Subgroups — C₁₉×#32838 #18038797

Order 4864.

666 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2868	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2863	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2100	C₂	1	2	yes	maximal
4	C₁₉×#2095	C₂	1	2	yes	maximal
5	C₁₉×#2083	C₂	1	2	yes	maximal
6	C₁₉×#2072	C₂	1	2	yes	maximal
7	C₁₉×#2057	C₂	1	2	yes	maximal
8	C₁₉×#2024	C₂	1	2	yes	maximal
9	C₁₉×#1839	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₉×#107	C₂²	1	4	yes	—
14	C₂×C₁₉×#103	C₂²	1	4	yes	—
15	C₁₉×#391	C₂²	1	1	yes	characteristic
16	C₁₉×#391	C₂²	1	1	yes	characteristic
17	C₁₉×#391	C₂²	1	1	yes	characteristic
18	C₁₉×#389	C₂²	1	1	yes	characteristic
19	C₁₉×#389	C₂²	1	1	yes	characteristic
20	C₁₉×#389	C₂²	1	1	yes	characteristic
21	C₁₉×#387	—	2	1	no	—
22	C₁₉×#387	—	2	1	no	—
23	C₁₉×#387	—	2	1	no	—
24	C₁₉×#384	—	2	1	no	—
25	C₁₉×#384	—	2	1	no	—
26	C₁₉×#384	C₂²	1	2	yes	—
27	C₁₉×#384	C₂²	1	2	yes	—
28	C₁₉×#383	—	2	1	no	—
29	C₄³×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#340	C₂²	1	4	yes	—
31	C₁₉×#340	C₂²	1	4	yes	—
32	C₁₉×#336	C₂²	1	2	yes	—
33	C₁₉×#336	C₂²	1	2	yes	—
34	C₁₉×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₉	—	2	2	no	—
36	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
38	C₁₉×C₄⋊Q₈	—	4	4	no	—
39	C₁₉×C₄⋊Q₈	—	4	4	no	—
40	C₁₉×C₄⋊Q₈	—	4	4	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₁₉×#107	—	4	4	no	—
61	C₁₉×#103	—	4	4	no	—
62	C₁₉×#95	—	2	1	no	—
63	C₁₉×#95	—	2	1	no	—
64	C₁₉×#95	—	2	1	no	—
65	C₁₉×#95	—	2	1	no	—
66	C₁₉×#95	—	2	1	no	—
67	C₁₉×#95	—	2	1	no	—
68	C₁₉×#95	—	2	1	no	—
69	C₁₉×#95	—	2	1	no	—
70	C₁₉×#95	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	4	4	no	—
75	C₂×Q₈×C₁₉	—	4	4	no	—
76	C₂×Q₈×C₁₉	—	4	4	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	1	no	—
79	C₂²×C₄×C₁₉	—	2	1	no	—
80	C₂²×C₄×C₁₉	—	2	1	no	—
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₉	—	2	4	no	—
90	C₂×C₈×C₁₉	—	2	4	no	—
91	C₂×C₈×C₁₉	—	2	4	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	4	4	no	—
98	C₄⋊C₄×C₁₉	—	4	4	no	—
99	C₄⋊C₄×C₁₉	—	4	4	no	—
100	C₄⋊C₄×C₁₉	—	4	4	no	—
101	C₄⋊C₄×C₁₉	—	4	2	no	—
102	C₄⋊C₄×C₁₉	—	4	2	no	—
103	C₄⋊C₄×C₁₉	—	4	4	no	—
104	C₄⋊C₄×C₁₉	—	4	2	no	—
105	C₄²×C₁₉	—	2	2	no	—
106	C₄²×C₁₉	—	2	2	no	—
107	C₄²×C₁₉	—	2	2	no	—
108	C₄²×C₁₉	—	2	2	no	—
109	C₄²×C₁₉	—	2	2	no	—
110	C₄²×C₁₉	—	2	2	no	—
111	C₄²×C₁₉	Q₁₆	1	4	yes	—
112	#32838	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₉	—	4	4	no	—
115	Q₈×C₁₉	—	8	8	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	4	4	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	4	no	—
136	C₂×C₄×C₁₉	—	4	4	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
144	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
145	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
146	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₉	#137	1	2	yes	—
149	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
150	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
151	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
152	C₈×C₁₉	—	4	4	no	—
153	#2868	C₂×C₁₉	1	1	yes	characteristic
154	#2863	C₂×C₁₉	1	1	yes	characteristic
155	#2100	C₂×C₁₉	1	2	yes	—
156	#2095	C₂×C₁₉	1	2	yes	—
157	#2083	C₂×C₁₉	1	2	yes	—
158	#2072	C₂×C₁₉	1	2	yes	—
159	#2057	C₂×C₁₉	1	2	yes	—
160	#2024	C₂×C₁₉	1	2	yes	—
161	#1839	C₂×C₁₉	1	1	yes	characteristic
162	C₂²×C₁₉	#535	1	1	yes	characteristic
163	C₂²×C₁₉	#536	1	1	yes	characteristic
164	C₂²×C₁₉	#536	1	1	yes	characteristic
165	C₂²×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
166	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	2	2	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	2	2	no	—
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	2	no	—
175	C₄×C₁₉	—	8	1	no	—
176	C₄×C₁₉	—	8	1	no	—
177	C₄×C₁₉	—	8	4	no	—
178	C₄×C₁₉	—	8	1	no	—
