Subgroups — C₁₉×#13322 #18019281

Order 4864.

520 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#473	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3059	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₉×#350	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#137	—	2	1	no	—
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
12	C₁₉×#536	—	2	1	no	—
13	C₂×C₁₉×Q₃₂	—	2	2	no	—
14	C₂×C₁₉×Q₃₂	—	2	2	no	—
15	C₂×C₁₉×Q₃₂	—	2	2	no	—
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#473	—	2	2	no	—
19	C₁₉×#473	—	2	2	no	—
20	C₁₉×#470	—	2	2	no	—
21	C₁₉×#467	—	2	2	no	—
22	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
23	C₁₉×#427	C₂²	1	1	yes	characteristic
24	C₁₉×#419	—	2	1	no	—
25	C₂×C₁₉×#101	—	2	1	no	—
26	C₁₉×#362	—	2	2	no	—
27	C₁₉×#358	—	2	2	no	—
28	C₁₉×#350	—	2	2	no	—
29	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂²×Q₈×C₁₉	—	2	1	no	—
32	C₁₉×#137	—	2	4	no	—
33	C₁₉×#137	—	4	2	no	—
34	C₂×Q₁₆×C₁₉	—	2	1	no	—
35	C₂×Q₁₆×C₁₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₉	—	2	1	no	—
37	C₂×Q₁₆×C₁₉	—	2	1	no	—
38	C₂×Q₁₆×C₁₉	—	2	1	no	—
39	C₂×Q₁₆×C₁₉	—	2	1	no	—
40	C₂×Q₁₆×C₁₉	—	2	1	no	—
41	C₂×Q₁₆×C₁₉	—	2	1	no	—
42	C₂×Q₁₆×C₁₉	—	2	1	no	—
43	C₂×Q₁₆×C₁₉	—	2	1	no	—
44	C₂×Q₁₆×C₁₉	—	4	1	no	—
45	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
46	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
47	C₂×SD₁₆×C₁₉	—	4	1	no	—
48	C₁₉×#131	—	2	1	no	—
49	C₁₉×#131	—	2	1	no	—
50	C₁₉×#131	—	2	1	no	—
51	C₁₉×#131	—	2	1	no	—
52	C₂×M₁₆×C₁₉	—	2	1	no	—
53	C₂×M₁₆×C₁₉	—	2	1	no	—
54	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₉	—	2	1	no	—
56	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×C₄⋊Q₈	—	4	1	no	—
58	C₁₉×#124	—	4	1	no	—
59	C₁₉×#123	—	4	2	no	—
60	C₁₉×C₂²⋊Q₈	—	2	4	no	—
61	C₁₉×#117	—	2	1	no	—
62	C₁₉×Q₃₂	—	2	8	no	—
63	C₁₉×M₃₂	—	2	1	no	—
64	C₁₉×M₃₂	—	2	1	no	—
65	C₂×C₁₆×C₁₉	—	2	1	no	—
66	C₂×C₁₆×C₁₉	—	2	1	no	—
67	C₂×C₁₆×C₁₉	—	2	1	no	—
68	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
69	C₁₉×#108	—	2	1	no	—
70	C₁₉×#108	D₄	1	2	yes	—
71	C₁₉×#107	D₄	1	2	yes	—
72	C₁₉×#106	—	2	1	no	—
73	C₁₉×#103	—	2	4	no	—
74	C₁₉×#101	—	2	4	no	—
75	#41×C₁₉	—	2	1	no	—
76	#41×C₁₉	—	2	1	no	—
77	#41×C₁₉	—	2	1	no	—
78	#41×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	2	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	4	1	no	—
82	C₂×Q₈×C₁₉	—	4	1	no	—
83	C₂×Q₈×C₁₉	—	4	1	no	—
84	C₂×Q₈×C₁₉	—	4	1	no	—
85	C₂×Q₈×C₁₉	—	4	1	no	—
86	C₂×Q₈×C₁₉	—	4	1	no	—
87	C₂×Q₈×C₁₉	—	4	1	no	—
88	C₂×Q₈×C₁₉	—	4	1	no	—
89	C₂×Q₈×C₁₉	—	4	1	no	—
90	C₂×Q₈×C₁₉	—	4	1	no	—
91	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
92	C₂×D₄×C₁₉	—	2	1	no	—
93	C₂×D₄×C₁₉	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	—	2	1	no	—
95	C₂²×C₄×C₁₉	—	4	1	no	—
96	C₂²×C₄×C₁₉	—	4	1	no	—
97	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₁₉	—	2	2	no	—
99	Q₁₆×C₁₉	—	2	2	no	—
100	Q₁₆×C₁₉	—	2	2	no	—
101	Q₁₆×C₁₉	—	2	2	no	—
102	Q₁₆×C₁₉	—	4	4	no	—
103	Q₁₆×C₁₉	—	4	1	no	—
104	Q₁₆×C₁₉	—	4	1	no	—
105	Q₁₆×C₁₉	—	4	1	no	—
106	Q₁₆×C₁₉	—	4	1	no	—
107	SD₁₆×C₁₉	—	4	4	no	—
108	M₁₆×C₁₉	—	2	2	no	—
109	M₁₆×C₁₉	—	2	1	no	—
110	M₁₆×C₁₉	—	2	1	no	—
111	M₁₆×C₁₉	—	4	1	no	—
112	M₁₆×C₁₉	—	4	1	no	—
113	M₁₆×C₁₉	—	4	1	no	—
114	C₂×C₈×C₁₉	—	2	1	no	—
115	C₂×C₈×C₁₉	—	4	1	no	—
116	C₂×C₈×C₁₉	—	4	1	no	—
117	C₂×C₈×C₁₉	—	4	1	no	—
118	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
124	C₄⋊C₄×C₁₉	—	2	2	no	—
