Subgroups — C₁₉×#18408 #18024367

Order 4864.

1170 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3497	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3226	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2546	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1610	C₂	1	8	yes	maximal, complemented
5	C₁₉×#1502	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×2₋¹⁺⁴	—	2	1	no	—
8	C₁₉×#550	—	2	2	no	—
9	D₄×Q₈×C₁₉	—	2	2	no	—
10	C₁₉×#548	—	2	4	no	—
11	C₁₉×#537	—	2	1	no	—
12	C₁₉×#537	C₂²	1	4	yes	—
13	C₁₉×#532	—	2	2	no	—
14	C₁₉×#532	C₂²	1	2	yes	—
15	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
16	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#500	—	2	2	no	—
19	C₁₉×#486	—	2	2	no	—
20	C₁₉×#480	C₂²	1	8	yes	—
21	C₁₉×#474	—	2	4	no	—
22	C₁₉×#431	C₂²	1	2	yes	—
23	C₁₉×#398	C₂²	1	4	yes	—
24	C₁₉×#391	—	2	1	no	—
25	C₁₉×#389	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#327	C₂²	1	8	yes	—
28	C₁₉×#324	C₂²	1	4	yes	—
29	C₁₉×2₋¹⁺⁴	—	4	4	no	—
30	C₁₉×2₋¹⁺⁴	—	4	4	no	—
31	C₂×#41×C₁₉	—	2	4	no	—
32	C₂×#41×C₁₉	—	2	4	no	—
33	C₂×#41×C₁₉	—	2	2	no	—
34	C₂²×Q₈×C₁₉	—	2	4	no	—
35	C₂²×Q₈×C₁₉	—	2	1	no	—
36	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
37	C₁₉×C₄⋊Q₈	—	2	2	no	—
38	C₁₉×C₄⋊Q₈	—	2	2	no	—
39	C₁₉×C₄⋊Q₈	—	2	2	no	—
40	C₁₉×C₄⋊Q₈	—	4	4	no	—
41	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
42	C₁₉×#127	—	2	2	no	—
43	C₁₉×#127	D₄	1	2	yes	—
44	C₁₉×#124	—	2	4	no	—
45	C₁₉×#124	—	2	2	no	—
46	C₁₉×C₂²⋊Q₈	—	2	4	no	—
47	C₁₉×C₂²⋊Q₈	—	2	8	no	—
48	C₁₉×#121	—	2	4	no	—
49	C₁₉×#121	—	2	8	no	—
50	C₁₉×#121	—	2	2	no	—
51	C₁₉×#121	—	2	4	no	—
52	C₁₉×#121	C₂³	1	4	yes	—
53	C₄×Q₈×C₁₉	—	2	2	no	—
54	C₄×Q₈×C₁₉	—	2	2	no	—
55	C₄×D₄×C₁₉	—	2	4	no	—
56	C₄×D₄×C₁₉	—	2	4	no	—
57	C₄×D₄×C₁₉	—	2	2	no	—
58	C₄×D₄×C₁₉	D₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
65	C₂×C₄²×C₁₉	—	2	1	no	—
66	C₂×C₄²×C₁₉	D₄	1	2	yes	—
67	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
68	C₁₉×#107	—	2	8	no	—
69	C₁₉×C₄⋊C₈	—	2	2	no	—
70	C₁₉×#102	—	2	8	no	—
71	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
72	C₄×C₈×C₁₉	—	2	2	no	—
73	C₁₉×#95	—	2	4	no	—
74	C₁₉×#95	C₂³	1	2	yes	—
75	#41×C₁₉	—	4	8	no	—
76	#41×C₁₉	—	4	16	no	—
77	#41×C₁₉	—	4	8	no	—
78	#41×C₁₉	—	4	4	no	—
79	#41×C₁₉	—	4	4	no	—
80	C₂×Q₈×C₁₉	—	2	4	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	2	2	no	—
83	C₂×Q₈×C₁₉	—	2	2	no	—
84	C₂×Q₈×C₁₉	—	4	16	no	—
85	C₂×Q₈×C₁₉	—	4	2	no	—
86	C₂×Q₈×C₁₉	—	4	2	no	—
87	C₂×Q₈×C₁₉	—	4	2	no	—
88	C₂×Q₈×C₁₉	D₈	1	4	yes	—
89	C₂×D₄×C₁₉	—	2	4	no	—
90	C₂×D₄×C₁₉	—	2	4	no	—
91	C₂×D₄×C₁₉	—	2	4	no	—
92	C₂×D₄×C₁₉	—	2	2	no	—
93	C₂×D₄×C₁₉	—	2	2	no	—
94	C₂×D₄×C₁₉	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₁₉	—	2	8	no	—
96	C₂²×C₄×C₁₉	—	2	2	no	—
97	C₂²×C₄×C₁₉	—	2	4	no	—
98	C₂²×C₄×C₁₉	—	2	2	no	—
99	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	2	4	no	—
106	C₄⋊C₄×C₁₉	—	2	4	no	—
107	C₄⋊C₄×C₁₉	—	2	2	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	2	4	no	—
110	C₄⋊C₄×C₁₉	—	2	4	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₄⋊C₄×C₁₉	—	4	1	no	—
113	C₄⋊C₄×C₁₉	—	4	8	no	—
114	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
115	C₂²⋊C₄×C₁₉	—	2	8	no	—
116	C₂²⋊C₄×C₁₉	—	2	4	no	—
117	C₂²⋊C₄×C₁₉	—	2	4	no	—
118	C₄²×C₁₉	—	2	2	no	—
119	C₄²×C₁₉	—	2	2	no	—
120	C₄²×C₁₉	—	4	2	no	—
121	C₄²×C₁₉	C₂×D₄	1	2	yes	—
