Subgroups — C₁₉×#62357 #18068316

Order 4864.

1926 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3497	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3392	C₂	1	4	yes	maximal, complemented
4	Q₈×SD₁₆×C₁₉	C₂	1	4	yes	maximal, complemented
5	C₁₉×#3362	C₂	1	4	yes	maximal, complemented
6	C₁₉×#3359	C₂	1	4	yes	maximal, complemented
7	C₁₉×#3246	C₂	1	6	yes	maximal, complemented
8	C₁₉×#3167	C₂	1	6	yes	maximal, complemented
9	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₉×2₋¹⁺⁴	—	2	1	no	—
11	Q₈²×C₁₉	C₂²	1	4	yes	—
12	C₁₉×#557	—	2	4	no	—
13	C₁₉×#557	C₂²	1	4	yes	—
14	C₁₉×#550	C₂²	1	4	yes	—
15	D₄×Q₈×C₁₉	C₂²	1	4	yes	—
16	C₁₉×#548	—	2	4	no	—
17	C₁₉×#544	—	2	3	no	—
18	C₁₉×#537	—	2	3	no	—
19	C₁₉×#533	C₂²	1	6	yes	—
20	C₁₉×#532	C₂²	1	6	yes	—
21	C₁₉×#519	—	2	1	no	—
22	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#498	C₂²	1	6	yes	—
24	C₁₉×#492	C₂²	1	6	yes	—
25	C₁₉×#487	C₂²	1	6	yes	—
26	C₁₉×#486	C₂²	1	6	yes	—
27	C₁₉×#476	C₂²	1	12	yes	—
28	C₁₉×#475	C₂²	1	12	yes	—
29	C₁₉×#465	C₂²	1	4	yes	—
30	C₁₉×#465	C₂²	1	12	yes	—
31	C₁₉×#461	C₂²	1	12	yes	—
32	C₁₉×#460	C₂²	1	12	yes	—
33	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
34	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
35	C₄×SD₁₆×C₁₉	C₂²	1	12	yes	—
36	C₄×SD₁₆×C₁₉	C₂²	1	12	yes	—
37	C₂×C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
38	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
39	C₁₉×2₋¹⁺⁴	—	4	8	no	—
40	C₂×#41×C₁₉	—	2	4	no	—
41	C₂×#41×C₁₉	—	2	6	no	—
42	C₂²×Q₈×C₁₉	—	2	4	no	—
43	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
44	C₂×SD₁₆×C₁₉	C₂³	1	4	yes	—
45	C₂×SD₁₆×C₁₉	C₂³	1	12	yes	—
46	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
48	C₁₉×C₄⋊Q₈	—	2	12	no	—
49	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
50	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
51	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
52	C₁₉×#127	C₂³	1	6	yes	—
53	C₁₉×#125	—	2	12	no	—
54	C₁₉×#125	—	2	6	no	—
55	C₁₉×#124	—	2	6	no	—
56	C₁₉×C₂²⋊Q₈	—	2	6	no	—
57	C₁₉×C₂²⋊Q₈	—	2	12	no	—
58	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
59	C₁₉×#121	—	2	6	no	—
60	C₁₉×#121	—	2	12	no	—
61	C₁₉×#121	C₂³	1	4	yes	—
62	C₄×Q₈×C₁₉	—	2	4	no	—
63	C₄×Q₈×C₁₉	—	2	4	no	—
64	C₄×Q₈×C₁₉	—	2	6	no	—
65	C₄×Q₈×C₁₉	D₄	1	4	yes	—
66	C₄×Q₈×C₁₉	C₂³	1	12	yes	—
67	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
68	C₄×D₄×C₁₉	—	2	6	no	—
69	C₄×D₄×C₁₉	C₂³	1	12	yes	—
70	C₁₉×#117	—	2	6	no	—
71	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
72	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
73	C₁₉×#106	C₂³	1	4	yes	—
74	C₁₉×#106	C₂³	1	12	yes	—
75	C₁₉×C₄⋊C₈	C₂³	1	12	yes	—
76	C₁₉×#103	C₂³	1	4	yes	—
77	C₁₉×#103	C₂³	1	12	yes	—
78	C₁₉×#102	C₂³	1	4	yes	—
79	C₁₉×#102	C₂³	1	12	yes	—
80	C₄×C₈×C₁₉	C₂³	1	6	yes	—
81	C₄×C₈×C₁₉	C₂³	1	6	yes	—
82	#41×C₁₉	—	4	16	no	—
83	#41×C₁₉	—	4	24	no	—