125	C₄⋊C₄×C₁₉	—	4	4	no	—
126	C₂²⋊C₄×C₁₉	—	4	4	no	—
127	C₂²⋊C₄×C₁₉	—	4	1	no	—
128	C₄²×C₁₉	—	4	1	no	—
129	C₁₆×C₁₉	—	2	2	no	—
130	C₁₆×C₁₉	—	4	1	no	—
131	#13322	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₁₉	—	2	1	no	—
133	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	Q₈×C₁₉	—	2	1	no	—
135	Q₈×C₁₉	—	2	1	no	—
136	Q₈×C₁₉	—	4	2	no	—
137	Q₈×C₁₉	—	4	2	no	—
138	Q₈×C₁₉	—	4	2	no	—
139	Q₈×C₁₉	—	4	2	no	—
140	Q₈×C₁₉	—	8	1	no	—
141	Q₈×C₁₉	—	8	1	no	—
142	Q₈×C₁₉	—	8	1	no	—
143	Q₈×C₁₉	—	8	1	no	—
144	D₄×C₁₉	—	2	1	no	—
145	D₄×C₁₉	—	2	1	no	—
146	D₄×C₁₉	—	4	1	no	—
147	D₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	2	1	no	—
149	C₂×C₄×C₁₉	—	4	1	no	—
150	C₂×C₄×C₁₉	—	4	1	no	—
151	C₂×C₄×C₁₉	—	4	2	no	—
152	C₂×C₄×C₁₉	—	4	2	no	—
153	C₂×C₄×C₁₉	—	4	2	no	—
154	C₂×C₄×C₁₉	—	8	1	no	—
155	C₂×C₄×C₁₉	—	8	1	no	—
156	C₂×C₄×C₁₉	—	8	1	no	—
157	C₂×C₄×C₁₉	—	8	1	no	—
158	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
160	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
163	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₈×C₁₉	—	4	1	no	—
165	C₈×C₁₉	—	4	2	no	—
166	C₈×C₁₉	#120	1	1	yes	characteristic
167	C₈×C₁₉	#121	1	1	yes	characteristic
168	C₈×C₁₉	#121	1	1	yes	characteristic
169	C₈×C₁₉	#127	1	1	yes	characteristic
170	C₂²×Q₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
171	C₂×#473	C₂×C₁₉	1	1	yes	characteristic
172	#3059	C₂×C₁₉	1	1	yes	characteristic
173	C₂×#362	C₂×C₁₉	1	1	yes	characteristic, complemented
174	#2141	C₂×C₁₉	1	1	yes	characteristic, complemented
175	C₂×#350	C₂×C₁₉	1	1	yes	characteristic
176	#1382	C₂×C₁₉	1	1	yes	characteristic, complemented
177	C₂²×C₁₉	—	2	2	no	—
178	C₂²×C₁₉	—	4	2	no	—
179	C₂²×C₁₉	—	4	1	no	—
180	C₂²×C₁₉	#447	1	1	yes	characteristic
181	C₂²×C₁₉	#449	1	1	yes	characteristic
182	C₂²×C₁₉	#469	1	1	yes	characteristic
183	C₄×C₁₉	—	4	1	no	—
184	C₄×C₁₉	—	8	2	no	—
185	C₄×C₁₉	—	8	2	no	—
186	C₄×C₁₉	—	8	2	no	—
187	C₄×C₁₉	#392	1	1	yes	characteristic
188	C₄×C₁₉	#447	1	1	yes	characteristic
189	C₄×C₁₉	#449	1	1	yes	characteristic
190	C₄×C₁₉	#466	1	1	yes	characteristic
191	C₂×#137	—	2	1	no	—
192	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
193	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
194	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
195	#536	—	2	1	no	—
196	C₂×Q₃₂	—	2	2	no	—
197	C₂×Q₃₂	—	2	2	no	—
198	C₂×Q₃₂	—	2	2	no	—
199	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
200	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
201	#473	—	2	2	no	—
202	#473	—	2	2	no	—
203	#470	—	2	2	no	—
204	#467	—	2	2	no	—
205	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
206	#427	C₂²×C₁₉	1	1	yes	characteristic
207	#419	—	2	1	no	—
208	C₂×#101	—	2	1	no	—
209	#362	—	2	2	no	—
210	#358	—	2	2	no	—
211	#350	—	2	2	no	—
212	C₂×C₁₉	—	2	1	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	8	1	no	—
215	C₂×C₁₉	#1999	1	1	yes	characteristic
216	C₂×C₁₉	#2195	1	2	yes	—
217	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×Q₈	—	2	1	no	—
220	#137	—	2	4	no	—
221	#137	—	4	2	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×Q₁₆	—	2	1	no	—
226	C₂×Q₁₆	—	2	1	no	—
227	C₂×Q₁₆	—	2	1	no	—
228	C₂×Q₁₆	—	2	1	no	—
229	C₂×Q₁₆	—	2	1	no	—
230	C₂×Q₁₆	—	2	1	no	—
231	C₂×Q₁₆	—	2	1	no	—
232	C₂×Q₁₆	—	4	1	no	—
233	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
234	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
235	C₂×SD₁₆	—	4	1	no	—
