Subgroups — C₁₉×#33664 #18039623

Order 4864.

546 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2236	C₂	1	2	yes	maximal
4	C₁₉×#2227	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2209	C₂	1	2	yes	maximal, complemented
6	C₁₉×#2188	C₂	1	2	yes	maximal
7	C₄×C₁₉×Q₃₂	C₂	1	4	yes	maximal, complemented
8	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#533	—	2	1	no	—
11	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
12	C₂×C₁₉×Q₃₂	C₂²	1	4	yes	—
13	C₁₉×#500	C₂²	1	1	yes	characteristic
14	C₁₉×#494	C₂²	1	1	yes	characteristic
15	C₁₉×#487	—	2	1	no	—
16	C₁₉×#467	—	2	1	no	—
17	C₁₉×#467	C₂²	1	2	yes	—
18	C₄×Q₁₆×C₁₉	—	2	1	no	—
19	C₄×Q₁₆×C₁₉	—	2	1	no	—
20	C₄×Q₁₆×C₁₉	—	2	1	no	—
21	C₄×Q₁₆×C₁₉	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₁₉	C₂²	1	2	yes	—
23	C₄×Q₁₆×C₁₉	C₂²	1	2	yes	—
24	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×#103	—	2	1	no	—
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#366	C₂²	1	4	yes	—
28	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₉×#358	C₂²	1	4	yes	—
30	C₁₉×#358	C₂²	1	4	yes	—
31	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₂×Q₁₆×C₁₉	—	2	4	no	—
35	C₂×Q₁₆×C₁₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₉	—	2	1	no	—
37	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	—	2	1	no	—
43	C₁₉×#125	—	4	1	no	—
44	C₁₉×C₂²⋊Q₈	—	2	2	no	—
45	C₁₉×C₂²⋊Q₈	—	4	1	no	—
46	C₄×Q₈×C₁₉	—	2	2	no	—
47	C₄×Q₈×C₁₉	—	2	2	no	—
48	C₄×Q₈×C₁₉	—	2	2	no	—
49	C₄×Q₈×C₁₉	—	4	1	no	—
50	C₄×Q₈×C₁₉	—	4	1	no	—
51	C₄×Q₈×C₁₉	—	4	1	no	—
52	C₁₉×#117	—	4	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
55	C₂×C₄²×C₁₉	—	4	1	no	—
56	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
57	C₁₉×Q₃₂	—	2	8	no	—
58	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
59	C₁₉×#107	—	2	1	no	—
60	C₁₉×#107	—	2	1	no	—
61	C₁₉×#107	C₂³	1	2	yes	—
62	C₁₉×#107	C₂³	1	2	yes	—
63	C₁₉×#103	—	2	2	no	—
64	C₁₉×#103	—	2	2	no	—
65	C₁₉×#103	—	2	2	no	—
66	C₁₉×#103	—	2	2	no	—
67	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₂×Q₈×C₁₉	—	2	4	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	4	2	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂²×C₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₁₉	—	2	4	no	—
83	Q₁₆×C₁₉	—	2	2	no	—
84	Q₁₆×C₁₉	—	2	4	no	—
85	Q₁₆×C₁₉	—	4	2	no	—
86	Q₁₆×C₁₉	#41	1	4	yes	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	4	2	no	—
97	C₄⋊C₄×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	2	no	—
101	C₄⋊C₄×C₁₉	—	4	2	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₂²⋊C₄×C₁₉	—	4	2	no	—
104	C₄²×C₁₉	—	4	2	no	—
105	C₄²×C₁₉	—	4	2	no	—
106	C₄²×C₁₉	—	4	2	no	—
107	C₄²×C₁₉	D₈	1	1	yes	characteristic
108	C₄²×C₁₉	D₈	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₁₆×C₁₉	—	2	4	no	—
112	#33664	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₁₉	—	2	4	no	—
115	Q₈×C₁₉	—	4	4	no	—
116	Q₈×C₁₉	—	4	2	no	—
117	Q₈×C₁₉	—	4	2	no	—
118	Q₈×C₁₉	—	4	2	no	—
119	Q₈×C₁₉	—	8	2	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	Q₃₂	1	4	yes	complemented
130	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	—	2	1	no	—
139	C₈×C₁₉	C₂×#41	1	2	yes	—
140	#3170	C₂×C₁₉	1	1	yes	characteristic
141	C₂×C₄×Q₁₆	C₂×C₁₉	1	1	yes	characteristic
142	#2236	C₂×C₁₉	1	2	yes	—
143	#2227	C₂×C₁₉	1	2	yes	complemented
144	#2209	C₂×C₁₉	1	2	yes	complemented
145	#2188	C₂×C₁₉	1	2	yes	—
146	C₄×Q₃₂	C₂×C₁₉	1	4	yes	complemented
147	#2162	C₂×C₁₉	1	1	yes	characteristic
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	C₂×Q₃₂	1	2	yes	—
151	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	4	4	no	—
156	C₄×C₁₉	—	8	2	no	—