84	C₂×Q₈×C₁₉	—	2	6	no	—
85	C₂×Q₈×C₁₉	—	2	6	no	—
86	C₂×Q₈×C₁₉	—	4	16	no	—
87	C₂×Q₈×C₁₉	SD₁₆	1	8	yes	complemented
88	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
89	C₂×Q₈×C₁₉	C₂⁴	1	4	yes	—
90	C₂×D₄×C₁₉	—	2	4	no	—
91	C₂×D₄×C₁₉	—	2	6	no	—
92	C₂×D₄×C₁₉	—	2	6	no	—
93	C₂×D₄×C₁₉	C₂⁴	1	4	yes	—
94	C₂²×C₄×C₁₉	—	2	12	no	—
95	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
96	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
97	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
98	SD₁₆×C₁₉	—	2	8	no	—
99	SD₁₆×C₁₉	—	4	12	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	4	3	no	—
102	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
103	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉	—	2	6	no	—
106	C₄⋊C₄×C₁₉	—	2	6	no	—
107	C₄⋊C₄×C₁₉	—	2	6	no	—
108	C₄⋊C₄×C₁₉	—	2	4	no	—
109	C₄⋊C₄×C₁₉	—	2	6	no	—
110	C₄⋊C₄×C₁₉	—	2	12	no	—
111	C₄⋊C₄×C₁₉	C₂×D₄	1	12	yes	—
112	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
113	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
114	C₂²⋊C₄×C₁₉	—	2	4	no	—
115	C₂²⋊C₄×C₁₉	—	2	4	no	—
116	C₂²⋊C₄×C₁₉	—	2	12	no	—
117	C₄²×C₁₉	—	2	12	no	—
118	C₄²×C₁₉	C₂×D₄	1	6	yes	—
119	C₄²×C₁₉	C₂⁴	1	6	yes	—
120	#62357	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₁₉	—	2	4	no	—
122	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₁₉	—	2	4	no	—
124	Q₈×C₁₉	—	2	4	no	—
125	Q₈×C₁₉	—	4	12	no	—
126	Q₈×C₁₉	—	8	6	no	—
127	Q₈×C₁₉	C₂×SD₁₆	1	16	yes	—
128	D₄×C₁₉	—	2	4	no	—
129	D₄×C₁₉	—	2	4	no	—
130	D₄×C₁₉	—	4	8	no	—
131	D₄×C₁₉	—	4	12	no	—
132	D₄×C₁₉	—	8	6	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	4	no	—
136	C₂×C₄×C₁₉	—	2	4	no	—
137	C₂×C₄×C₁₉	—	2	4	no	—
138	C₂×C₄×C₁₉	—	4	3	no	—
139	C₂×C₄×C₁₉	—	4	24	no	—
140	C₂×C₄×C₁₉	C₂×SD₁₆	1	12	yes	—
141	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
143	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
144	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	—	2	6	no	—
147	C₈×C₁₉	2₋¹⁺⁴	1	2	yes	complemented
148	#3502	C₂×C₁₉	1	1	yes	characteristic, complemented
149	#3497	C₂×C₁₉	1	1	yes	characteristic
150	#3392	C₂×C₁₉	1	4	yes	complemented
151	Q₈×SD₁₆	C₂×C₁₉	1	4	yes	complemented
152	#3362	C₂×C₁₉	1	4	yes	complemented
153	#3359	C₂×C₁₉	1	4	yes	complemented
154	#3246	C₂×C₁₉	1	6	yes	complemented
155	#3167	C₂×C₁₉	1	6	yes	complemented
156	C₂×C₈×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	4	4	no	—
160	C₂²×C₁₉	—	4	4	no	—
161	C₂²×C₁₉	—	4	4	no	—
162	C₂²×C₁₉	C₂²×SD₁₆	1	2	yes	—
163	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
164	C₄×C₁₉	—	2	6	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	4	4	no	—
167	C₄×C₁₉	—	4	4	no	—
168	C₄×C₁₉	—	4	4	no	—
169	C₄×C₁₉	C₂²×SD₁₆	1	12	yes	—
170	C₄×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
171	C₄×C₁₉	#585	1	1	yes	characteristic
172	C₂×2₋¹⁺⁴	—	2	1	no	—
173	Q₈²	C₂²×C₁₉	1	4	yes	—
174	#557	—	2	4	no	—
