Subgroups — C₁₉×#9326 #18015285

Order 4864.

574 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3112	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2276	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1980	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1505	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1300	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#532	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
10	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#480	—	2	1	no	—
12	C₁₉×#477	C₄	1	2	yes	complemented
13	C₁₉×#460	C₄	1	2	yes	complemented
14	C₁₉×#448	—	2	1	no	—
15	C₁₉×#440	—	2	2	no	—
16	C₁₉×#432	C₂²	1	1	yes	characteristic
17	C₁₉×#432	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#104	—	2	2	no	—
19	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
21	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
22	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#376	—	2	1	no	—
25	C₁₉×#376	C₂²	1	1	yes	characteristic
26	C₁₉×#327	—	2	1	no	—
27	C₁₉×#327	—	2	1	no	—
28	C₁₉×Q₈⋊C₈	—	2	2	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#124	—	2	1	no	—
36	C₁₉×#121	—	2	1	no	—
37	C₁₉×#121	—	2	1	no	—
38	C₄×Q₈×C₁₉	—	2	8	no	—
39	C₄×Q₈×C₁₉	—	2	1	no	—
40	C₄×Q₈×C₁₉	—	2	1	no	—
41	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₁₉	D₄	1	2	yes	—
43	C₄×D₄×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	—	2	2	no	—
50	C₂×C₄²×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	—	2	2	no	—
53	C₂×C₄²×C₁₉	D₄	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#107	—	2	2	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
58	C₁₉×#104	—	4	4	no	—
59	C₁₉×#103	—	2	2	no	—
60	C₁₉×#102	—	2	2	no	—
61	C₁₉×C₂²⋊C₈	—	2	1	no	—
62	C₁₉×C₂²⋊C₈	—	2	1	no	—
63	C₁₉×#97	—	2	1	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₁₉×#95	—	2	1	no	—
69	C₁₉×#95	—	2	1	no	—
70	C₁₉×#95	D₄	1	1	yes	characteristic
71	C₁₉×#95	D₄	1	1	yes	characteristic
72	#41×C₁₉	—	4	2	no	—
73	#41×C₁₉	—	4	2	no	—
74	C₂×Q₈×C₁₉	—	2	4	no	—
75	C₂×Q₈×C₁₉	—	2	1	no	—
76	C₂×Q₈×C₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×D₄×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	—	2	1	no	—
83	C₂²×C₄×C₁₉	—	2	1	no	—
84	C₂²×C₄×C₁₉	—	2	1	no	—
85	C₂²×C₄×C₁₉	—	2	1	no	—
86	C₂²×C₄×C₁₉	—	2	2	no	—
87	C₂²×C₄×C₁₉	—	4	1	no	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	SD₁₆×C₁₉	—	4	4	no	—
92	M₁₆×C₁₉	—	2	2	no	—
93	M₁₆×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	2	8	no	—
98	C₄⋊C₄×C₁₉	—	2	4	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₁₉	—	4	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₄²×C₁₉	—	2	8	no	—
108	C₄²×C₁₉	—	2	4	no	—
109	C₄²×C₁₉	—	2	2	no	—
110	C₄²×C₁₉	—	2	2	no	—
111	C₄²×C₁₉	—	4	2	no	—
112	C₄²×C₁₉	—	8	2	no	—
113	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₉	#41	1	1	yes	characteristic
116	C₄²×C₁₉	#41	1	1	yes	characteristic
117	#9326	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₁₉	—	4	1	no	—
119	C₂³×C₁₉	#120	1	1	yes	characteristic
120	Q₈×C₁₉	—	2	4	no	—
121	Q₈×C₁₉	—	2	2	no	—
122	Q₈×C₁₉	—	4	2	no	—
123	Q₈×C₁₉	#104	1	4	yes	complemented
124	D₄×C₁₉	—	4	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	8	1	no	—
128	D₄×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	2	4	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	2	4	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	1	no	—
147	C₂×C₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	8	2	no	—
149	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
155	C₈×C₁₉	—	4	2	no	—
156	C₈×C₁₉	—	4	2	no	—
157	C₈×C₁₉	—	8	1	no	—
158	#3112	C₂×C₁₉	1	1	yes	characteristic
159	#2276	C₂×C₁₉	1	1	yes	characteristic, complemented
160	#1980	C₂×C₁₉	1	1	yes	characteristic
161	#1505	C₂×C₁₉	1	1	yes	characteristic
162	#1491	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1300	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#1285	C₂×C₁₉	1	1	yes	characteristic, complemented
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	8	1	no	—
168	C₂²×C₁₉	—	8	1	no	—
169	C₂²×C₁₉	—	8	1	no	—
170	C₂²×C₁₉	#386	1	1	yes	characteristic
171	C₂²×C₁₉	#453	1	1	yes	characteristic
172	C₂²×C₁₉	#454	1	1	yes	characteristic
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	4	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	4	4	no	—
180	C₄×C₁₉	—	4	2	no	—
181	C₄×C₁₉	—	4	2	no	—
