Subgroups — C₁₉×#62373 #18068332

Order 4864.

1812 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3548	C₂	1	2	yes	maximal, complemented
2	C₁₉×#3391	C₂	1	4	yes	maximal, complemented
3	C₁₉×#3380	C₂	1	4	yes	maximal, complemented
4	C₁₉×#3366	C₂	1	8	yes	maximal, complemented
5	C₁₉×#3259	C₂	1	4	yes	maximal, complemented
6	C₁₉×#3253	C₂	1	2	yes	maximal, complemented
7	C₁₉×#3176	C₂	1	4	yes	maximal, complemented
8	C₁₉×#3169	C₂	1	2	yes	maximal, complemented
9	C₁₉×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₉×#562	—	2	4	no	—
11	C₁₉×#554	C₂²	1	8	yes	—
12	C₁₉×#550	C₂²	1	8	yes	—
13	C₁₉×#547	—	2	4	no	—
14	C₁₉×#540	C₂²	1	8	yes	—
15	C₁₉×#537	—	2	8	no	—
16	C₁₉×#533	C₂²	1	1	yes	characteristic
17	C₁₉×#532	C₂²	1	4	yes	—
18	C₁₉×#500	C₂²	1	2	yes	—
19	C₁₉×#499	C₂²	1	4	yes	—
20	C₁₉×#493	C₂²	1	2	yes	—
21	C₁₉×#493	C₂²	1	4	yes	—
22	C₁₉×#490	C₂²	1	8	yes	—
23	C₁₉×#486	C₂²	1	4	yes	—
24	C₁₉×#478	C₂²	1	16	yes	—
25	C₁₉×#476	C₂²	1	8	yes	—
26	C₁₉×#469	C₂²	1	8	yes	—
27	C₁₉×#466	C₂²	1	4	yes	—
28	C₁₉×#466	C₂²	1	4	yes	—
29	C₁₉×#459	C₂²	1	8	yes	—
30	C₁₉×#459	C₂²	1	16	yes	—
31	C₁₉×#446	C₂²	1	4	yes	—
32	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
33	C₁₉×#442	C₂²	1	8	yes	—
34	C₄×D₈×C₁₉	C₂²	1	4	yes	—
35	C₄×D₈×C₁₉	C₂²	1	4	yes	—
36	C₄×D₈×C₁₉	C₂²	1	8	yes	—
37	C₁₉×#424	C₂²	1	2	yes	—
38	C₁₉×#423	C₂²	1	1	yes	characteristic
39	C₁₉×#405	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
41	C₂×#41×C₁₉	—	2	8	no	—
42	C₂×#41×C₁₉	—	2	4	no	—
43	C₂×D₈×C₁₉	C₂³	1	4	yes	—
44	C₂×D₈×C₁₉	C₂³	1	4	yes	—
45	C₂×D₈×C₁₉	C₂³	1	8	yes	—
46	C₂×M₁₆×C₁₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
50	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
51	C₁₉×#127	C₂³	1	4	yes	—
52	C₁₉×#127	C₂³	1	8	yes	—
53	C₁₉×#125	D₄	1	2	yes	—
54	C₁₉×#125	C₂³	1	8	yes	—
55	C₁₉×#124	—	2	4	no	—
56	C₁₉×#123	—	2	16	no	—
57	C₁₉×C₂²⋊Q₈	—	2	8	no	—
58	C₁₉×C₂²⋊Q₈	—	2	16	no	—
59	C₁₉×C₂²⋊Q₈	D₄	1	4	yes	—
60	C₁₉×#121	—	2	4	no	—
61	C₁₉×#121	—	2	8	no	—
62	C₁₉×#121	—	2	16	no	—
63	C₁₉×#121	—	2	16	no	—
64	C₁₉×#121	C₂³	1	8	yes	—
65	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
66	C₄×D₄×C₁₉	—	2	4	no	—
67	C₄×D₄×C₁₉	C₂³	1	8	yes	—
68	C₄×D₄×C₁₉	C₂³	1	16	yes	—
69	C₁₉×#117	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
71	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
72	C₁₉×#107	C₂³	1	4	yes	—
73	C₁₉×#107	C₂³	1	4	yes	—
74	C₁₉×#106	C₂³	1	8	yes	—
75	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
76	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
77	C₁₉×#102	C₂³	1	8	yes	—
78	C₁₉×#102	C₂³	1	8	yes	—
79	C₁₉×#102	C₂³	1	16	yes	—
80	C₁₉×#97	C₂³	1	4	yes	—
81	C₄×C₈×C₁₉	C₂³	1	2	yes	—
