Subgroups — C₁₉×#17741 #18023700

Order 4864.

1866 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	C₂²×C₁₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2920	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1555	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1545	C₂	1	8	yes	maximal, complemented
5	C₂³×D₄×C₁₉	—	2	2	no	—
6	C₂³×D₄×C₁₉	C₄	1	2	yes	—
7	C₂×C₁₉×#127	—	2	2	no	—
8	C₂×C₁₉×#127	—	2	2	no	—
9	C₂×C₁₉×#127	—	2	2	no	—
10	C₂×C₁₉×#127	C₄	1	2	yes	—
11	C₂×C₁₉×#127	C₄	1	4	yes	—
12	C₂×C₁₉×#127	C₂²	1	4	yes	—
13	C₂×C₁₉×#127	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#432	—	2	2	no	—
16	C₁₉×#432	C₂²	1	4	yes	complemented
17	C₁₉×#431	—	2	2	no	—
18	C₁₉×#431	C₂²	1	4	yes	complemented
19	C₁₉×#423	C₂²	1	4	yes	complemented
20	C₁₉×#407	—	2	2	no	—
21	C₄×M₁₆×C₁₉	—	2	2	no	—
22	C₁₉×#331	C₂²	1	16	yes	complemented
23	C₂⁵×C₁₉	—	2	2	no	—
24	C₂²×D₄×C₁₉	—	2	4	no	—
25	C₂²×D₄×C₁₉	—	2	2	no	—
26	C₂²×D₄×C₁₉	—	2	4	no	—
27	C₂²×D₄×C₁₉	—	2	4	no	—
28	C₂²×D₄×C₁₉	—	2	4	no	—
29	C₂²×D₄×C₁₉	—	2	4	no	—
30	C₂²×D₄×C₁₉	—	2	4	no	—
31	C₂²×D₄×C₁₉	—	2	4	no	—
32	C₂²×D₄×C₁₉	—	4	16	no	—
33	C₂²×D₄×C₁₉	—	4	8	no	—
34	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
35	C₂²×D₄×C₁₉	C₂×C₄	1	8	yes	—
36	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₉	D₄	1	2	yes	—
38	C₂×M₁₆×C₁₉	—	2	4	no	—
39	C₁₉×#127	—	2	2	no	—
40	C₁₉×#127	—	2	4	no	—
41	C₁₉×#127	—	2	4	no	—
42	C₁₉×#127	—	2	2	no	—
43	C₁₉×#127	—	2	4	no	—
44	C₁₉×#127	—	2	4	no	—
45	C₁₉×#127	—	2	4	no	—
46	C₁₉×#127	C₂×C₄	1	8	yes	—
47	C₁₉×#127	C₂×C₄	1	4	yes	—
48	C₁₉×#127	C₂³	1	4	yes	—
49	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₉	D₄	1	2	yes	—
52	C₂×C₄²×C₁₉	D₄	1	2	yes	—
53	C₂×C₄²×C₁₉	D₄	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	—	2	4	no	—
57	C₁₉×C₄⋊C₈	C₂³	1	8	yes	complemented
58	C₁₉×C₂²⋊C₈	—	2	4	no	—
59	C₁₉×C₂²⋊C₈	—	2	4	no	—
60	C₁₉×#97	—	2	4	no	—
61	C₄×C₈×C₁₉	—	2	4	no	—
62	C₂⁴×C₁₉	—	2	4	no	—
63	C₂⁴×C₁₉	—	2	8	no	—
64	C₂⁴×C₁₉	—	4	8	no	—
65	C₂⁴×C₁₉	—	4	16	no	—
66	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₄×C₁₉	—	2	4	no	—
68	C₂×D₄×C₁₉	—	2	8	no	—
69	C₂×D₄×C₁₉	—	2	8	no	—
70	C₂×D₄×C₁₉	—	2	8	no	—
71	C₂×D₄×C₁₉	—	2	8	no	—
72	C₂×D₄×C₁₉	—	2	8	no	—
73	C₂×D₄×C₁₉	—	4	8	no	—
74	C₂×D₄×C₁₉	—	4	16	no	—
75	C₂×D₄×C₁₉	—	4	16	no	—
76	C₂×D₄×C₁₉	—	4	16	no	—
77	C₂×D₄×C₁₉	—	4	16	no	—
78	C₂×D₄×C₁₉	—	4	16	no	—
79	C₂×D₄×C₁₉	—	4	16	no	—
80	C₂×D₄×C₁₉	—	4	16	no	—
81	C₂×D₄×C₁₉	—	4	8	no	—
82	C₂×D₄×C₁₉	—	4	16	no	—
83	C₂×D₄×C₁₉	C₂²×C₄	1	8	yes	—
84	C₂²×C₄×C₁₉	—	2	4	no	—
85	C₂²×C₄×C₁₉	—	2	8	no	—
86	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
90	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
96	M₁₆×C₁₉	—	4	8	no	—
97	C₂×C₈×C₁₉	—	2	8	no	—
98	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₉	C₂×D₄	1	2	yes	—
105	C₄²×C₁₉	C₂×D₄	1	2	yes	—
106	C₄²×C₁₉	C₂×D₄	1	2	yes	—
107	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
108	#17741	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₁₉	—	2	2	no	—
110	C₂³×C₁₉	—	2	4	no	—
111	C₂³×C₁₉	—	2	8	no	—
112	C₂³×C₁₉	—	4	16	no	—
113	C₂³×C₁₉	—	4	16	no	—
114	C₂³×C₁₉	—	4	16	no	—
115	C₂³×C₁₉	—	4	8	no	—
116	C₂³×C₁₉	—	4	16	no	—
117	C₂³×C₁₉	—	8	32	no	—