179	C₄×C₁₉	—	8	1	no	—
180	C₄×C₁₉	#563	1	4	yes	—
181	C₂×C₄⋊Q₈	—	2	1	no	—
182	C₂×C₄⋊Q₈	—	2	1	no	—
183	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
184	C₂×#107	C₂²×C₁₉	1	4	yes	—
185	C₂×#103	C₂²×C₁₉	1	4	yes	—
186	#391	C₂²×C₁₉	1	1	yes	characteristic
187	#391	C₂²×C₁₉	1	1	yes	characteristic
188	#391	C₂²×C₁₉	1	1	yes	characteristic
189	#389	C₂²×C₁₉	1	1	yes	characteristic
190	#389	C₂²×C₁₉	1	1	yes	characteristic
191	#389	C₂²×C₁₉	1	1	yes	characteristic
192	#387	—	2	1	no	—
193	#387	—	2	1	no	—
194	#387	—	2	1	no	—
195	#384	—	2	1	no	—
196	#384	—	2	1	no	—
197	#384	C₂²×C₁₉	1	2	yes	—
198	#384	C₂²×C₁₉	1	2	yes	—
199	#383	—	2	1	no	—
200	C₄³	C₂²×C₁₉	1	1	yes	characteristic
201	#340	C₂²×C₁₉	1	4	yes	—
202	#340	C₂²×C₁₉	1	4	yes	—
203	#336	C₂²×C₁₉	1	2	yes	—
204	#336	C₂²×C₁₉	1	2	yes	—
205	#336	C₂²×C₁₉	1	2	yes	—
206	C₂×C₁₉	#2821	1	1	yes	characteristic
207	C₂×C₁₉	#3209	1	1	yes	characteristic
208	C₂×C₁₉	#3217	1	1	yes	characteristic
209	C₂×C₁₉	#3230	1	1	yes	characteristic
210	C₂×C₁₉	#3230	1	1	yes	characteristic
211	C₂×C₁₉	#3238	1	1	yes	characteristic
212	C₂×C₁₉	#3239	1	1	yes	characteristic
213	C₂²×Q₈	—	2	2	no	—
214	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
216	C₄⋊Q₈	—	4	4	no	—
217	C₄⋊Q₈	—	4	4	no	—
218	C₄⋊Q₈	—	4	4	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
228	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
232	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
233	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
234	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
238	#107	—	4	4	no	—
239	#103	—	4	4	no	—
240	#95	—	2	1	no	—
241	#95	—	2	1	no	—
242	#95	—	2	1	no	—
243	#95	—	2	1	no	—
244	#95	—	2	1	no	—
245	#95	—	2	1	no	—
246	#95	—	2	1	no	—
247	#95	—	2	1	no	—
248	#95	—	2	1	no	—
249	C₁₉	#32838	1	1	yes	characteristic, direct product factor, Sylow
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	—	4	4	no	—
255	C₂×Q₈	—	4	4	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×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	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	Q₁₆×C₁₉	1	4	yes	—
291	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
292	Q₈	—	4	4	no	—
293	Q₈	—	8	8	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
322	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
323	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
324	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
325	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
326	C₂×C₄	C₁₉×#137	1	2	yes	—
327	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
328	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
329	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
330	C₈	—	4	4	no	—
331	C₂²	C₁₉×#535	1	1	yes	characteristic
332	C₂²	C₁₉×#536	1	1	yes	characteristic
333	C₂²	C₁₉×#536	1	1	yes	characteristic
334	C₂²	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
335	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
336	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
337	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	4	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	C₁₉×#563	1	4	yes	—
350	C₂	C₁₉×#2821	1	1	yes	characteristic
351	C₂	C₁₉×#3209	1	1	yes	characteristic
352	C₂	C₁₉×#3217	1	1	yes	characteristic
353	C₂	C₁₉×#3230	1	1	yes	characteristic
354	C₂	C₁₉×#3230	1	1	yes	characteristic
355	C₂	C₁₉×#3238	1	1	yes	characteristic
356	C₂	C₁₉×#3239	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
112. #32838	C₁₉	direct (N × H)
249. C₁₉	#32838	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×#2868	2
C₁₉×#2863	2
C₁₉×#2100	2
C₁₉×#2095	2
C₁₉×#2083	2
C₁₉×#2072	2
C₁₉×#2057	2
C₁₉×#2024	2
C₁₉×#1839	2
#32838	19

Subgroups — C19×#32838 #18038797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32838 #18038797