82	C₄×C₈×C₁₉	C₂³	1	2	yes	—
83	C₄×C₈×C₁₉	C₂³	1	4	yes	—
84	#41×C₁₉	—	8	8	no	—
85	#41×C₁₉	—	8	16	no	—
86	C₂×Q₈×C₁₉	—	2	4	no	—
87	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
88	C₂×D₄×C₁₉	—	2	8	no	—
89	C₂×D₄×C₁₉	—	2	4	no	—
90	C₂×D₄×C₁₉	—	2	16	no	—
91	C₂×D₄×C₁₉	—	2	4	no	—
92	C₂×D₄×C₁₉	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₁₉	—	2	8	no	—
94	C₂²×C₄×C₁₉	—	2	16	no	—
95	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
97	D₈×C₁₉	—	2	8	no	—
98	D₈×C₁₉	—	4	4	no	—
99	D₈×C₁₉	—	4	8	no	—
100	M₁₆×C₁₉	—	4	2	no	—
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₂×C₈×C₁₉	—	4	1	no	—
103	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
104	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₁₉	C₂⁴	1	4	yes	—
106	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₁₉	—	2	4	no	—
108	C₄⋊C₄×C₁₉	—	2	16	no	—
109	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
110	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
111	C₄⋊C₄×C₁₉	C₂×D₄	1	8	yes	—
112	C₄⋊C₄×C₁₉	C₂⁴	1	8	yes	—
113	C₂²⋊C₄×C₁₉	—	2	8	no	—
114	C₂²⋊C₄×C₁₉	—	2	16	no	—
115	C₂²⋊C₄×C₁₉	—	2	8	no	—
116	C₂²⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
117	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₉	C₂⁴	1	2	yes	—
120	C₄²×C₁₉	C₂⁴	1	4	yes	—
121	#62373	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₁₉	—	2	8	no	—
123	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₁₉	—	2	8	no	—
125	Q₈×C₁₉	—	8	4	no	—
126	D₄×C₁₉	—	2	8	no	—
127	D₄×C₁₉	—	2	8	no	—
128	D₄×C₁₉	—	4	8	no	—
129	D₄×C₁₉	—	8	4	no	—
130	D₄×C₁₉	—	8	16	no	—
131	D₄×C₁₉	—	8	8	no	—
132	C₂×C₄×C₁₉	—	2	4	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	8	no	—
136	C₂×C₄×C₁₉	—	4	16	no	—
137	C₂×C₄×C₁₉	—	8	16	no	—
138	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
141	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
143	C₈×C₁₉	—	2	1	no	—
144	C₈×C₁₉	—	2	2	no	—
145	C₈×C₁₉	—	2	4	no	—
146	C₈×C₁₉	2₋¹⁺⁴	1	2	yes	—
147	#3548	C₂×C₁₉	1	2	yes	complemented
148	#3391	C₂×C₁₉	1	4	yes	complemented
149	#3380	C₂×C₁₉	1	4	yes	complemented
150	#3366	C₂×C₁₉	1	8	yes	complemented
151	#3259	C₂×C₁₉	1	4	yes	complemented
152	#3253	C₂×C₁₉	1	2	yes	complemented
153	#3176	C₂×C₁₉	1	4	yes	complemented
154	#3169	C₂×C₁₉	1	2	yes	complemented
155	#2960	C₂×C₁₉	1	1	yes	characteristic, complemented
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	4	8	no	—
159	C₂²×C₁₉	—	4	8	no	—
160	C₂²×C₁₉	—	4	8	no	—
161	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	8	no	—
165	C₄×C₁₉	—	4	8	no	—
166	C₄×C₁₉	#576	1	4	yes	—
167	C₄×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
168	C₄×C₁₉	#585	1	1	yes	characteristic
169	#562	—	2	4	no	—