118	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
120	D₄×C₁₉	—	4	8	no	—
121	D₄×C₁₉	—	4	8	no	—
122	D₄×C₁₉	—	4	16	no	—
123	D₄×C₁₉	—	4	16	no	—
124	D₄×C₁₉	—	4	16	no	—
125	D₄×C₁₉	—	4	16	no	—
126	D₄×C₁₉	—	4	16	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	2	4	no	—
129	C₂×C₄×C₁₉	—	2	4	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	2	4	no	—
133	C₂×C₄×C₁₉	—	2	4	no	—
134	C₂×C₄×C₁₉	—	2	4	no	—
135	C₂×C₄×C₁₉	#100	1	4	yes	—
136	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	8	yes	—
142	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
144	C₈×C₁₉	—	4	8	no	—
145	C₂²×#127	C₂×C₁₉	1	1	yes	characteristic
146	#2920	C₂×C₁₉	1	2	yes	complemented
147	#1555	C₂×C₁₉	1	4	yes	complemented
148	#1545	C₂×C₁₉	1	8	yes	complemented
149	C₂²×C₁₉	—	2	8	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	—	4	8	no	—
154	C₂²×C₁₉	—	4	16	no	—
155	C₂²×C₁₉	—	8	16	no	—
156	C₂²×C₁₉	—	8	32	no	—
157	C₂²×C₁₉	C₂×#100	1	2	yes	—
158	C₂²×C₁₉	#418	1	4	yes	—
159	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	C₂×#100	1	4	yes	—
165	C₄×C₁₉	#418	1	4	yes	—
166	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
167	C₂³×D₄	—	2	2	no	—
168	C₂³×D₄	C₄×C₁₉	1	2	yes	—
169	C₂×#127	—	2	2	no	—
170	C₂×#127	—	2	2	no	—
171	C₂×#127	—	2	2	no	—
172	C₂×#127	C₄×C₁₉	1	2	yes	—
173	C₂×#127	C₄×C₁₉	1	4	yes	—
174	C₂×#127	C₂²×C₁₉	1	4	yes	—
175	C₂×#127	C₂²×C₁₉	1	2	yes	—
176	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
177	#432	—	2	2	no	—
178	#432	C₂²×C₁₉	1	4	yes	complemented
179	#431	—	2	2	no	—
180	#431	C₂²×C₁₉	1	4	yes	complemented
181	#423	C₂²×C₁₉	1	4	yes	complemented
182	#407	—	2	2	no	—
183	C₄×M₁₆	—	2	2	no	—
184	#331	C₂²×C₁₉	1	16	yes	complemented
185	C₂×C₁₉	—	2	2	no	—
186	C₂×C₁₉	—	2	4	no	—
187	C₂×C₁₉	—	8	8	no	—
188	C₂×C₁₉	C₂²×#100	1	1	yes	characteristic
189	C₂×C₁₉	C₂×#418	1	2	yes	—
190	C₂⁵	—	2	2	no	—
191	C₂²×D₄	—	2	4	no	—
192	C₂²×D₄	—	2	2	no	—
193	C₂²×D₄	—	2	4	no	—
194	C₂²×D₄	—	2	4	no	—
195	C₂²×D₄	—	2	4	no	—
196	C₂²×D₄	—	2	4	no	—
197	C₂²×D₄	—	2	4	no	—
198	C₂²×D₄	—	2	4	no	—
199	C₂²×D₄	—	4	16	no	—
200	C₂²×D₄	—	4	8	no	—
201	C₂²×D₄	C₂×C₄×C₁₉	1	4	yes	—
202	C₂²×D₄	C₂×C₄×C₁₉	1	8	yes	—
203	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
204	C₂³×C₄	D₄×C₁₉	1	2	yes	—
205	C₂×M₁₆	—	2	4	no	—
206	#127	—	2	2	no	—
207	#127	—	2	4	no	—
208	#127	—	2	4	no	—
209	#127	—	2	2	no	—
210	#127	—	2	4	no	—
211	#127	—	2	4	no	—
212	#127	—	2	4	no	—
213	#127	C₂×C₄×C₁₉	1	8	yes	—
214	#127	C₂×C₄×C₁₉	1	4	yes	—
215	#127	C₂³×C₁₉	1	4	yes	—
216	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
218	C₂×C₄²	D₄×C₁₉	1	2	yes	—
219	C₂×C₄²	D₄×C₁₉	1	2	yes	—
220	C₂×C₄²	D₄×C₁₉	1	2	yes	—
221	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
222	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	C₂³×C₁₉	1	8	yes	complemented
225	C₂²⋊C₈	—	2	4	no	—
226	C₂²⋊C₈	—	2	4	no	—
227	#97	—	2	4	no	—
228	C₄×C₈	—	2	4	no	—
229	C₁₉	#17741	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	—	2	4	no	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	4	8	no	—
233	C₂⁴	—	4	16	no	—
234	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	8	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	8	no	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	4	8	no	—
