Subgroups — C₁₉×#20343 #18026302

Order 4864.

2130 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#568	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3428	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1733	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#339	C₂	1	6	yes	maximal, complemented
6	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
7	C₁₉×#568	—	2	4	no	—
8	C₁₉×#568	C₄	1	8	yes	complemented
9	C₂×C₁₉×#131	—	2	4	no	—
10	C₂²×M₁₆×C₁₉	C₂²	1	6	yes	—
11	C₁₉×#520	C₄	1	8	yes	complemented
12	C₁₉×#519	—	2	1	no	—
13	C₁₉×#518	—	2	3	no	—
14	C₄×#41×C₁₉	—	2	4	no	—
15	C₄×#41×C₁₉	—	2	4	no	—
16	C₄×#41×C₁₉	C₄	1	8	yes	complemented
17	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
18	C₂×C₄×D₄×C₁₉	—	2	3	no	—
19	C₂×C₁₉×#117	C₂²	1	6	yes	—
20	C₂×C₁₉×#117	C₂²	1	3	yes	—
21	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
22	C₁₉×#339	C₂²	1	4	yes	—
23	C₁₉×#339	C₂²	1	12	yes	—
24	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
25	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×#41×C₁₉	D₄	1	4	yes	—
28	C₂×#41×C₁₉	D₄	1	4	yes	—
29	C₂×#41×C₁₉	Q₈	1	4	yes	—
30	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₉	D₄	1	3	yes	—
32	C₂³×C₄×C₁₉	—	2	3	no	—
33	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
34	C₁₉×#131	—	2	32	no	—
35	C₂×M₁₆×C₁₉	—	2	6	no	—
36	C₂×M₁₆×C₁₉	—	2	12	no	—
37	C₂×M₁₆×C₁₉	C₂×C₄	1	24	yes	complemented
38	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
39	C₂²×C₈×C₁₉	—	2	6	no	—
40	C₄×Q₈×C₁₉	—	2	8	no	—
41	C₄×Q₈×C₁₉	—	2	8	no	—
42	C₄×D₄×C₁₉	—	2	24	no	—
43	C₄×D₄×C₁₉	—	2	24	no	—
44	C₁₉×#117	—	2	6	no	—
45	C₁₉×#117	—	2	6	no	—
46	C₁₉×#117	—	2	6	no	—
47	C₁₉×#117	C₂×C₄	1	12	yes	complemented
48	C₁₉×#117	C₂×C₄	1	12	yes	complemented
49	C₁₉×#117	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
51	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	12	yes	complemented
52	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
53	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
54	C₂×C₄²×C₁₉	—	2	3	no	—
55	C₂×C₄²×C₁₉	—	2	6	no	—
56	C₂×C₄²×C₁₉	—	2	6	no	—
57	C₂×C₄²×C₁₉	C₂×C₄	1	12	yes	complemented
58	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
60	C₂⁴×C₁₉	C₂×D₄	1	3	yes	—
61	#41×C₁₉	C₄²	1	16	yes	—
62	#41×C₁₉	C₂²⋊C₄	1	16	yes	—
63	#41×C₁₉	C₄⋊C₄	1	16	yes	—
64	#41×C₁₉	C₄⋊C₄	1	16	yes	—
65	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
66	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
67	C₂×Q₈×C₁₉	C₄⋊C₄	1	4	yes	—
68	C₂×D₄×C₁₉	C₂²⋊C₄	1	12	yes	—
69	C₂×D₄×C₁₉	C₂²⋊C₄	1	12	yes	—
70	C₂×D₄×C₁₉	C₄⋊C₄	1	12	yes	—
71	C₂²×C₄×C₁₉	—	2	1	no	—
72	C₂²×C₄×C₁₉	—	2	12	no	—
73	C₂²×C₄×C₁₉	—	2	3	no	—
74	C₂²×C₄×C₁₉	—	2	12	no	—
75	C₂²×C₄×C₁₉	—	4	12	no	—
76	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
77	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
82	C₂²×C₄×C₁₉	C₂×Q₈	1	6	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₁₉	—	2	24	no	—
85	M₁₆×C₁₉	—	2	4	no	—
86	M₁₆×C₁₉	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₁₉	—	2	24	no	—
88	C₂×C₈×C₁₉	—	2	4	no	—
89	C₄⋊C₄×C₁₉	—	2	12	no	—
90	C₄⋊C₄×C₁₉	—	2	12	no	—
91	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
92	C₂²⋊C₄×C₁₉	—	2	12	no	—
93	C₂²⋊C₄×C₁₉	—	2	12	no	—
94	C₂²⋊C₄×C₁₉	—	2	4	no	—
95	C₄²×C₁₉	—	2	12	no	—
96	C₄²×C₁₉	—	2	12	no	—
97	C₄²×C₁₉	—	2	1	no	—
98	C₄²×C₁₉	—	2	3	no	—
99	C₄²×C₁₉	—	2	4	no	—
100	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
102	#20343	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	4	6	no	—
104	C₂³×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂³×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