157	C₄×C₁₉	—	8	2	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	C₂×Q₃₂	1	2	yes	—
160	C₄×C₁₉	C₂×Q₃₂	1	2	yes	—
161	C₄×C₁₉	#525	1	1	yes	characteristic
162	C₄×C₁₉	#575	1	1	yes	characteristic
163	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
164	#533	—	2	1	no	—
165	C₂×C₄×Q₈	—	2	1	no	—
166	C₂×Q₃₂	C₂²×C₁₉	1	4	yes	—
167	#500	C₂²×C₁₉	1	1	yes	characteristic
168	#494	C₂²×C₁₉	1	1	yes	characteristic
169	#487	—	2	1	no	—
170	#467	—	2	1	no	—
171	#467	C₂²×C₁₉	1	2	yes	—
172	C₄×Q₁₆	—	2	1	no	—
173	C₄×Q₁₆	—	2	1	no	—
174	C₄×Q₁₆	—	2	1	no	—
175	C₄×Q₁₆	C₂²×C₁₉	1	2	yes	—
176	C₄×Q₁₆	C₂²×C₁₉	1	2	yes	—
177	C₄×Q₁₆	C₂²×C₁₉	1	2	yes	—
178	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×#103	—	2	1	no	—
180	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
181	#366	C₂²×C₁₉	1	4	yes	—
182	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
183	#358	C₂²×C₁₉	1	4	yes	—
184	#358	C₂²×C₁₉	1	4	yes	—
185	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
186	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
187	C₂×C₁₉	—	2	2	no	—
188	C₂×C₁₉	#3100	1	1	yes	characteristic
189	C₂×C₁₉	C₂²×Q₃₂	1	1	yes	characteristic
190	C₂×C₁₉	#3414	1	1	yes	characteristic
191	C₂²×Q₈	—	2	1	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
196	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
197	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	#125	—	4	1	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	4	1	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	4	1	no	—
208	C₄×Q₈	—	4	1	no	—
209	C₄×Q₈	—	4	1	no	—
210	#117	—	4	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	4	1	no	—
213	C₂×C₄²	—	4	1	no	—
214	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
215	Q₃₂	—	2	8	no	—
216	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	C₂³×C₁₉	1	2	yes	—
220	#107	C₂³×C₁₉	1	2	yes	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
226	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
229	C₁₉	#33664	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
239	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	4	no	—
244	Q₁₆	—	4	2	no	—
245	Q₁₆	#41×C₁₉	1	4	yes	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
251	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	D₈×C₁₉	1	1	yes	characteristic
267	C₄²	D₈×C₁₉	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
270	C₁₆	—	2	4	no	—
271	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	C₁₉×Q₃₂	1	4	yes	complemented
288	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	C₂×#41×C₁₉	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	C₂×C₁₉×Q₃₂	1	2	yes	—
301	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×C₁₉×Q₃₂	1	2	yes	—
310	C₄	C₂×C₁₉×Q₃₂	1	2	yes	—
311	C₄	C₁₉×#525	1	1	yes	characteristic
312	C₄	C₁₉×#575	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₁₉×#3100	1	1	yes	characteristic
315	C₂	C₂²×C₁₉×Q₃₂	1	1	yes	characteristic
316	C₂	C₁₉×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×D₈
prime radical	C₂³×C₁₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
4. C₁₉×#2227	C₂	semidirect
5. C₁₉×#2209	C₂	semidirect
7. C₄×C₁₉×Q₃₂	C₂	semidirect
112. #33664	C₁₉	direct (N × H)
129. C₂×C₄×C₁₉	Q₃₂	semidirect
143. #2227	C₂×C₁₉	semidirect
144. #2209	C₂×C₁₉	semidirect
146. C₄×Q₃₂	C₂×C₁₉	semidirect
229. C₁₉	#33664	direct (N × H)
287. C₂×C₄	C₁₉×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3170	2
C₂×C₄×Q₁₆×C₁₉	2
C₁₉×#2236	2
C₁₉×#2227	2
C₁₉×#2209	2
C₁₉×#2188	2
C₄×C₁₉×Q₃₂	2
C₁₉×#2162	2
#33664	19

Subgroups — C19×#33664 #18039623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33664 #18039623