242	C₂×D₄	—	4	16	no	—
243	C₂×D₄	—	4	16	no	—
244	C₂×D₄	—	4	16	no	—
245	C₂×D₄	—	4	16	no	—
246	C₂×D₄	—	4	16	no	—
247	C₂×D₄	—	4	16	no	—
248	C₂×D₄	—	4	16	no	—
249	C₂×D₄	—	4	16	no	—
250	C₂×D₄	—	4	8	no	—
251	C₂×D₄	C₂²×C₄×C₁₉	1	8	yes	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
255	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
256	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
257	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
258	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
264	M₁₆	—	4	8	no	—
265	C₂×C₈	—	2	8	no	—
266	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
267	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
268	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
269	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
270	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
271	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
272	C₄²	C₂×D₄×C₁₉	1	2	yes	—
273	C₄²	C₂×D₄×C₁₉	1	2	yes	—
274	C₄²	C₂×D₄×C₁₉	1	2	yes	—
275	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	8	no	—
279	C₂³	—	4	16	no	—
280	C₂³	—	4	16	no	—
281	C₂³	—	4	16	no	—
282	C₂³	—	4	8	no	—
283	C₂³	—	4	16	no	—
284	C₂³	—	8	32	no	—
285	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
286	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
287	D₄	—	4	8	no	—
288	D₄	—	4	8	no	—
289	D₄	—	4	16	no	—
290	D₄	—	4	16	no	—
291	D₄	—	4	16	no	—
292	D₄	—	4	16	no	—
293	D₄	—	4	16	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	C₁₉×#100	1	4	yes	—
303	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
305	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
308	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	8	yes	—
309	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
311	C₈	—	4	8	no	—
312	C₂²	—	2	8	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	8	no	—
317	C₂²	—	4	16	no	—
318	C₂²	—	8	16	no	—
319	C₂²	—	8	32	no	—
320	C₂²	C₂×C₁₉×#100	1	2	yes	—
321	C₂²	C₁₉×#418	1	4	yes	—
322	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	C₂×C₁₉×#100	1	4	yes	—
328	C₄	C₁₉×#418	1	4	yes	—
329	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	4	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	2	4	no	—
332	C₂	—	8	8	no	—
333	C₂	C₂²×C₁₉×#100	1	1	yes	characteristic
334	C₂	C₂×C₁₉×#418	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂²×C₁₉×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2920	C₂	semidirect
3. C₁₉×#1555	C₂	semidirect
4. C₁₉×#1545	C₂	semidirect
16. C₁₉×#432	C₂²	semidirect
18. C₁₉×#431	C₂²	semidirect
19. C₁₉×#423	C₂²	semidirect
22. C₁₉×#331	C₂²	semidirect
57. C₁₉×C₄⋊C₈	C₂³	semidirect
108. #17741	C₁₉	direct (N × H)
146. #2920	C₂×C₁₉	semidirect
147. #1555	C₂×C₁₉	semidirect
148. #1545	C₂×C₁₉	semidirect
178. #432	C₂²×C₁₉	semidirect
180. #431	C₂²×C₁₉	semidirect
181. #423	C₂²×C₁₉	semidirect
184. #331	C₂²×C₁₉	semidirect
224. C₄⋊C₈	C₂³×C₁₉	semidirect
229. C₁₉	#17741	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#127	2
C₁₉×#2920	2
C₁₉×#1555	2
C₁₉×#1545	2
#17741	19

Subgroups — C19×#17741 #18023700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17741 #18023700