Subgroups — C₁₉×#20599 #18026558

Order 4864.

3838 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#419	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×#407	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2280	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1787	C₂	1	8	yes	maximal, complemented
6	C₂³×Q₈×C₁₉	—	2	16	no	—
7	C₂³×Q₈×C₁₉	—	2	4	no	—
8	C₂³×Q₈×C₁₉	—	2	2	no	—
9	C₂³×Q₈×C₁₉	—	2	1	no	—
10	C₂³×Q₈×C₁₉	C₄	1	8	yes	complemented
11	C₂³×Q₈×C₁₉	C₂²	1	4	yes	—
12	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₁₉	—	2	1	no	—
14	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂×C₁₉×#117	C₂²	1	2	yes	—
17	C₁₉×#419	—	2	16	no	—
18	C₂×C₁₉×#103	C₂²	1	16	yes	complemented
19	C₁₉×#407	—	2	4	no	—
20	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₁₉×#381	—	2	1	no	—
22	C₁₉×#380	C₂²	1	4	yes	—
23	C₁₉×#379	—	2	1	no	—
24	C₄×C₂²⋊C₄×C₁₉	—	2	2	no	—
25	C₂⁵×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₁₉	—	2	32	no	—
27	C₂²×Q₈×C₁₉	—	2	64	no	—
28	C₂²×Q₈×C₁₉	—	2	16	no	—
29	C₂²×Q₈×C₁₉	—	2	64	no	—
30	C₂²×Q₈×C₁₉	—	2	32	no	—
31	C₂²×Q₈×C₁₉	—	2	16	no	—
32	C₂²×Q₈×C₁₉	—	2	8	no	—
33	C₂²×Q₈×C₁₉	—	2	8	no	—
34	C₂²×Q₈×C₁₉	—	2	4	no	—
35	C₂²×Q₈×C₁₉	—	2	8	no	—
36	C₂²×Q₈×C₁₉	—	2	4	no	—
37	C₂²×Q₈×C₁₉	—	2	2	no	—
38	C₂²×Q₈×C₁₉	C₂×C₄	1	16	yes	—
39	C₂²×Q₈×C₁₉	C₂×C₄	1	8	yes	—
40	C₂²×Q₈×C₁₉	D₄	1	16	yes	—
41	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
42	C₂³×C₄×C₁₉	—	2	16	no	—
43	C₂³×C₄×C₁₉	—	2	8	no	—
44	C₂³×C₄×C₁₉	—	2	4	no	—
45	C₂³×C₄×C₁₉	—	2	2	no	—
46	C₂³×C₄×C₁₉	—	2	8	no	—
47	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₁₉	D₄	1	2	yes	—
49	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₁₉	D₄	1	4	yes	—
51	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
52	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₁₉	—	2	4	no	—
54	C₂×M₁₆×C₁₉	—	2	8	no	—
55	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
56	C₁₉×#117	—	2	8	no	—
57	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
58	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
60	C₂×C₄²×C₁₉	—	2	2	no	—
61	C₁₉×#103	—	2	32	no	—
62	C₁₉×C₂²⋊C₈	—	2	8	no	—
63	C₁₉×#95	—	2	4	no	—
64	C₂⁴×C₁₉	—	2	4	no	—
65	C₂⁴×C₁₉	—	2	8	no	—
66	C₂⁴×C₁₉	C₂²⋊C₄	1	4	yes	—
67	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂⁴×C₁₉	C₂×D₄	1	2	yes	—
69	C₂×Q₈×C₁₉	—	2	64	no	—
70	C₂×Q₈×C₁₉	—	2	128	no	—
71	C₂×Q₈×C₁₉	—	2	32	no	—
72	C₂×Q₈×C₁₉	—	2	64	no	—
73	C₂×Q₈×C₁₉	—	2	64	no	—
74	C₂×Q₈×C₁₉	—	2	16	no	—
75	C₂×Q₈×C₁₉	—	2	8	no	—
76	C₂×Q₈×C₁₉	—	2	16	no	—
77	C₂×Q₈×C₁₉	—	2	16	no	—
78	C₂×Q₈×C₁₉	—	2	32	no	—
79	C₂×Q₈×C₁₉	—	2	4	no	—
80	C₂×Q₈×C₁₉	—	2	8	no	—
81	C₂×Q₈×C₁₉	C₂²⋊C₄	1	32	yes	complemented
82	C₂×Q₈×C₁₉	C₂²×C₄	1	8	yes	—
83	C₂×Q₈×C₁₉	C₂×D₄	1	16	yes	—
84	C₂²×C₄×C₁₉	—	2	32	no	—
85	C₂²×C₄×C₁₉	—	2	64	no	—
86	C₂²×C₄×C₁₉	—	2	16	no	—
87	C₂²×C₄×C₁₉	—	2	8	no	—
