Subgroups — C₁₉×#8697 #18014656

Order 4864.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2966	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1509	C₂	1	2	yes	maximal
5	C₁₉×#1285	C₂	1	2	yes	maximal, complemented
6	C₁₉×#529	C₂²	1	1	yes	characteristic
7	C₁₉×#446	C₄	1	1	yes	characteristic, complemented
8	C₈×Q₈×C₁₉	C₄	1	1	yes	characteristic, complemented
9	C₁₉×#436	C₄	1	1	yes	characteristic, complemented
10	C₁₉×#435	—	2	1	no	—
11	C₈×D₄×C₁₉	—	2	1	no	—
12	C₈×D₄×C₁₉	C₄	1	1	yes	characteristic, complemented
13	C₁₉×#433	—	2	1	no	—
14	C₁₉×#432	C₂²	1	2	yes	—
15	C₁₉×#432	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#107	—	2	1	no	—
17	C₁₉×#405	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₁₉×#340	—	2	1	no	—
21	C₁₉×#335	—	2	1	no	—
22	C₁₉×#334	—	2	1	no	—
23	C₁₉×#328	—	2	2	no	—
24	C₁₉×#327	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₁₉×#126	—	2	1	no	—
30	C₁₉×#125	Q₈	1	1	yes	characteristic
31	C₁₉×#124	Q₈	1	1	yes	characteristic
32	C₁₉×#123	—	2	1	no	—
33	C₁₉×C₂²⋊Q₈	D₄	1	1	yes	characteristic
34	C₁₉×#121	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₁₉	—	2	1	no	—
37	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₁₉×#117	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₄²×C₁₉	—	2	2	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₁₉×#107	—	2	4	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	—	2	2	no	—
46	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₁₉×C₂²⋊C₈	—	2	1	no	—
50	C₁₉×C₂²⋊C₈	—	2	1	no	—
51	C₁₉×C₂²⋊C₈	—	2	2	no	—
52	C₁₉×#97	—	2	1	no	—
53	C₁₉×#97	—	2	1	no	—
54	C₁₉×#97	—	2	2	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
60	C₁₉×#95	—	2	2	no	—
61	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	4	2	no	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
69	M₁₆×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	4	no	—
85	C₄⋊C₄×C₁₉	—	4	2	no	—
86	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₂²⋊C₄×C₁₉	—	2	1	no	—
92	C₂²⋊C₄×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	—	8	2	no	—
95	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₁₉	#41	1	1	yes	characteristic
99	#8697	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	#121	1	1	yes	characteristic
102	Q₈×C₁₉	—	4	1	no	—
103	D₄×C₁₉	—	4	1	no	—
104	D₄×C₁₉	—	4	1	no	—
105	D₄×C₁₉	—	4	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	—	8	2	no	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₁₉	—	2	1	no	—
126	C₈×C₁₉	—	2	1	no	—
127	C₈×C₁₉	—	2	1	no	—
128	C₈×C₁₉	—	4	1	no	—
129	C₈×C₁₉	—	4	1	no	—
130	C₈×C₁₉	—	8	2	no	—
131	C₈×C₁₉	#104	1	2	yes	—
132	#2966	C₂×C₁₉	1	1	yes	characteristic
133	C₄×#107	C₂×C₁₉	1	1	yes	characteristic, complemented
134	#1568	C₂×C₁₉	1	1	yes	characteristic, complemented
135	#1509	C₂×C₁₉	1	2	yes	—
136	#1285	C₂×C₁₉	1	2	yes	complemented
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	#380	1	1	yes	characteristic
143	C₂²×C₁₉	#472	1	1	yes	characteristic
144	C₂²×C₁₉	#473	1	1	yes	characteristic
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	2	1	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	4	1	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	8	4	no	—
154	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
155	C₄×C₁₉	#421	1	1	yes	characteristic
156	#529	C₂²×C₁₉	1	1	yes	characteristic
157	#446	C₄×C₁₉	1	1	yes	characteristic, complemented
158	C₈×Q₈	C₄×C₁₉	1	1	yes	characteristic, complemented
159	#436	C₄×C₁₉	1	1	yes	characteristic, complemented
160	#435	—	2	1	no	—
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	C₄×C₁₉	1	1	yes	characteristic, complemented
163	#433	—	2	1	no	—
164	#432	C₂²×C₁₉	1	2	yes	—
165	#432	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×#107	—	2	1	no	—
167	#405	—	2	1	no	—
168	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
169	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
170	#340	—	2	1	no	—
171	#335	—	2	1	no	—
172	#334	—	2	1	no	—
173	#328	—	2	2	no	—
174	#327	—	2	2	no	—
175	C₂×C₁₉	—	2	1	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	8	1	no	—
178	C₂×C₁₉	#1806	1	1	yes	characteristic
179	C₂×C₁₉	#1812	1	1	yes	characteristic
180	C₂×C₁₉	#1816	1	1	yes	characteristic
181	C₂×M₁₆	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
185	#126	—	2	1	no	—
186	#125	Q₈×C₁₉	1	1	yes	characteristic
187	#124	Q₈×C₁₉	1	1	yes	characteristic
188	#123	—	2	1	no	—
189	C₂²⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
190	#121	D₄×C₁₉	1	1	yes	characteristic
191	C₄×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
198	#107	—	2	4	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
203	C₄⋊C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	2	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
213	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
214	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
215	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
216	#95	—	2	2	no	—
217	C₁₉	#8697	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×Q₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
244	C₄⋊C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
245	C₂²⋊C₄	—	2	1	no	—
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	8	2	no	—
252	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
254	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
255	C₄²	#41×C₁₉	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	C₁₉×#121	1	1	yes	characteristic
258	Q₈	—	4	1	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	1	no	—
261	D₄	—	4	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
277	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
278	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
279	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	8	2	no	—
287	C₈	C₁₉×#104	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	C₁₉×#380	1	1	yes	characteristic
294	C₂²	C₁₉×#472	1	1	yes	characteristic
295	C₂²	C₁₉×#473	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	8	4	no	—
305	C₄	C₂×C₁₉×#104	1	1	yes	characteristic
306	C₄	C₁₉×#421	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	8	1	no	—
310	C₂	C₁₉×#1806	1	1	yes	characteristic
311	C₂	C₁₉×#1812	1	1	yes	characteristic
312	C₂	C₁₉×#1816	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×#121	D₄
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. C₄×C₁₉×#107	C₂	semidirect
3. C₁₉×#1568	C₂	semidirect
5. C₁₉×#1285	C₂	semidirect
7. C₁₉×#446	C₄	semidirect
8. C₈×Q₈×C₁₉	C₄	semidirect
9. C₁₉×#436	C₄	semidirect
12. C₈×D₄×C₁₉	C₄	semidirect
99. #8697	C₁₉	direct (N × H)
133. C₄×#107	C₂×C₁₉	semidirect
134. #1568	C₂×C₁₉	semidirect
136. #1285	C₂×C₁₉	semidirect
157. #446	C₄×C₁₉	semidirect
158. C₈×Q₈	C₄×C₁₉	semidirect
159. #436	C₄×C₁₉	semidirect
162. C₈×D₄	C₄×C₁₉	semidirect
217. C₁₉	#8697	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×#2966	2
C₄×C₁₉×#107	2
C₁₉×#1568	2
C₁₉×#1509	2
C₁₉×#1285	2
#8697	19

Subgroups — C19×#8697 #18014656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8697 #18014656