170	#554	C₂²×C₁₉	1	8	yes	—
171	#550	C₂²×C₁₉	1	8	yes	—
172	#547	—	2	4	no	—
173	#540	C₂²×C₁₉	1	8	yes	—
174	#537	—	2	8	no	—
175	#533	C₂²×C₁₉	1	1	yes	characteristic
176	#532	C₂²×C₁₉	1	4	yes	—
177	#500	C₂²×C₁₉	1	2	yes	—
178	#499	C₂²×C₁₉	1	4	yes	—
179	#493	C₂²×C₁₉	1	2	yes	—
180	#493	C₂²×C₁₉	1	4	yes	—
181	#490	C₂²×C₁₉	1	8	yes	—
182	#486	C₂²×C₁₉	1	4	yes	—
183	#478	C₂²×C₁₉	1	16	yes	—
184	#476	C₂²×C₁₉	1	8	yes	—
185	#469	C₂²×C₁₉	1	8	yes	—
186	#466	C₂²×C₁₉	1	4	yes	—
187	#466	C₂²×C₁₉	1	4	yes	—
188	#459	C₂²×C₁₉	1	8	yes	—
189	#459	C₂²×C₁₉	1	16	yes	—
190	#446	C₂²×C₁₉	1	4	yes	—
191	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
192	#442	C₂²×C₁₉	1	8	yes	—
193	C₄×D₈	C₂²×C₁₉	1	4	yes	—
194	C₄×D₈	C₂²×C₁₉	1	4	yes	—
195	C₄×D₈	C₂²×C₁₉	1	8	yes	—
196	#424	C₂²×C₁₉	1	2	yes	—
197	#423	C₂²×C₁₉	1	1	yes	characteristic
198	#405	C₂²×C₁₉	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
200	C₂×C₁₉	—	4	1	no	—
201	C₂×C₁₉	—	8	8	no	—
202	C₂×C₁₉	#3497	1	1	yes	characteristic
203	C₂×C₁₉	C₂×#576	1	2	yes	—
204	C₂×#41	—	2	8	no	—
205	C₂×#41	—	2	4	no	—
206	C₂×D₈	C₂³×C₁₉	1	4	yes	—
207	C₂×D₈	C₂³×C₁₉	1	4	yes	—
208	C₂×D₈	C₂³×C₁₉	1	8	yes	—
209	C₂×M₁₆	C₂³×C₁₉	1	2	yes	—
210	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
212	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
214	#127	C₂³×C₁₉	1	4	yes	—
215	#127	C₂³×C₁₉	1	8	yes	—
216	#125	D₄×C₁₉	1	2	yes	—
217	#125	C₂³×C₁₉	1	8	yes	—
218	#124	—	2	4	no	—
219	#123	—	2	16	no	—
220	C₂²⋊Q₈	—	2	8	no	—
221	C₂²⋊Q₈	—	2	16	no	—
222	C₂²⋊Q₈	D₄×C₁₉	1	4	yes	—
223	#121	—	2	4	no	—
224	#121	—	2	8	no	—
225	#121	—	2	16	no	—
226	#121	—	2	16	no	—
227	#121	C₂³×C₁₉	1	8	yes	—
228	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
229	C₄×D₄	—	2	4	no	—
230	C₄×D₄	C₂³×C₁₉	1	8	yes	—
231	C₄×D₄	C₂³×C₁₉	1	16	yes	—
232	#117	C₂³×C₁₉	1	2	yes	—
233	C₂×C₄⋊C₄	—	2	8	no	—
234	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
235	#107	C₂³×C₁₉	1	4	yes	—
236	#107	C₂³×C₁₉	1	4	yes	—
237	#106	C₂³×C₁₉	1	8	yes	—
238	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
239	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
240	#102	C₂³×C₁₉	1	8	yes	—
241	#102	C₂³×C₁₉	1	8	yes	—
242	#102	C₂³×C₁₉	1	16	yes	—
243	#97	C₂³×C₁₉	1	4	yes	—
244	C₄×C₈	C₂³×C₁₉	1	2	yes	—
245	C₄×C₈	C₂³×C₁₉	1	2	yes	—
246	C₄×C₈	C₂³×C₁₉	1	4	yes	—
247	C₁₉	#62373	1	1	yes	characteristic, direct product factor, Sylow
248	#41	—	8	8	no	—
249	#41	—	8	16	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	16	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	C₂⁴×C₁₉	1	8	yes	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	16	no	—
259	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
