Subgroups — C₁₉×#22307 #18028266

Order 4864.

714 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1984	C₂	1	4	yes	maximal
3	C₁₉×#1939	C₂	1	2	yes	maximal
4	C₁₉×#1916	C₂	1	4	yes	maximal
5	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊C₄×C₁₉	C₂	1	2	yes	maximal
7	C₁₉×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
10	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₉	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#391	C₄	1	2	yes	—
16	C₁₉×#389	C₄	1	2	yes	—
17	C₁₉×#389	C₄	1	2	yes	—
18	C₁₉×#384	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
21	C₄³×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#336	C₂²	1	2	yes	—
23	C₁₉×#336	C₂²	1	2	yes	—
24	C₁₉×#336	C₂²	1	4	yes	—
25	C₁₉×#336	C₂²	1	4	yes	—
26	C₁₉×#336	C₂²	1	2	yes	—
27	C₁₉×#336	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
32	C₁₉×C₄⋊Q₈	C₈	1	8	yes	complemented
33	C₄×Q₈×C₁₉	—	4	4	no	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×C₄⋊C₈	—	2	4	no	—
49	C₁₉×C₄⋊C₈	—	4	4	no	—
50	C₄×C₈×C₁₉	—	2	2	no	—
51	C₄×C₈×C₁₉	—	4	4	no	—
52	C₄×C₈×C₁₉	Q₈	1	4	yes	complemented
53	C₁₉×#95	C₂×C₄	1	4	yes	—
54	C₁₉×#95	C₂×C₄	1	4	yes	—
55	C₂×Q₈×C₁₉	—	4	2	no	—
56	C₂×Q₈×C₁₉	—	4	2	no	—
57	C₂×Q₈×C₁₉	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄×C₁₉	—	2	4	no	—
77	C₄⋊C₄×C₁₉	—	4	2	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	4	no	—
80	C₄⋊C₄×C₁₉	C₂×C₈	1	8	yes	—
81	C₄⋊C₄×C₁₉	C₂×C₈	1	8	yes	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	4	no	—
85	C₄²×C₁₉	—	2	4	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	C₂×C₈	1	4	yes	—
90	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₁₉	#41	1	2	yes	—
92	#22307	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₁₉	—	4	8	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²×C₈	1	8	yes	—
117	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₉	#131	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
123	C₈×C₁₉	—	2	4	no	—
124	C₈×C₁₉	—	4	2	no	—
125	C₈×C₁₉	—	4	4	no	—
126	C₄×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
127	#1984	C₂×C₁₉	1	4	yes	—
128	#1939	C₂×C₁₉	1	2	yes	—
129	#1916	C₂×C₁₉	1	4	yes	—
130	#1831	C₂×C₁₉	1	1	yes	characteristic
131	C₈×C₄⋊C₄	C₂×C₁₉	1	2	yes	—
132	#1744	C₂×C₁₉	1	1	yes	characteristic
133	C₂²×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂²×C₁₉	#518	1	1	yes	characteristic
135	C₂²×C₁₉	#519	1	1	yes	characteristic
136	C₂²×C₁₉	C₂³×C₈	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×#131	1	1	yes	characteristic
138	C₂²×C₁₉	#568	1	1	yes	characteristic
139	C₂²×C₁₉	#568	1	1	yes	characteristic
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	2	8	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	4	1	no	—
145	C₄×C₁₉	—	4	4	no	—
146	C₄×C₁₉	—	4	1	no	—
147	C₄×C₁₉	C₈×Q₈	1	4	yes	—
148	C₄×C₁₉	#542	1	2	yes	—
149	C₄×C₁₉	#544	1	2	yes	—
150	C₂×C₄⋊Q₈	C₄×C₁₉	1	1	yes	characteristic
151	C₂×C₄×Q₈	C₂²×C₁₉	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
154	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₉	1	4	yes	—
156	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
157	#391	C₄×C₁₉	1	2	yes	—
158	#389	C₄×C₁₉	1	2	yes	—
159	#389	C₄×C₁₉	1	2	yes	—
160	#384	C₄×C₁₉	1	1	yes	characteristic
