Subgroups — C₁₉×#12731 #18018690

Order 4864.

1190 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#357	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1808	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₁₉	—	2	1	no	—
6	C₂²×D₈×C₁₉	—	2	8	no	—
7	C₂²×D₈×C₁₉	—	2	1	no	—
8	C₂²×D₈×C₁₉	—	2	2	no	—
9	C₂²×D₈×C₁₉	C₄	1	4	yes	complemented
10	C₂²×D₈×C₁₉	C₂²	1	2	yes	—
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	complemented
13	C₂×C₁₉×#107	C₂²	1	2	yes	complemented
14	C₁₉×#380	—	2	1	no	—
15	C₁₉×#357	—	2	8	no	—
16	C₁₉×C₂²⋊C₁₆	—	2	2	no	—
17	C₁₉×#340	—	2	1	no	—
18	C₂⁵×C₁₉	—	4	1	no	—
19	C₂²×D₄×C₁₉	—	2	8	no	—
20	C₂²×D₄×C₁₉	—	2	4	no	—
21	C₂²×D₄×C₁₉	—	2	2	no	—
22	C₂²×D₄×C₁₉	—	4	4	no	—
23	C₂²×D₄×C₁₉	—	4	1	no	—
24	C₂²×D₄×C₁₉	—	4	2	no	—
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂×D₈×C₁₉	—	2	8	no	—
27	C₂×D₈×C₁₉	—	2	16	no	—
28	C₂×D₈×C₁₉	—	2	16	no	—
29	C₂×D₈×C₁₉	—	2	4	no	—
30	C₂×D₈×C₁₉	—	2	4	no	—
31	C₂×D₈×C₁₉	—	2	2	no	—
32	C₂×D₈×C₁₉	C₂×C₄	1	4	yes	—
33	C₂×D₈×C₁₉	D₄	1	8	yes	—
34	C₂²×C₈×C₁₉	—	2	4	no	—
35	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	D₄	1	2	yes	—
38	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
41	C₂×C₁₆×C₁₉	—	2	2	no	—
42	C₂×C₁₆×C₁₉	—	2	4	no	—
43	C₁₉×#107	—	2	4	no	—
44	C₁₉×#95	—	4	1	no	—
45	C₂⁴×C₁₉	—	4	8	no	—
46	C₂⁴×C₁₉	—	4	4	no	—
47	C₂⁴×C₁₉	—	4	2	no	—
48	C₂⁴×C₁₉	—	8	4	no	—
49	C₂⁴×C₁₉	—	8	2	no	—
50	C₂⁴×C₁₉	—	8	2	no	—
51	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
52	C₂×D₄×C₁₉	—	2	16	no	—
53	C₂×D₄×C₁₉	—	2	4	no	—
54	C₂×D₄×C₁₉	—	2	8	no	—
55	C₂×D₄×C₁₉	—	4	8	no	—
56	C₂×D₄×C₁₉	—	4	8	no	—
57	C₂×D₄×C₁₉	—	4	16	no	—
58	C₂×D₄×C₁₉	—	4	4	no	—
59	C₂×D₄×C₁₉	—	4	2	no	—
60	C₂×D₄×C₁₉	—	4	4	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	D₈×C₁₉	—	2	16	no	—
69	D₈×C₁₉	—	2	4	no	—
70	D₈×C₁₉	—	2	8	no	—
71	D₈×C₁₉	C₂²⋊C₄	1	8	yes	complemented
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₂²⋊C₄×C₁₉	—	8	2	no	—
82	C₁₆×C₁₉	—	4	2	no	—
83	#12731	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₁₉	—	2	2	no	—
85	C₂³×C₁₉	—	2	2	no	—
86	C₂³×C₁₉	—	4	16	no	—
87	C₂³×C₁₉	—	4	4	no	—
88	C₂³×C₁₉	—	4	8	no	—
89	C₂³×C₁₉	—	8	8	no	—
90	C₂³×C₁₉	—	8	16	no	—
91	C₂³×C₁₉	—	8	16	no	—
92	C₂³×C₁₉	—	8	8	no	—
93	C₂³×C₁₉	—	8	2	no	—
94	C₂³×C₁₉	—	8	2	no	—
95	C₂³×C₁₉	—	8	4	no	—
96	C₂³×C₁₉	#102	1	2	yes	—
97	C₂³×C₁₉	D₁₆	1	2	yes	—
98	C₂³×C₁₉	SD₃₂	1	2	yes	—
99	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
100	D₄×C₁₉	—	2	8	no	—
101	D₄×C₁₉	—	4	16	no	—
102	D₄×C₁₉	—	4	4	no	—
103	D₄×C₁₉	—	4	8	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	8	2	no	—
109	C₂×C₄×C₁₉	—	8	2	no	—
110	C₂×C₄×C₁₉	—	8	2	no	—
111	C₂×C₄×C₁₉	#102	1	2	yes	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#120	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
116	C₈×C₁₉	—	2	2	no	—
