Subgroups — C₁₉×#9375 #18015334

Order 4864.

418 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	C₁₉×#3238	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2992	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1988	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1558	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1509	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1290	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₁₉×#529	C₂²	1	1	yes	characteristic
10	C₁₉×#501	C₄	1	2	yes	complemented
11	C₁₉×#489	C₄	1	2	yes	complemented
12	C₁₉×#483	—	2	1	no	—
13	C₁₉×#480	—	2	1	no	—
14	C₁₉×#477	—	2	1	no	—
15	C₁₉×#476	—	2	1	no	—
16	C₁₉×#446	—	2	1	no	—
17	C₁₉×#435	—	2	1	no	—
18	C₁₉×#435	—	2	1	no	—
19	C₈×D₄×C₁₉	—	2	1	no	—
20	C₁₉×#433	—	2	1	no	—
21	C₁₉×#432	C₂²	1	1	yes	characteristic
22	C₁₉×#432	C₂²	1	1	yes	characteristic
23	C₁₉×#432	C₂²	1	1	yes	characteristic
24	C₁₉×#424	—	2	1	no	—
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#340	—	2	1	no	—
28	C₁₉×#340	—	2	1	no	—
29	C₁₉×#333	—	2	1	no	—
30	C₁₉×#333	—	2	1	no	—
31	C₁₉×#328	—	2	1	no	—
32	C₁₉×#327	—	2	1	no	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₂²×C₈×C₁₉	—	2	1	no	—
36	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	—	2	1	no	—
40	C₁₉×C₄⋊Q₈	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₁₉×#126	—	2	1	no	—
43	C₁₉×#125	C₂×C₄	1	1	yes	characteristic
44	C₁₉×#124	C₂×C₄	1	1	yes	characteristic
45	C₁₉×#123	—	2	1	no	—
46	C₁₉×C₂²⋊Q₈	D₄	1	1	yes	characteristic
47	C₁₉×#121	D₄	1	1	yes	characteristic
48	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
49	C₄×D₄×C₁₉	—	2	1	no	—
50	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
51	C₁₉×#117	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄²×C₁₉	—	2	1	no	—
56	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×#107	—	2	2	no	—
58	C₁₉×#106	—	2	2	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₄⋊C₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	—	2	1	no	—
63	C₁₉×C₄⋊C₈	—	2	1	no	—
64	C₁₉×C₄⋊C₈	—	2	1	no	—
65	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₁₉×#103	—	2	2	no	—
68	C₁₉×#102	—	2	2	no	—
69	C₁₉×C₂²⋊C₈	—	2	1	no	—
70	C₁₉×C₂²⋊C₈	—	2	1	no	—
71	C₁₉×C₂²⋊C₈	—	2	1	no	—
72	C₁₉×C₂²⋊C₈	—	2	1	no	—
73	C₁₉×C₂²⋊C₈	—	2	1	no	—
74	C₁₉×#97	—	2	1	no	—
75	C₁₉×#97	—	2	1	no	—
76	C₁₉×#97	—	2	1	no	—
77	C₁₉×#97	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₁₉	—	2	1	no	—
79	C₁₉×#95	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	4	1	no	—
82	C₂×Q₈×C₁₉	—	8	1	no	—
83	C₂×Q₈×C₁₉	—	8	1	no	—
84	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×D₄×C₁₉	—	2	1	no	—
86	C₂×D₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	—	2	1	no	—
88	C₂²×C₄×C₁₉	—	4	1	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
93	M₁₆×C₁₉	—	8	1	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	1	no	—
98	C₂×C₈×C₁₉	—	2	1	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₂×C₈×C₁₉	—	4	1	no	—
101	C₂×C₈×C₁₉	—	4	1	no	—
102	C₂×C₈×C₁₉	—	8	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	2	1	no	—
106	C₄⋊C₄×C₁₉	—	2	1	no	—
107	C₄⋊C₄×C₁₉	—	2	2	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	4	1	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₄⋊C₄×C₁₉	—	4	1	no	—
113	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
114	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
115	C₂²⋊C₄×C₁₉	—	2	1	no	—
116	C₂²⋊C₄×C₁₉	—	2	1	no	—
117	C₂²⋊C₄×C₁₉	—	2	1	no	—
118	C₂²⋊C₄×C₁₉	—	2	1	no	—
119	C₂²⋊C₄×C₁₉	—	2	1	no	—
120	C₄²×C₁₉	—	2	1	no	—
121	C₄²×C₁₉	—	8	1	no	—
122	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	#41	1	1	yes	characteristic
126	#9375	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₁₉	—	2	1	no	—
128	C₂³×C₁₉	#121	1	1	yes	characteristic
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	8	2	no	—
131	Q₈×C₁₉	—	8	2	no	—
132	D₄×C₁₉	—	4	1	no	—
133	D₄×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	2	1	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	2	1	no	—
141	C₂×C₄×C₁₉	—	2	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	8	1	no	—
148	C₂×C₄×C₁₉	—	8	1	no	—
149	C₂×C₄×C₁₉	—	8	1	no	—
150	C₂×C₄×C₁₉	—	8	1	no	—
151	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
157	C₈×C₁₉	—	4	2	no	—
158	C₈×C₁₉	—	4	1	no	—
159	C₈×C₁₉	—	4	1	no	—
160	C₈×C₁₉	—	4	1	no	—
161	C₈×C₁₉	—	4	1	no	—
162	C₈×C₁₉	—	8	1	no	—
163	#3238	C₂×C₁₉	1	1	yes	characteristic
164	#2992	C₂×C₁₉	1	1	yes	characteristic
165	#1988	C₂×C₁₉	1	1	yes	characteristic, complemented
166	#1558	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#1509	C₂×C₁₉	1	1	yes	characteristic
168	#1290	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#1285	C₂×C₁₉	1	1	yes	characteristic, complemented