182	C₄×C₁₉	—	4	2	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	C₂×#104	1	2	yes	—
185	#532	C₂²×C₁₉	1	1	yes	characteristic
186	C₂×C₄×Q₈	—	2	1	no	—
187	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
188	#480	—	2	1	no	—
189	#477	C₄×C₁₉	1	2	yes	complemented
190	#460	C₄×C₁₉	1	2	yes	complemented
191	#448	—	2	1	no	—
192	#440	—	2	2	no	—
193	#432	C₂²×C₁₉	1	1	yes	characteristic
194	#432	C₂²×C₁₉	1	1	yes	characteristic
195	C₂×#104	—	2	2	no	—
196	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
197	C₄×C₄⋊C₄	—	2	1	no	—
198	C₄×C₄⋊C₄	—	2	2	no	—
199	C₄×C₄⋊C₄	—	2	1	no	—
200	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
201	#376	—	2	1	no	—
202	#376	C₂²×C₁₉	1	1	yes	characteristic
203	#327	—	2	1	no	—
204	#327	—	2	1	no	—
205	Q₈⋊C₈	—	2	2	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	—	16	1	no	—
209	C₂×C₁₉	#1883	1	1	yes	characteristic
210	C₂×C₁₉	#1896	1	1	yes	characteristic
211	C₂×C₁₉	#1905	1	1	yes	characteristic
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
214	C₂×SD₁₆	—	2	2	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
217	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
218	#124	—	2	1	no	—
219	#121	—	2	1	no	—
220	#121	—	2	1	no	—
221	C₄×Q₈	—	2	8	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	C₂×C₄×C₁₉	1	2	yes	—
225	C₄×Q₈	D₄×C₁₉	1	2	yes	—
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	D₄×C₁₉	1	2	yes	—
237	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
238	#107	—	2	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
241	#104	—	4	4	no	—
242	#103	—	2	2	no	—
243	#102	—	2	2	no	—
244	C₂²⋊C₈	—	2	1	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	#97	—	2	1	no	—
247	#97	—	2	1	no	—
248	#97	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	#95	—	2	2	no	—
251	#95	—	2	1	no	—
252	#95	—	2	1	no	—
253	#95	D₄×C₁₉	1	1	yes	characteristic
254	#95	D₄×C₁₉	1	1	yes	characteristic
255	C₁₉	#9326	1	1	yes	characteristic, direct product factor, Sylow
256	#41	—	4	2	no	—
257	#41	—	4	2	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	C₂²⋊C₄×C₁₉	1	2	yes	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	SD₁₆	—	4	4	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	4	1	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₄⋊C₄	—	2	8	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₄²	—	2	8	no	—
292	C₄²	—	2	4	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	4	2	no	—
296	C₄²	—	8	2	no	—
297	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
299	C₄²	#41×C₁₉	1	1	yes	characteristic
300	C₄²	#41×C₁₉	1	1	yes	characteristic
301	C₂³	—	4	1	no	—
302	C₂³	C₁₉×#120	1	1	yes	characteristic
303	Q₈	—	2	4	no	—
304	Q₈	—	2	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	C₁₉×#104	1	4	yes	complemented
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	8	1	no	—
311	D₄	—	8	1	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
333	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
334	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
335	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
336	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
337	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	8	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	C₁₉×#386	1	1	yes	characteristic
347	C₂²	C₁₉×#453	1	1	yes	characteristic
348	C₂²	C₁₉×#454	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	8	1	no	—
360	C₄	C₂×C₁₉×#104	1	2	yes	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	16	1	no	—
364	C₂	C₁₉×#1883	1	1	yes	characteristic
365	C₂	C₁₉×#1896	1	1	yes	characteristic
366	C₂	C₁₉×#1905	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
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. C₁₉×#2276	C₂	semidirect
5. C₁₉×#1491	C₂	semidirect
6. C₁₉×#1300	C₂	semidirect
7. C₁₉×#1285	C₂	semidirect
12. C₁₉×#477	C₄	semidirect
13. C₁₉×#460	C₄	semidirect
117. #9326	C₁₉	direct (N × H)
123. Q₈×C₁₉	#104	semidirect
159. #2276	C₂×C₁₉	semidirect
162. #1491	C₂×C₁₉	semidirect
163. #1300	C₂×C₁₉	semidirect
164. #1285	C₂×C₁₉	semidirect
189. #477	C₄×C₁₉	semidirect
190. #460	C₄×C₁₉	semidirect
255. C₁₉	#9326	direct (N × H)
306. Q₈	C₁₉×#104	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3112	2
C₁₉×#2276	2
C₁₉×#1980	2
C₁₉×#1505	2
C₁₉×#1491	2
C₁₉×#1300	2
C₁₉×#1285	2
#9326	19

Subgroups — C19×#9326 #18015285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9326 #18015285