Subgroups — C₁₉×#8354 #18014313

Order 4864.

950 conjugacy classes of subgroups.

The subgroup lattice has 348 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×#1984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1801	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1547	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×2₋¹⁺⁴	—	2	1	no	—
7	C₁₉×#550	—	2	2	no	—
8	D₄×Q₈×C₁₉	—	2	2	no	—
9	C₁₉×#548	—	2	4	no	—
10	C₁₉×#537	—	2	1	no	—
11	C₁₉×#537	C₄	1	4	yes	—
12	C₁₉×#532	—	2	2	no	—
13	C₁₉×#532	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#423	C₂²	1	2	yes	—
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#101	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#343	—	2	1	no	—
20	C₁₉×#336	—	2	2	no	—
21	C₁₉×#332	—	2	2	no	—
22	C₁₉×#331	—	2	2	no	—
23	C₁₉×D₄⋊C₈	—	2	4	no	—
24	C₁₉×2₋¹⁺⁴	—	4	4	no	—
25	C₁₉×2₋¹⁺⁴	—	4	2	no	—
26	C₁₉×2₋¹⁺⁴	—	4	2	no	—
27	C₂×#41×C₁₉	—	2	4	no	—
28	C₂×#41×C₁₉	—	2	4	no	—
29	C₂×#41×C₁₉	—	2	2	no	—
30	C₂²×Q₈×C₁₉	—	2	4	no	—
31	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
37	C₁₉×#127	—	2	2	no	—
38	C₁₉×#127	D₄	1	2	yes	—
39	C₁₉×#124	—	2	4	no	—
40	C₁₉×#124	—	2	2	no	—
41	C₁₉×C₂²⋊Q₈	—	2	4	no	—
42	C₁₉×C₂²⋊Q₈	—	2	8	no	—
43	C₁₉×#121	—	2	4	no	—
44	C₁₉×#121	—	2	8	no	—
45	C₁₉×#121	—	2	4	no	—
46	C₁₉×#121	—	2	2	no	—
47	C₁₉×#121	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₁₉	—	2	2	no	—
49	C₄×Q₈×C₁₉	—	2	2	no	—
50	C₄×D₄×C₁₉	—	2	4	no	—
51	C₄×D₄×C₁₉	—	2	4	no	—
52	C₄×D₄×C₁₉	—	2	2	no	—
53	C₄×D₄×C₁₉	D₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	D₄	1	2	yes	—
57	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₁₉×C₄⋊C₈	—	2	2	no	—
59	C₁₉×C₄⋊C₈	—	2	2	no	—
60	C₁₉×C₄⋊C₈	—	2	2	no	—
61	C₁₉×#101	—	4	2	no	—
62	C₄×C₈×C₁₉	—	2	2	no	—
63	#41×C₁₉	—	4	8	no	—
64	#41×C₁₉	—	4	8	no	—
65	#41×C₁₉	—	4	8	no	—
66	#41×C₁₉	—	4	8	no	—
67	#41×C₁₉	—	4	2	no	—
68	#41×C₁₉	—	4	2	no	—
69	#41×C₁₉	—	4	2	no	—
70	#41×C₁₉	—	4	2	no	—
71	C₂×Q₈×C₁₉	—	2	4	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	4	8	no	—
76	C₂×Q₈×C₁₉	—	4	8	no	—
77	C₂×Q₈×C₁₉	D₈	1	2	yes	—
78	C₂×Q₈×C₁₉	SD₁₆	1	2	yes	—
79	C₂×D₄×C₁₉	—	2	4	no	—
80	C₂×D₄×C₁₉	—	2	4	no	—
81	C₂×D₄×C₁₉	—	2	4	no	—
82	C₂×D₄×C₁₉	—	2	2	no	—
83	C₂×D₄×C₁₉	—	2	2	no	—
84	C₂×D₄×C₁₉	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₁₉	—	2	8	no	—
86	C₂²×C₄×C₁₉	—	2	4	no	—
87	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	M₁₆×C₁₉	—	4	2	no	—
93	M₁₆×C₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	4	2	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	4	2	no	—
100	C₄⋊C₄×C₁₉	—	2	4	no	—
101	C₄⋊C₄×C₁₉	—	2	4	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
105	C₂²⋊C₄×C₁₉	—	2	8	no	—
106	C₂²⋊C₄×C₁₉	—	2	4	no	—
107	C₂²⋊C₄×C₁₉	—	2	4	no	—
108	C₄²×C₁₉	—	2	2	no	—
109	C₄²×C₁₉	—	2	2	no	—
110	C₄²×C₁₉	C₂×D₄	1	2	yes	—
111	C₄²×C₁₉	C₂×D₄	1	2	yes	—
112	#8354	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	—	2	4	no	—
114	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	Q₈×C₁₉	—	2	4	no	—
116	Q₈×C₁₉	—	2	4	no	—
117	Q₈×C₁₉	—	4	8	no	—
118	Q₈×C₁₉	—	4	2	no	—
119	Q₈×C₁₉	—	4	2	no	—
120	D₄×C₁₉	—	4	4	no	—
121	D₄×C₁₉	—	4	8	no	—
122	D₄×C₁₉	—	4	4	no	—
123	D₄×C₁₉	—	4	4	no	—
124	D₄×C₁₉	—	4	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	2	no	—
127	D₄×C₁₉	—	4	2	no	—
128	D₄×C₁₉	—	8	4	no	—
129	C₂×C₄×C₁₉	—	2	2	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	4	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	16	no	—
137	C₂×C₄×C₁₉	—	4	4	no	—
138	C₂×C₄×C₁₉	—	4	4	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	#102	1	4	yes	—
141	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#120	1	2	yes	—
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₁₉	—	4	2	no	—
149	C₈×C₁₉	—	8	2	no	—
150	C₈×C₁₉	—	8	1	no	—
151	#3497	C₂×C₁₉	1	1	yes	characteristic
152	#1984	C₂×C₁₉	1	1	yes	characteristic, complemented
153	#1801	C₂×C₁₉	1	1	yes	characteristic, complemented
154	#1547	C₂×C₁₉	1	2	yes	complemented
155	#1492	C₂×C₁₉	1	2	yes	complemented
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	4	4	no	—
159	C₂²×C₁₉	—	4	4	no	—
160	C₂²×C₁₉	—	4	4	no	—
161	C₂²×C₁₉	#379	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
163	C₂²×C₁₉	#418	1	1	yes	characteristic
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	4	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	4	4	no	—
168	C₄×C₁₉	—	4	2	no	—
169	C₄×C₁₉	#447	1	2	yes	—