236	#131	—	2	1	no	—
237	#131	—	2	1	no	—
238	#131	—	2	1	no	—
239	#131	—	2	1	no	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	2	1	no	—
242	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
245	C₄⋊Q₈	—	4	1	no	—
246	#124	—	4	1	no	—
247	#123	—	4	2	no	—
248	C₂²⋊Q₈	—	2	4	no	—
249	#117	—	2	1	no	—
250	Q₃₂	—	2	8	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	2	1	no	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	2	1	no	—
255	C₂×C₁₆	—	2	1	no	—
256	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
257	#108	—	2	1	no	—
258	#108	D₄×C₁₉	1	2	yes	—
259	#107	D₄×C₁₉	1	2	yes	—
260	#106	—	2	1	no	—
261	#103	—	2	4	no	—
262	#101	—	2	4	no	—
263	C₁₉	#13322	1	1	yes	characteristic, direct product factor, Sylow
264	#41	—	2	1	no	—
265	#41	—	2	1	no	—
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	D₈×C₁₉	1	1	yes	characteristic
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	4	4	no	—
292	Q₁₆	—	4	1	no	—
293	Q₁₆	—	4	1	no	—
294	Q₁₆	—	4	1	no	—
295	Q₁₆	—	4	1	no	—
296	SD₁₆	—	4	4	no	—
297	M₁₆	—	2	1	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	2	1	no	—
300	M₁₆	—	4	1	no	—
301	M₁₆	—	4	1	no	—
302	M₁₆	—	4	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
308	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
312	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	4	no	—
315	C₂²⋊C₄	—	4	4	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₄²	—	4	1	no	—
318	C₁₆	—	2	2	no	—
319	C₁₆	—	4	1	no	—
320	C₂³	—	2	1	no	—
321	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	2	no	—
326	Q₈	—	4	2	no	—
327	Q₈	—	4	2	no	—
328	Q₈	—	8	1	no	—
329	Q₈	—	8	1	no	—
330	Q₈	—	8	1	no	—
331	Q₈	—	8	1	no	—
332	D₄	—	2	1	no	—
333	D₄	—	2	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	8	1	no	—
346	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
347	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
348	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
349	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
350	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
351	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₈	—	4	1	no	—
353	C₈	—	4	2	no	—
354	C₈	C₁₉×#120	1	1	yes	characteristic
355	C₈	C₁₉×#121	1	1	yes	characteristic
356	C₈	C₁₉×#121	1	1	yes	characteristic
357	C₈	C₁₉×#127	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	1	no	—
361	C₂²	C₁₉×#447	1	1	yes	characteristic
362	C₂²	C₁₉×#449	1	1	yes	characteristic
363	C₂²	C₁₉×#469	1	1	yes	characteristic
364	C₄	—	4	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	2	no	—
368	C₄	C₁₉×#392	1	1	yes	characteristic
369	C₄	C₁₉×#447	1	1	yes	characteristic
370	C₄	C₁₉×#449	1	1	yes	characteristic
371	C₄	C₁₉×#466	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	8	1	no	—
375	C₂	C₁₉×#1999	1	1	yes	characteristic
376	C₂	C₁₉×#2195	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₉×Q₃₂	C₂	semidirect
4. C₂×C₁₉×#362	C₂	semidirect
5. C₁₉×#2141	C₂	semidirect
7. C₁₉×#1382	C₂	semidirect
131. #13322	C₁₉	direct (N × H)
170. C₂²×Q₃₂	C₂×C₁₉	semidirect
173. C₂×#362	C₂×C₁₉	semidirect
174. #2141	C₂×C₁₉	semidirect
176. #1382	C₂×C₁₉	semidirect
263. C₁₉	#13322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×Q₃₂	2
C₂×C₁₉×#473	2
C₁₉×#3059	2
C₂×C₁₉×#362	2
C₁₉×#2141	2
C₂×C₁₉×#350	2
C₁₉×#1382	2
#13322	19

Subgroups — C19×#13322 #18019281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13322 #18019281