C₁₉×#2112	2
C₁₉×#1807	2
C₁₉×#1386	2
#12870	19

Subgroups — C19×#12870 #18018829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12870 #18018829