117	C₈×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
118	C₈×C₁₉	#120	1	2	yes	—
119	C₂³×D₈	C₂×C₁₉	1	1	yes	characteristic
120	C₂×#357	C₂×C₁₉	1	4	yes	complemented
121	C₂×C₂²⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
122	#1808	C₂×C₁₉	1	1	yes	characteristic, complemented
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	4	no	—
125	C₂²×C₁₉	—	2	4	no	—
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	4	8	no	—
128	C₂²×C₁₉	—	8	32	no	—
129	C₂²×C₁₉	—	8	4	no	—
130	C₂²×C₁₉	—	8	4	no	—
131	C₂²×C₁₉	—	8	8	no	—
132	C₂²×C₁₉	—	8	8	no	—
133	C₂²×C₁₉	#357	1	4	yes	complemented
134	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₉	#447	1	2	yes	—
136	C₂²×C₁₉	C₂×D₁₆	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×SD₃₂	1	1	yes	characteristic
138	C₂²×C₁₉	#509	1	2	yes	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	16	2	no	—
141	C₄×C₁₉	#379	1	1	yes	characteristic
142	C₄×C₁₉	#418	1	1	yes	characteristic
143	C₄×C₁₉	#450	1	2	yes	—
144	C₂³×D₄	—	2	1	no	—
145	C₂²×D₈	—	2	8	no	—
146	C₂²×D₈	—	2	1	no	—
147	C₂²×D₈	—	2	2	no	—
148	C₂²×D₈	C₄×C₁₉	1	4	yes	complemented
149	C₂²×D₈	C₂²×C₁₉	1	2	yes	—
150	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	complemented
152	C₂×#107	C₂²×C₁₉	1	2	yes	complemented
153	#380	—	2	1	no	—
154	#357	—	2	8	no	—
155	C₂²⋊C₁₆	—	2	2	no	—
156	#340	—	2	1	no	—
157	C₂×C₁₉	—	2	4	no	—
158	C₂×C₁₉	—	8	8	no	—
159	C₂×C₁₉	#1786	1	1	yes	characteristic
160	C₂×C₁₉	C₂×#357	1	1	yes	characteristic
161	C₂×C₁₉	#2140	1	1	yes	characteristic
162	C₂×C₁₉	#2184	1	2	yes	—
163	C₂×C₁₉	#2187	1	2	yes	—
164	C₂⁵	—	4	1	no	—
165	C₂²×D₄	—	2	8	no	—
166	C₂²×D₄	—	2	4	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	4	4	no	—
169	C₂²×D₄	—	4	1	no	—
170	C₂²×D₄	—	4	2	no	—
171	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
172	C₂×D₈	—	2	8	no	—
173	C₂×D₈	—	2	16	no	—
174	C₂×D₈	—	2	16	no	—
175	C₂×D₈	—	2	4	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	C₂×C₄×C₁₉	1	4	yes	—
179	C₂×D₈	D₄×C₁₉	1	8	yes	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
182	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₁₉	1	2	yes	—
184	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	4	1	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	2	4	no	—
189	#107	—	2	4	no	—
190	#95	—	4	1	no	—
191	C₁₉	#12731	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	—	4	8	no	—
193	C₂⁴	—	4	2	no	—
194	C₂⁴	—	4	4	no	—
195	C₂⁴	—	8	4	no	—
196	C₂⁴	—	8	2	no	—
197	C₂⁴	—	8	2	no	—
198	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
199	C₂×D₄	—	2	16	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	8	no	—
202	C₂×D₄	—	4	8	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂×D₄	—	4	16	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
212	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