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	4	1	no	—
173	C₂²×C₁₉	—	4	1	no	—
174	C₂²×C₁₉	—	4	1	no	—
175	C₂²×C₁₉	#386	1	1	yes	characteristic
176	C₂²×C₁₉	#471	1	1	yes	characteristic
177	C₂²×C₁₉	#472	1	1	yes	characteristic
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	4	1	no	—
184	C₄×C₁₉	—	4	1	no	—
185	C₄×C₁₉	—	4	1	no	—
186	C₄×C₁₉	—	8	2	no	—
187	C₄×C₁₉	—	8	2	no	—
188	C₄×C₁₉	#455	1	1	yes	characteristic
189	C₄×C₁₉	#456	1	1	yes	characteristic
190	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
191	#529	C₂²×C₁₉	1	1	yes	characteristic
192	#501	C₄×C₁₉	1	2	yes	complemented
193	#489	C₄×C₁₉	1	2	yes	complemented
194	#483	—	2	1	no	—
195	#480	—	2	1	no	—
196	#477	—	2	1	no	—
197	#476	—	2	1	no	—
198	#446	—	2	1	no	—
199	#435	—	2	1	no	—
200	#435	—	2	1	no	—
201	C₈×D₄	—	2	1	no	—
202	#433	—	2	1	no	—
203	#432	C₂²×C₁₉	1	1	yes	characteristic
204	#432	C₂²×C₁₉	1	1	yes	characteristic
205	#432	C₂²×C₁₉	1	1	yes	characteristic
206	#424	—	2	1	no	—
207	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
208	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
209	#340	—	2	1	no	—
210	#340	—	2	1	no	—
211	#333	—	2	1	no	—
212	#333	—	2	1	no	—
213	#328	—	2	1	no	—
214	#327	—	2	1	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	8	1	no	—
218	C₂×C₁₉	#1882	1	1	yes	characteristic
219	C₂×C₁₉	#1907	1	1	yes	characteristic
220	C₂×C₁₉	#1909	1	1	yes	characteristic
221	C₂²×Q₈	—	2	1	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
230	#126	—	2	1	no	—
231	#125	C₂×C₄×C₁₉	1	1	yes	characteristic
232	#124	C₂×C₄×C₁₉	1	1	yes	characteristic
233	#123	—	2	1	no	—
234	C₂²⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
235	#121	D₄×C₁₉	1	1	yes	characteristic
236	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
239	#117	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
245	#107	—	2	2	no	—
246	#106	—	2	2	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
254	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
255	#103	—	2	2	no	—
256	#102	—	2	2	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	2	1	no	—
261	C₂²⋊C₈	—	2	1	no	—
262	#97	—	2	1	no	—
263	#97	—	2	1	no	—
264	#97	—	2	1	no	—
265	#97	C₂×C₄×C₁₉	1	2	yes	—
266	C₄×C₈	—	2	1	no	—
267	#95	—	2	1	no	—
268	C₁₉	#9375	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	8	1	no	—
272	C₂×Q₈	—	8	1	no	—
273	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
281	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
282	M₁₆	—	8	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	8	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
303	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	2	1	no	—
306	C₂²⋊C₄	—	2	1	no	—
307	C₂²⋊C₄	—	2	1	no	—
308	C₂²⋊C₄	—	2	1	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	8	1	no	—
311	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
313	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
314	C₄²	#41×C₁₉	1	1	yes	characteristic
315	C₂³	—	2	1	no	—
316	C₂³	C₁₉×#121	1	1	yes	characteristic
317	Q₈	—	4	1	no	—
318	Q₈	—	8	2	no	—
319	Q₈	—	8	2	no	—
320	D₄	—	4	1	no	—
321	D₄	—	4	1	no	—
322	D₄	—	4	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
340	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
341	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
342	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
343	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
344	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
345	C₈	—	4	2	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	1	no	—
350	C₈	—	8	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	C₁₉×#386	1	1	yes	characteristic
357	C₂²	C₁₉×#471	1	1	yes	characteristic
358	C₂²	C₁₉×#472	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	8	2	no	—
369	C₄	C₁₉×#455	1	1	yes	characteristic
370	C₄	C₁₉×#456	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	8	1	no	—
374	C₂	C₁₉×#1882	1	1	yes	characteristic
375	C₂	C₁₉×#1907	1	1	yes	characteristic
376	C₂	C₁₉×#1909	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#121	D₄
LCS	G ▷ C₄² ▷ 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
3. C₁₉×#1988	C₂	semidirect
4. C₁₉×#1558	C₂	semidirect
6. C₁₉×#1290	C₂	semidirect
7. C₁₉×#1285	C₂	semidirect
10. C₁₉×#501	C₄	semidirect
11. C₁₉×#489	C₄	semidirect
126. #9375	C₁₉	direct (N × H)
165. #1988	C₂×C₁₉	semidirect
166. #1558	C₂×C₁₉	semidirect
168. #1290	C₂×C₁₉	semidirect
169. #1285	C₂×C₁₉	semidirect
192. #501	C₄×C₁₉	semidirect
193. #489	C₄×C₁₉	semidirect
268. C₁₉	#9375	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×#3238	2
C₁₉×#2992	2
C₁₉×#1988	2
C₁₉×#1558	2
C₁₉×#1509	2
C₁₉×#1290	2
C₁₉×#1285	2
#9375	19

Subgroups — C19×#9375 #18015334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9375 #18015334