88	C₂²×C₄×C₁₉	—	2	16	no	—
89	C₂²×C₄×C₁₉	—	2	4	no	—
90	C₂²×C₄×C₁₉	—	2	4	no	—
91	C₂²×C₄×C₁₉	—	2	16	no	—
92	C₂²×C₄×C₁₉	—	2	8	no	—
93	C₂²×C₄×C₁₉	—	2	16	no	—
94	C₂²×C₄×C₁₉	—	2	8	no	—
95	C₂²×C₄×C₁₉	—	2	4	no	—
96	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
97	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
98	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
102	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₉	C₂×D₄	1	8	yes	—
104	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₁₉	—	4	8	no	—
107	C₂×C₈×C₁₉	—	2	4	no	—
108	C₂×C₈×C₁₉	—	2	8	no	—
109	C₄⋊C₄×C₁₉	—	2	8	no	—
110	C₂²⋊C₄×C₁₉	—	4	4	no	—
111	C₂²⋊C₄×C₁₉	—	4	8	no	—
112	C₄²×C₁₉	—	4	4	no	—
113	#20599	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₁₉	—	2	32	no	—
115	C₂³×C₁₉	—	2	4	no	—
116	C₂³×C₁₉	—	2	4	no	—
117	C₂³×C₁₉	—	2	4	no	—
118	C₂³×C₁₉	—	2	8	no	—
119	C₂³×C₁₉	—	2	16	no	—
120	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂³×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂³×C₁₉	#120	1	4	yes	—
123	C₂³×C₁₉	#137	1	8	yes	—
124	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₁₉	—	2	64	no	—
126	Q₈×C₁₉	—	2	8	no	—
127	Q₈×C₁₉	—	2	16	no	—
128	Q₈×C₁₉	—	2	32	no	—
129	Q₈×C₁₉	C₂×C₂²⋊C₄	1	16	yes	—
130	C₂×C₄×C₁₉	—	2	64	no	—
131	C₂×C₄×C₁₉	—	2	8	no	—
132	C₂×C₄×C₁₉	—	2	16	no	—
133	C₂×C₄×C₁₉	—	2	4	no	—
134	C₂×C₄×C₁₉	—	2	32	no	—
135	C₂×C₄×C₁₉	—	2	8	no	—
136	C₂×C₄×C₁₉	—	2	8	no	—
137	C₂×C₄×C₁₉	—	2	8	no	—
138	C₂×C₄×C₁₉	—	2	16	no	—
139	C₂×C₄×C₁₉	—	4	4	no	—
140	C₂×C₄×C₁₉	—	4	8	no	—
141	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
142	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
143	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₉	#120	1	8	yes	—
145	C₂×C₄×C₁₉	#120	1	4	yes	—
146	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
148	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₁₉	—	4	4	no	—
150	C₂⁴×Q₈	C₂×C₁₉	1	1	yes	characteristic
151	C₂×#419	C₂×C₁₉	1	4	yes	complemented
152	C₂×#407	C₂×C₁₉	1	1	yes	characteristic
153	#2280	C₂×C₁₉	1	1	yes	characteristic
154	#1787	C₂×C₁₉	1	8	yes	complemented
155	C₂²×C₁₉	—	2	32	no	—
156	C₂²×C₁₉	—	2	4	no	—
157	C₂²×C₁₉	—	2	4	no	—
158	C₂²×C₁₉	—	2	8	no	—
159	C₂²×C₁₉	—	2	16	no	—
160	C₂²×C₁₉	—	2	4	no	—
161	C₂²×C₁₉	#379	1	4	yes	—
162	C₂²×C₁₉	#419	1	8	yes	complemented
163	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₁₉	C₂×#120	1	2	yes	—
165	C₂²×C₁₉	C₂×#137	1	4	yes	—
166	C₄×C₁₉	—	2	16	no	—
167	C₄×C₁₉	—	2	4	no	—
168	C₄×C₁₉	—	8	4	no	—
169	C₄×C₁₉	#379	1	4	yes	—
170	C₄×C₁₉	C₂²×C₂²⋊C₄	1	2	yes	—
171	C₄×C₁₉	C₂×#120	1	2	yes	—
172	C₂³×Q₈	—	2	16	no	—
173	C₂³×Q₈	—	2	4	no	—
174	C₂³×Q₈	—	2	2	no	—
175	C₂³×Q₈	—	2	1	no	—
176	C₂³×Q₈	C₄×C₁₉	1	8	yes	complemented
177	C₂³×Q₈	C₂²×C₁₉	1	4	yes	—
178	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
179	C₂⁴×C₄	—	2	1	no	—