213	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
215	D₈	—	2	16	no	—
216	D₈	—	2	4	no	—
217	D₈	—	2	8	no	—
218	D₈	C₂²⋊C₄×C₁₉	1	8	yes	complemented
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
225	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	8	2	no	—
229	C₁₆	—	4	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	4	16	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	8	no	—
235	C₂³	—	8	16	no	—
236	C₂³	—	8	16	no	—
237	C₂³	—	8	8	no	—
238	C₂³	—	8	8	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₁₉×#102	1	2	yes	—
243	C₂³	C₁₉×D₁₆	1	2	yes	—
244	C₂³	C₁₉×SD₃₂	1	2	yes	—
245	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
246	D₄	—	2	8	no	—
247	D₄	—	4	16	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₁₉×#102	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
259	C₂×C₄	C₁₉×#120	1	2	yes	—
260	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
261	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
262	C₈	—	2	2	no	—
263	C₈	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
264	C₈	C₁₉×#120	1	2	yes	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	8	no	—
270	C₂²	—	8	32	no	—
271	C₂²	—	8	8	no	—
272	C₂²	—	8	8	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	C₁₉×#357	1	4	yes	complemented
276	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
277	C₂²	C₁₉×#447	1	2	yes	—
278	C₂²	C₂×C₁₉×D₁₆	1	1	yes	characteristic
279	C₂²	C₂×C₁₉×SD₃₂	1	1	yes	characteristic
280	C₂²	C₁₉×#509	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	16	2	no	—
283	C₄	C₁₉×#379	1	1	yes	characteristic
284	C₄	C₁₉×#418	1	1	yes	characteristic
285	C₄	C₁₉×#450	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	—	8	8	no	—
288	C₂	C₁₉×#1786	1	1	yes	characteristic
289	C₂	C₂×C₁₉×#357	1	1	yes	characteristic
290	C₂	C₁₉×#2140	1	1	yes	characteristic
291	C₂	C₁₉×#2184	1	2	yes	—
292	C₂	C₁₉×#2187	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×D₈×C₁₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#357	C₂	semidirect
3. C₂×C₁₉×C₂²⋊C₁₆	C₂	semidirect
4. C₁₉×#1808	C₂	semidirect
9. C₂²×D₈×C₁₉	C₄	semidirect
12. C₂²×C₁₆×C₁₉	C₂²	semidirect
13. C₂×C₁₉×#107	C₂²	semidirect
71. D₈×C₁₉	C₂²⋊C₄	semidirect
83. #12731	C₁₉	direct (N × H)
120. C₂×#357	C₂×C₁₉	semidirect
121. C₂×C₂²⋊C₁₆	C₂×C₁₉	semidirect
122. #1808	C₂×C₁₉	semidirect
133. C₂²×C₁₉	#357	semidirect
148. C₂²×D₈	C₄×C₁₉	semidirect
151. C₂²×C₁₆	C₂²×C₁₉	semidirect
152. C₂×#107	C₂²×C₁₉	semidirect
191. C₁₉	#12731	direct (N × H)
218. D₈	C₂²⋊C₄×C₁₉	semidirect
275. C₂²	C₁₉×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₁₉	2
C₂×C₁₉×#357	2
C₂×C₁₉×C₂²⋊C₁₆	2
C₁₉×#1808	2
#12731	19

Subgroups — C19×#12731 #18018690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12731 #18018690