170	C₄×C₁₉	#450	1	2	yes	—
171	C₄×C₁₉	#454	1	1	yes	characteristic
172	C₄×C₁₉	#456	1	1	yes	characteristic
173	C₂×2₋¹⁺⁴	—	2	1	no	—
174	#550	—	2	2	no	—
175	D₄×Q₈	—	2	2	no	—
176	#548	—	2	4	no	—
177	#537	—	2	1	no	—
178	#537	C₄×C₁₉	1	4	yes	—
179	#532	—	2	2	no	—
180	#532	C₂²×C₁₉	1	2	yes	—
181	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
182	#423	C₂²×C₁₉	1	2	yes	—
183	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×#101	—	2	1	no	—
185	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
186	#343	—	2	1	no	—
187	#336	—	2	2	no	—
188	#332	—	2	2	no	—
189	#331	—	2	2	no	—
190	D₄⋊C₈	—	2	4	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	8	4	no	—
194	C₂×C₁₉	#1786	1	1	yes	characteristic
195	C₂×C₁₉	#1791	1	1	yes	characteristic
196	C₂×C₁₉	#1793	1	1	yes	characteristic
197	2₋¹⁺⁴	—	4	4	no	—
198	2₋¹⁺⁴	—	4	2	no	—
199	2₋¹⁺⁴	—	4	2	no	—
200	C₂×#41	—	2	4	no	—
201	C₂×#41	—	2	4	no	—
202	C₂×#41	—	2	2	no	—
203	C₂²×Q₈	—	2	4	no	—
204	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
205	C₂×M₁₆	—	2	2	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
210	#127	—	2	2	no	—
211	#127	D₄×C₁₉	1	2	yes	—
212	#124	—	2	4	no	—
213	#124	—	2	2	no	—
214	C₂²⋊Q₈	—	2	4	no	—
215	C₂²⋊Q₈	—	2	8	no	—
216	#121	—	2	4	no	—
217	#121	—	2	8	no	—
218	#121	—	2	4	no	—
219	#121	—	2	2	no	—
220	#121	C₂×C₄×C₁₉	1	4	yes	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×D₄	—	2	4	no	—
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	D₄×C₁₉	1	4	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
228	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₁₉	1	2	yes	—
230	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	#101	—	4	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₁₉	#8354	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	4	8	no	—
238	#41	—	4	8	no	—
239	#41	—	4	8	no	—
240	#41	—	4	2	no	—
241	#41	—	4	2	no	—
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	#41	—	4	8	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	8	no	—
250	C₂×Q₈	—	4	8	no	—
251	C₂×Q₈	D₈×C₁₉	1	2	yes	—
252	C₂×Q₈	SD₁₆×C₁₉	1	2	yes	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	C₂²⋊C₄×C₁₉	1	4	yes	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
262	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
266	M₁₆	—	4	2	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
279	C₂²⋊C₄	—	2	8	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂×D₄×C₁₉	1	2	yes	—
285	C₄²	C₂×D₄×C₁₉	1	2	yes	—
286	C₂³	—	2	4	no	—
287	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	2	4	no	—
290	Q₈	—	4	8	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	8	no	—
295	D₄	—	4	4	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	16	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	C₁₉×#102	1	4	yes	—
314	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
315	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
316	C₂×C₄	C₁₉×#120	1	2	yes	—
317	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
318	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
319	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
320	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
321	C₈	—	4	2	no	—
322	C₈	—	8	2	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	4	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	4	4	no	—
329	C₂²	C₁₉×#379	1	1	yes	characteristic
330	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
331	C₂²	C₁₉×#418	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	4	no	—
336	C₄	—	4	2	no	—
337	C₄	C₁₉×#447	1	2	yes	—
338	C₄	C₁₉×#450	1	2	yes	—
339	C₄	C₁₉×#454	1	1	yes	characteristic
340	C₄	C₁₉×#456	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	8	4	no	—
344	C₂	C₁₉×#1786	1	1	yes	characteristic
345	C₂	C₁₉×#1791	1	1	yes	characteristic
346	C₂	C₁₉×#1793	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×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×#1984	C₂	semidirect
3. C₁₉×#1801	C₂	semidirect
4. C₁₉×#1547	C₂	semidirect
5. C₁₉×#1492	C₂	semidirect
112. #8354	C₁₉	direct (N × H)
152. #1984	C₂×C₁₉	semidirect
153. #1801	C₂×C₁₉	semidirect
154. #1547	C₂×C₁₉	semidirect
155. #1492	C₂×C₁₉	semidirect
236. C₁₉	#8354	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×#1984	2
C₁₉×#1801	2
C₁₉×#1547	2
C₁₉×#1492	2
#8354	19

Subgroups — C19×#8354 #18014313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8354 #18014313