180	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
181	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
182	C₂×#117	C₂²×C₁₉	1	2	yes	—
183	#419	—	2	16	no	—
184	C₂×#103	C₂²×C₁₉	1	16	yes	complemented
185	#407	—	2	4	no	—
186	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
187	#381	—	2	1	no	—
188	#380	C₂²×C₁₉	1	4	yes	—
189	#379	—	2	1	no	—
190	C₄×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₁₉	—	2	4	no	—
192	C₂×C₁₉	—	2	8	no	—
193	C₂×C₁₉	C₂×#379	1	1	yes	characteristic
194	C₂×C₁₉	C₂×#419	1	2	yes	—
195	C₂×C₁₉	#3004	1	4	yes	—
196	C₂⁵	D₄×C₁₉	1	1	yes	characteristic
197	C₂²×Q₈	—	2	64	no	—
198	C₂²×Q₈	—	2	32	no	—
199	C₂²×Q₈	—	2	16	no	—
200	C₂²×Q₈	—	2	64	no	—
201	C₂²×Q₈	—	2	32	no	—
202	C₂²×Q₈	—	2	16	no	—
203	C₂²×Q₈	—	2	8	no	—
204	C₂²×Q₈	—	2	8	no	—
205	C₂²×Q₈	—	2	4	no	—
206	C₂²×Q₈	—	2	8	no	—
207	C₂²×Q₈	—	2	4	no	—
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	C₂×C₄×C₁₉	1	16	yes	—
210	C₂²×Q₈	C₂×C₄×C₁₉	1	8	yes	—
211	C₂²×Q₈	D₄×C₁₉	1	16	yes	—
212	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
213	C₂³×C₄	—	2	16	no	—
214	C₂³×C₄	—	2	8	no	—
215	C₂³×C₄	—	2	4	no	—
216	C₂³×C₄	—	2	2	no	—
217	C₂³×C₄	—	2	8	no	—
218	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
219	C₂³×C₄	D₄×C₁₉	1	2	yes	—
220	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
221	C₂³×C₄	D₄×C₁₉	1	4	yes	—
222	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
223	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
224	C₂×M₁₆	—	2	4	no	—
225	C₂×M₁₆	—	2	8	no	—
226	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
227	#117	—	2	8	no	—
228	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₄²	—	2	2	no	—
232	#103	—	2	32	no	—
233	C₂²⋊C₈	—	2	8	no	—
234	#95	—	2	4	no	—
235	C₁₉	#20599	1	1	yes	characteristic, direct product factor, Sylow
236	C₂⁴	—	2	4	no	—
237	C₂⁴	—	2	8	no	—
238	C₂⁴	C₂²⋊C₄×C₁₉	1	4	yes	—
239	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
240	C₂⁴	C₂×D₄×C₁₉	1	2	yes	—
241	C₂×Q₈	—	2	128	no	—
242	C₂×Q₈	—	2	64	no	—
243	C₂×Q₈	—	2	64	no	—
244	C₂×Q₈	—	2	32	no	—
245	C₂×Q₈	—	2	64	no	—
246	C₂×Q₈	—	2	16	no	—
247	C₂×Q₈	—	2	16	no	—
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	—	2	32	no	—
250	C₂×Q₈	—	2	16	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	8	no	—
253	C₂×Q₈	C₂²⋊C₄×C₁₉	1	32	yes	complemented
254	C₂×Q₈	C₂²×C₄×C₁₉	1	8	yes	—
255	C₂×Q₈	C₂×D₄×C₁₉	1	16	yes	—
256	C₂²×C₄	—	2	64	no	—
257	C₂²×C₄	—	2	32	no	—
258	C₂²×C₄	—	2	16	no	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	16	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	16	no	—
264	C₂²×C₄	—	2	16	no	—
265	C₂²×C₄	—	2	8	no	—
266	C₂²×C₄	—	2	8	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
269	C₂²×C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
270	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