161	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
163	C₄³	C₂²×C₁₉	1	1	yes	characteristic
164	#336	C₂²×C₁₉	1	2	yes	—
165	#336	C₂²×C₁₉	1	2	yes	—
166	#336	C₂²×C₁₉	1	2	yes	—
167	#336	C₂²×C₁₉	1	4	yes	—
168	#336	C₂²×C₁₉	1	4	yes	—
169	#336	C₂²×C₁₉	1	2	yes	—
170	C₂×C₁₉	#2331	1	1	yes	characteristic
171	C₂×C₁₉	C₂×C₈×Q₈	1	1	yes	characteristic
172	C₂×C₁₉	#2961	1	1	yes	characteristic
173	C₂×C₁₉	#2972	1	1	yes	characteristic
174	C₂×C₁₉	#2981	1	1	yes	characteristic
175	C₂×C₁₉	#2982	1	1	yes	characteristic
176	C₂×C₁₉	#2985	1	1	yes	characteristic
177	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
180	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
181	C₄⋊Q₈	C₈×C₁₉	1	8	yes	complemented
182	C₄×Q₈	—	4	4	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
189	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
195	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	4	4	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	4	4	no	—
201	C₄×C₈	Q₈×C₁₉	1	4	yes	complemented
202	#95	C₂×C₄×C₁₉	1	4	yes	—
203	#95	C₂×C₄×C₁₉	1	4	yes	—
204	C₁₉	#22307	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	C₂×C₈×C₁₉	1	8	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	C₂×C₈×C₁₉	1	8	yes	—
231	C₄⋊C₄	C₂×C₈×C₁₉	1	8	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×C₈×C₁₉	1	4	yes	—
240	C₄²	C₂×Q₈×C₁₉	1	2	yes	—
241	C₄²	#41×C₁₉	1	2	yes	—
242	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
262	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
263	C₂×C₄	C₂²×C₈×C₁₉	1	2	yes	—
264	C₂×C₄	C₂²×C₈×C₁₉	1	2	yes	—
265	C₂×C₄	C₂²×C₈×C₁₉	1	8	yes	—
266	C₂×C₄	C₂²×C₈×C₁₉	1	2	yes	—
267	C₂×C₄	C₁₉×#131	1	2	yes	—
268	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
270	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
271	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	4	2	no	—
274	C₈	—	4	4	no	—
275	C₂²	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
276	C₂²	C₁₉×#518	1	1	yes	characteristic
277	C₂²	C₁₉×#519	1	1	yes	characteristic
278	C₂²	C₂³×C₈×C₁₉	1	1	yes	characteristic
279	C₂²	C₂×C₁₉×#131	1	1	yes	characteristic
280	C₂²	C₁₉×#568	1	1	yes	characteristic
281	C₂²	C₁₉×#568	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	8	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	1	no	—
289	C₄	C₈×Q₈×C₁₉	1	4	yes	—
290	C₄	C₁₉×#542	1	2	yes	—
291	C₄	C₁₉×#544	1	2	yes	—
292	C₂	C₁₉×#2331	1	1	yes	characteristic
293	C₂	C₂×C₈×Q₈×C₁₉	1	1	yes	characteristic
294	C₂	C₁₉×#2961	1	1	yes	characteristic
295	C₂	C₁₉×#2972	1	1	yes	characteristic
296	C₂	C₁₉×#2981	1	1	yes	characteristic
297	C₂	C₁₉×#2982	1	1	yes	characteristic
298	C₂	C₁₉×#2985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle, prime radical	C₂³×C₁₉	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
32. C₁₉×C₄⋊Q₈	C₈	semidirect
52. C₄×C₈×C₁₉	Q₈	semidirect
92. #22307	C₁₉	direct (N × H)
181. C₄⋊Q₈	C₈×C₁₉	semidirect
201. C₄×C₈	Q₈×C₁₉	semidirect
204. C₁₉	#22307	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₉×C₄⋊Q₈	2
C₁₉×#1984	2
C₁₉×#1939	2
C₁₉×#1916	2
C₁₉×#1831	2
C₈×C₄⋊C₄×C₁₉	2
C₁₉×#1744	2
#22307	19

Subgroups — C19×#22307 #18028266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#22307 #18028266