260	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
261	D₈	—	2	8	no	—
262	D₈	—	4	4	no	—
263	D₈	—	4	8	no	—
264	M₁₆	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
268	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
269	C₂×C₈	C₂⁴×C₁₉	1	4	yes	—
270	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	16	no	—
273	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
274	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
275	C₄⋊C₄	C₂×D₄×C₁₉	1	8	yes	—
276	C₄⋊C₄	C₂⁴×C₁₉	1	8	yes	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	2	16	no	—
279	C₂²⋊C₄	—	2	8	no	—
280	C₂²⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
281	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
283	C₄²	C₂⁴×C₁₉	1	2	yes	—
284	C₄²	C₂⁴×C₁₉	1	4	yes	—
285	C₂³	—	2	8	no	—
286	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	Q₈	—	8	4	no	—
289	D₄	—	2	8	no	—
290	D₄	—	2	8	no	—
291	D₄	—	4	8	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	16	no	—
294	D₄	—	8	8	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	4	16	no	—
300	C₂×C₄	—	8	16	no	—
301	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
304	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
305	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	4	no	—
309	C₈	C₁₉×2₋¹⁺⁴	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	8	no	—
313	C₂²	—	4	8	no	—
314	C₂²	—	4	8	no	—
315	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	8	no	—
319	C₄	—	4	8	no	—
320	C₄	C₁₉×#576	1	4	yes	—
321	C₄	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
322	C₄	C₁₉×#585	1	1	yes	characteristic
323	C₂	—	4	1	no	—
324	C₂	—	8	8	no	—
325	C₂	C₁₉×#3497	1	1	yes	characteristic
326	C₂	C₂×C₁₉×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₁₉×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3548	C₂	semidirect
2. C₁₉×#3391	C₂	semidirect
3. C₁₉×#3380	C₂	semidirect
4. C₁₉×#3366	C₂	semidirect
5. C₁₉×#3259	C₂	semidirect
6. C₁₉×#3253	C₂	semidirect
7. C₁₉×#3176	C₂	semidirect
8. C₁₉×#3169	C₂	semidirect
9. C₁₉×#2960	C₂	semidirect
121. #62373	C₁₉	direct (N × H)
147. #3548	C₂×C₁₉	semidirect
148. #3391	C₂×C₁₉	semidirect
149. #3380	C₂×C₁₉	semidirect
150. #3366	C₂×C₁₉	semidirect
151. #3259	C₂×C₁₉	semidirect
152. #3253	C₂×C₁₉	semidirect
153. #3176	C₂×C₁₉	semidirect
154. #3169	C₂×C₁₉	semidirect
155. #2960	C₂×C₁₉	semidirect
247. C₁₉	#62373	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×#3548	2
C₁₉×#3391	2
C₁₉×#3380	2
C₁₉×#3366	2
C₁₉×#3259	2
C₁₉×#3253	2
C₁₉×#3176	2
C₁₉×#3169	2
C₁₉×#2960	2
#62373	19

Subgroups — C19×#62373 #18068332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62373 #18068332