271	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
272	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₁₉	1	8	yes	—
276	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
277	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
278	M₁₆	—	4	8	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	8	no	—
281	C₄⋊C₄	—	2	8	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	8	no	—
284	C₄²	—	4	4	no	—
285	C₂³	—	2	32	no	—
286	C₂³	—	2	8	no	—
287	C₂³	—	2	16	no	—
288	C₂³	—	2	4	no	—
289	C₂³	—	2	4	no	—
290	C₂³	—	2	4	no	—
291	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
292	C₂³	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
293	C₂³	C₁₉×#120	1	4	yes	—
294	C₂³	C₁₉×#137	1	8	yes	—
295	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
296	Q₈	—	2	64	no	—
297	Q₈	—	2	32	no	—
298	Q₈	—	2	8	no	—
299	Q₈	—	2	16	no	—
300	Q₈	C₂×C₂²⋊C₄×C₁₉	1	16	yes	—
301	C₂×C₄	—	2	64	no	—
302	C₂×C₄	—	2	32	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	8	no	—
305	C₂×C₄	—	2	16	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	16	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	8	no	—
312	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
314	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
315	C₂×C₄	C₁₉×#120	1	4	yes	—
316	C₂×C₄	C₁₉×#120	1	8	yes	—
317	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
319	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
320	C₈	—	4	4	no	—
321	C₂²	—	2	32	no	—
322	C₂²	—	2	8	no	—
323	C₂²	—	2	16	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	4	no	—
327	C₂²	C₁₉×#379	1	4	yes	—
328	C₂²	C₁₉×#419	1	8	yes	complemented
329	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
330	C₂²	C₂×C₁₉×#120	1	2	yes	—
331	C₂²	C₂×C₁₉×#137	1	4	yes	—
332	C₄	—	2	16	no	—
333	C₄	—	2	4	no	—
334	C₄	—	8	4	no	—
335	C₄	C₁₉×#379	1	4	yes	—
336	C₄	C₂²×C₂²⋊C₄×C₁₉	1	2	yes	—
337	C₄	C₂×C₁₉×#120	1	2	yes	—
338	C₂	—	2	4	no	—
339	C₂	—	2	8	no	—
340	C₂	C₂×C₁₉×#379	1	1	yes	characteristic
341	C₂	C₂×C₁₉×#419	1	2	yes	—
342	C₂	C₁₉×#3004	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁵×C₁₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂⁴×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#419	C₂	semidirect
5. C₁₉×#1787	C₂	semidirect
10. C₂³×Q₈×C₁₉	C₄	semidirect
18. C₂×C₁₉×#103	C₂²	semidirect
81. C₂×Q₈×C₁₉	C₂²⋊C₄	semidirect
113. #20599	C₁₉	direct (N × H)
151. C₂×#419	C₂×C₁₉	semidirect
154. #1787	C₂×C₁₉	semidirect
162. C₂²×C₁₉	#419	semidirect
176. C₂³×Q₈	C₄×C₁₉	semidirect
184. C₂×#103	C₂²×C₁₉	semidirect
235. C₁₉	#20599	direct (N × H)
253. C₂×Q₈	C₂²⋊C₄×C₁₉	semidirect
328. C₂²	C₁₉×#419	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₉	2
C₂×C₁₉×#419	2
C₂×C₁₉×#407	2
C₁₉×#2280	2
C₁₉×#1787	2
#20599	19

Subgroups — C19×#20599 #18026558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20599 #18026558