122	C₄²×C₁₉	C₂×D₄	1	2	yes	—
123	#18408	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₁₉	—	2	4	no	—
125	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₁₉	—	2	8	no	—
127	Q₈×C₁₉	—	4	8	no	—
128	Q₈×C₁₉	—	4	4	no	—
129	Q₈×C₁₉	—	8	4	no	—
130	D₄×C₁₉	—	4	4	no	—
131	D₄×C₁₉	—	4	8	no	—
132	D₄×C₁₉	—	4	4	no	—
133	D₄×C₁₉	—	4	4	no	—
134	D₄×C₁₉	—	4	4	no	—
135	D₄×C₁₉	—	4	4	no	—
136	D₄×C₁₉	—	8	4	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	2	2	no	—
140	C₂×C₄×C₁₉	—	2	4	no	—
141	C₂×C₄×C₁₉	—	2	2	no	—
142	C₂×C₄×C₁₉	—	2	4	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	16	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	4	8	no	—
148	C₂×C₄×C₁₉	—	4	2	no	—
149	C₂×C₄×C₁₉	—	4	2	no	—
150	C₂×C₄×C₁₉	—	4	2	no	—
151	C₂×C₄×C₁₉	—	4	2	no	—
152	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#120	1	2	yes	—
154	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
155	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
156	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
158	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
159	C₈×C₁₉	—	4	4	no	—
160	#3497	C₂×C₁₉	1	1	yes	characteristic
161	#3226	C₂×C₁₉	1	2	yes	complemented
162	#2546	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1610	C₂×C₁₉	1	8	yes	complemented
164	#1502	C₂×C₁₉	1	2	yes	complemented
165	#1461	C₂×C₁₉	1	1	yes	characteristic, complemented
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	4	4	no	—
169	C₂²×C₁₉	—	4	4	no	—
170	C₂²×C₁₉	—	4	4	no	—
171	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
172	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
173	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₄×C₁₉	—	2	4	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	4	4	no	—
178	C₄×C₁₉	—	4	4	no	—
179	C₄×C₁₉	—	4	2	no	—
180	C₄×C₁₉	—	8	2	no	—
181	C₄×C₁₉	#447	1	4	yes	—
182	C₄×C₁₉	#456	1	2	yes	—
183	C₂×2₋¹⁺⁴	—	2	1	no	—
184	#550	—	2	2	no	—
185	D₄×Q₈	—	2	2	no	—
186	#548	—	2	4	no	—
187	#537	—	2	1	no	—
188	#537	C₂²×C₁₉	1	4	yes	—
189	#532	—	2	2	no	—
190	#532	C₂²×C₁₉	1	2	yes	—
191	C₂×C₄⋊Q₈	—	2	1	no	—
192	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
194	#500	—	2	2	no	—
195	#486	—	2	2	no	—
196	#480	C₂²×C₁₉	1	8	yes	—
197	#474	—	2	4	no	—
198	#431	C₂²×C₁₉	1	2	yes	—
199	#398	C₂²×C₁₉	1	4	yes	—
200	#391	—	2	1	no	—
201	#389	—	2	2	no	—
202	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
203	#327	C₂²×C₁₉	1	8	yes	—
204	#324	C₂²×C₁₉	1	4	yes	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	8	4	no	—
208	C₂×C₁₉	C₂×#447	1	1	yes	characteristic
209	C₂×C₁₉	C₂×#456	1	1	yes	characteristic
210	C₂×C₁₉	#3028	1	1	yes	characteristic
211	2₋¹⁺⁴	—	4	4	no	—
212	2₋¹⁺⁴	—	4	4	no	—
213	C₂×#41	—	2	4	no	—
214	C₂×#41	—	2	4	no	—
215	C₂×#41	—	2	2	no	—
216	C₂²×Q₈	—	2	4	no	—
217	C₂²×Q₈	—	2	1	no	—
218	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	4	no	—
223	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
224	#127	—	2	2	no	—
225	#127	D₄×C₁₉	1	2	yes	—
226	#124	—	2	4	no	—
227	#124	—	2	2	no	—
228	C₂²⋊Q₈	—	2	4	no	—
229	C₂²⋊Q₈	—	2	8	no	—
230	#121	—	2	4	no	—
231	#121	—	2	8	no	—
232	#121	—	2	2	no	—
233	#121	—	2	4	no	—
234	#121	C₂³×C₁₉	1	4	yes	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	2	4	no	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	D₄×C₁₉	1	4	yes	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	4	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	D₄×C₁₉	1	2	yes	—