175	#557	C₂²×C₁₉	1	4	yes	—
176	#550	C₂²×C₁₉	1	4	yes	—
177	D₄×Q₈	C₂²×C₁₉	1	4	yes	—
178	#548	—	2	4	no	—
179	#544	—	2	3	no	—
180	#537	—	2	3	no	—
181	#533	C₂²×C₁₉	1	6	yes	—
182	#532	C₂²×C₁₉	1	6	yes	—
183	#519	—	2	1	no	—
184	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
185	#498	C₂²×C₁₉	1	6	yes	—
186	#492	C₂²×C₁₉	1	6	yes	—
187	#487	C₂²×C₁₉	1	6	yes	—
188	#486	C₂²×C₁₉	1	6	yes	—
189	#476	C₂²×C₁₉	1	12	yes	—
190	#475	C₂²×C₁₉	1	12	yes	—
191	#465	C₂²×C₁₉	1	4	yes	—
192	#465	C₂²×C₁₉	1	12	yes	—
193	#461	C₂²×C₁₉	1	12	yes	—
194	#460	C₂²×C₁₉	1	12	yes	—
195	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
196	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
197	C₄×SD₁₆	C₂²×C₁₉	1	12	yes	—
198	C₄×SD₁₆	C₂²×C₁₉	1	12	yes	—
199	C₂×C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
200	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
201	C₂×C₁₉	—	2	2	no	—
202	C₂×C₁₉	—	8	4	no	—
203	C₂×C₁₉	#3497	1	1	yes	characteristic
204	C₂×C₁₉	C₂³×SD₁₆	1	1	yes	characteristic
205	C₂×C₁₉	#3584	1	1	yes	characteristic
206	2₋¹⁺⁴	—	4	8	no	—
207	C₂×#41	—	2	4	no	—
208	C₂×#41	—	2	6	no	—
209	C₂²×Q₈	—	2	4	no	—
210	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
211	C₂×SD₁₆	C₂³×C₁₉	1	4	yes	—
212	C₂×SD₁₆	C₂³×C₁₉	1	12	yes	—
213	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
215	C₄⋊Q₈	—	2	12	no	—
216	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
217	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
218	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
219	#127	C₂³×C₁₉	1	6	yes	—
220	#125	—	2	12	no	—
221	#125	—	2	6	no	—
222	#124	—	2	6	no	—
223	C₂²⋊Q₈	—	2	6	no	—
224	C₂²⋊Q₈	—	2	12	no	—
225	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
226	#121	—	2	6	no	—
227	#121	—	2	12	no	—
228	#121	C₂³×C₁₉	1	4	yes	—
229	C₄×Q₈	—	2	4	no	—
230	C₄×Q₈	—	2	4	no	—
231	C₄×Q₈	—	2	6	no	—
232	C₄×Q₈	D₄×C₁₉	1	4	yes	—
233	C₄×Q₈	C₂³×C₁₉	1	12	yes	—
234	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
235	C₄×D₄	—	2	6	no	—
236	C₄×D₄	C₂³×C₁₉	1	12	yes	—
237	#117	—	2	6	no	—
238	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
239	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
240	#106	C₂³×C₁₉	1	4	yes	—
241	#106	C₂³×C₁₉	1	12	yes	—
242	C₄⋊C₈	C₂³×C₁₉	1	12	yes	—
243	#103	C₂³×C₁₉	1	4	yes	—
244	#103	C₂³×C₁₉	1	12	yes	—
245	#102	C₂³×C₁₉	1	4	yes	—
246	#102	C₂³×C₁₉	1	12	yes	—
247	C₄×C₈	C₂³×C₁₉	1	6	yes	—
248	C₄×C₈	C₂³×C₁₉	1	6	yes	—
249	C₁₉	#62357	1	1	yes	characteristic, direct product factor, Sylow
250	#41	—	4	16	no	—
251	#41	—	4	24	no	—
252	C₂×Q₈	—	2	6	no	—
253	C₂×Q₈	—	2	6	no	—
254	C₂×Q₈	—	4	16	no	—
255	C₂×Q₈	SD₁₆×C₁₉	1	8	yes	complemented
256	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
257	C₂×Q₈	C₂⁴×C₁₉	1	4	yes	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	2	6	no	—
260	C₂×D₄	—	2	6	no	—
261	C₂×D₄	C₂⁴×C₁₉	1	4	yes	—
262	C₂²×C₄	—	2	12	no	—
263	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
264	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
265	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
266	SD₁₆	—	2	8	no	—
267	SD₁₆	—	4	12	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	3	no	—
270	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
271	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
272	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	6	no	—
274	C₄⋊C₄	—	2	6	no	—
275	C₄⋊C₄	—	2	6	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	6	no	—
278	C₄⋊C₄	—	2	12	no	—
279	C₄⋊C₄	C₂×D₄×C₁₉	1	12	yes	—
280	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
281	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	—	2	12	no	—
285	C₄²	—	2	12	no	—
286	C₄²	C₂×D₄×C₁₉	1	6	yes	—
287	C₄²	C₂⁴×C₁₉	1	6	yes	—
288	C₂³	—	2	4	no	—
289	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	2	4	no	—
292	Q₈	—	4	12	no	—
293	Q₈	—	8	6	no	—
294	Q₈	C₂×SD₁₆×C₁₉	1	16	yes	—
295	D₄	—	2	4	no	—
296	D₄	—	2	4	no	—
297	D₄	—	4	8	no	—
298	D₄	—	4	12	no	—
299	D₄	—	8	6	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	24	no	—
307	C₂×C₄	C₂×SD₁₆×C₁₉	1	12	yes	—
308	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
310	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
311	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	6	no	—
314	C₈	C₁₉×2₋¹⁺⁴	1	2	yes	complemented
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	C₂²×SD₁₆×C₁₉	1	2	yes	—
321	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	6	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	4	no	—
327	C₄	C₂²×SD₁₆×C₁₉	1	12	yes	—
328	C₄	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
329	C₄	C₁₉×#585	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₁₉×#3497	1	1	yes	characteristic
333	C₂	C₂³×SD₁₆×C₁₉	1	1	yes	characteristic
334	C₂	C₁₉×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₁₉×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3502	C₂	semidirect
3. C₁₉×#3392	C₂	semidirect
4. Q₈×SD₁₆×C₁₉	C₂	semidirect
5. C₁₉×#3362	C₂	semidirect
6. C₁₉×#3359	C₂	semidirect
7. C₁₉×#3246	C₂	semidirect
8. C₁₉×#3167	C₂	semidirect
9. C₂×C₈×Q₈×C₁₉	C₂	semidirect
87. C₂×Q₈×C₁₉	SD₁₆	semidirect
120. #62357	C₁₉	direct (N × H)
147. C₈×C₁₉	2₋¹⁺⁴	semidirect
148. #3502	C₂×C₁₉	semidirect
150. #3392	C₂×C₁₉	semidirect
151. Q₈×SD₁₆	C₂×C₁₉	semidirect
152. #3362	C₂×C₁₉	semidirect
153. #3359	C₂×C₁₉	semidirect
154. #3246	C₂×C₁₉	semidirect
155. #3167	C₂×C₁₉	semidirect
156. C₂×C₈×Q₈	C₂×C₁₉	semidirect
249. C₁₉	#62357	direct (N × H)
255. C₂×Q₈	SD₁₆×C₁₉	semidirect
314. C₈	C₁₉×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3502	2
C₁₉×#3497	2
C₁₉×#3392	2
Q₈×SD₁₆×C₁₉	2
C₁₉×#3362	2
C₁₉×#3359	2
C₁₉×#3246	2
C₁₉×#3167	2
C₂×C₈×Q₈×C₁₉	2
#62357	19

Subgroups — C19×#62357 #18068316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62357 #18068316