249	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
250	#107	—	2	8	no	—
251	C₄⋊C₈	—	2	2	no	—
252	#102	—	2	8	no	—
253	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
254	C₄×C₈	—	2	2	no	—
255	#95	—	2	4	no	—
256	#95	C₂³×C₁₉	1	2	yes	—
257	C₁₉	#18408	1	1	yes	characteristic, direct product factor, Sylow
258	#41	—	4	8	no	—
259	#41	—	4	16	no	—
260	#41	—	4	8	no	—
261	#41	—	4	4	no	—
262	#41	—	4	4	no	—
263	C₂×Q₈	—	2	4	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	4	16	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	4	2	no	—
271	C₂×Q₈	D₈×C₁₉	1	4	yes	—
272	C₂×D₄	—	2	4	no	—
273	C₂×D₄	—	2	4	no	—
274	C₂×D₄	—	2	4	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	C₂×D₄×C₁₉	1	4	yes	—
278	C₂²×C₄	—	2	8	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
284	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
286	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
287	C₂×C₈	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	8	no	—
297	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
298	C₂²⋊C₄	—	2	8	no	—
299	C₂²⋊C₄	—	2	4	no	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	4	2	no	—
304	C₄²	C₂×D₄×C₁₉	1	2	yes	—
305	C₄²	C₂×D₄×C₁₉	1	2	yes	—
306	C₂³	—	2	4	no	—
307	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
308	Q₈	—	2	8	no	—
309	Q₈	—	4	8	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	8	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	8	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	4	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	8	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	16	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	8	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
335	C₂×C₄	C₁₉×#120	1	2	yes	—
336	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
337	C₂×C₄	C₂×D₈×C₁₉	1	4	yes	—
338	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
339	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
340	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
341	C₈	—	4	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	4	4	no	—
347	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
348	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
349	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	2	no	—
356	C₄	—	8	2	no	—
357	C₄	C₁₉×#447	1	4	yes	—
358	C₄	C₁₉×#456	1	2	yes	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	4	no	—
362	C₂	C₂×C₁₉×#447	1	1	yes	characteristic
363	C₂	C₂×C₁₉×#456	1	1	yes	characteristic
364	C₂	C₁₉×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₁₉×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ 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×#3226	C₂	semidirect
3. C₁₉×#2546	C₂	semidirect
4. C₁₉×#1610	C₂	semidirect
5. C₁₉×#1502	C₂	semidirect
6. C₁₉×#1461	C₂	semidirect
123. #18408	C₁₉	direct (N × H)
161. #3226	C₂×C₁₉	semidirect
162. #2546	C₂×C₁₉	semidirect
163. #1610	C₂×C₁₉	semidirect
164. #1502	C₂×C₁₉	semidirect
165. #1461	C₂×C₁₉	semidirect
257. C₁₉	#18408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3497	2
C₁₉×#3226	2
C₁₉×#2546	2
C₁₉×#1610	2
C₁₉×#1502	2
C₁₉×#1461	2
#18408	19

Subgroups — C19×#18408 #18024367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18408 #18024367