106	C₂³×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
107	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₁₉	#95	1	16	yes	—
109	D₄×C₁₉	#95	1	48	yes	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	8	no	—
113	C₂×C₄×C₁₉	—	2	6	no	—
114	C₂×C₄×C₁₉	—	4	24	no	—
115	C₂×C₄×C₁₉	—	4	6	no	—
116	C₂×C₄×C₁₉	—	8	12	no	—
117	C₂×C₄×C₁₉	C₂×C₄²	1	12	yes	—
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	12	yes	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
124	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
130	C₈×C₁₉	—	2	8	no	—
131	C₂×#568	C₂×C₁₉	1	1	yes	characteristic
132	#3428	C₂×C₁₉	1	1	yes	characteristic
133	C₂×C₄×#41	C₂×C₁₉	1	1	yes	characteristic
134	#1733	C₂×C₁₉	1	6	yes	complemented
135	C₂×#339	C₂×C₁₉	1	6	yes	complemented
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	12	no	—
138	C₂²×C₁₉	—	4	6	no	—
139	C₂²×C₁₉	—	4	6	no	—
140	C₂²×C₁₉	C₂×#95	1	12	yes	—
141	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₁₉	—	2	4	no	—
145	C₄×C₁₉	—	4	6	no	—
146	C₄×C₁₉	—	4	8	no	—
147	C₄×C₁₉	C₂×#95	1	12	yes	—
148	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
149	C₄×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
153	#568	—	2	4	no	—
154	#568	C₄×C₁₉	1	8	yes	complemented
155	C₂×#131	—	2	4	no	—
156	C₂²×M₁₆	C₂²×C₁₉	1	6	yes	—
157	#520	C₄×C₁₉	1	8	yes	complemented
158	#519	—	2	1	no	—
159	#518	—	2	3	no	—
160	C₄×#41	—	2	4	no	—
161	C₄×#41	—	2	4	no	—
162	C₄×#41	C₄×C₁₉	1	8	yes	complemented
163	C₂×C₄×Q₈	—	2	1	no	—
164	C₂×C₄×D₄	—	2	3	no	—
165	C₂×#117	C₂²×C₁₉	1	6	yes	—
166	C₂×#117	C₂²×C₁₉	1	3	yes	—
167	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
168	#339	C₂²×C₁₉	1	4	yes	—
169	#339	C₂²×C₁₉	1	12	yes	—
170	C₂×C₁₉	—	2	1	no	—
171	C₂×C₁₉	—	2	1	no	—
172	C₂×C₁₉	—	2	12	no	—
173	C₂×C₁₉	C₂²×#95	1	1	yes	characteristic
174	C₂×C₁₉	#2901	1	2	yes	—
175	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
176	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
177	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
178	C₂×#41	D₄×C₁₉	1	4	yes	—
179	C₂×#41	D₄×C₁₉	1	4	yes	—
180	C₂×#41	Q₈×C₁₉	1	4	yes	—
181	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
182	C₂²×D₄	D₄×C₁₉	1	3	yes	—
183	C₂³×C₄	—	2	3	no	—
184	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
185	#131	—	2	32	no	—
186	C₂×M₁₆	—	2	6	no	—
187	C₂×M₁₆	—	2	12	no	—
188	C₂×M₁₆	C₂×C₄×C₁₉	1	24	yes	complemented
189	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
190	C₂²×C₈	—	2	6	no	—
191	C₄×Q₈	—	2	8	no	—
192	C₄×Q₈	—	2	8	no	—
193	C₄×D₄	—	2	24	no	—
194	C₄×D₄	—	2	24	no	—
195	#117	—	2	6	no	—
196	#117	—	2	6	no	—
197	#117	—	2	6	no	—
198	#117	C₂×C₄×C₁₉	1	12	yes	complemented
199	#117	C₂×C₄×C₁₉	1	12	yes	complemented
200	#117	C₂³×C₁₉	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	3	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	12	yes	complemented
203	C₂×C₂²⋊C₄	—	2	3	no	—
204	C₂×C₂²⋊C₄	—	2	6	no	—
205	C₂×C₄²	—	2	6	no	—
206	C₂×C₄²	—	2	3	no	—
207	C₂×C₄²	—	2	6	no	—
208	C₂×C₄²	C₂×C₄×C₁₉	1	12	yes	complemented
209	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
210	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
211	C₁₉	#20343	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	C₂×D₄×C₁₉	1	3	yes	—
213	#41	C₄²×C₁₉	1	16	yes	—
214	#41	C₂²⋊C₄×C₁₉	1	16	yes	—
215	#41	C₄⋊C₄×C₁₉	1	16	yes	—
216	#41	C₄⋊C₄×C₁₉	1	16	yes	—
217	C₂×Q₈	C₂²⋊C₄×C₁₉	1	4	yes	—
218	C₂×Q₈	C₂²⋊C₄×C₁₉	1	4	yes	—
219	C₂×Q₈	C₄⋊C₄×C₁₉	1	4	yes	—
220	C₂×D₄	C₂²⋊C₄×C₁₉	1	12	yes	—
221	C₂×D₄	C₂²⋊C₄×C₁₉	1	12	yes	—
222	C₂×D₄	C₄⋊C₄×C₁₉	1	12	yes	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	12	no	—
225	C₂²×C₄	—	2	3	no	—
226	C₂²×C₄	—	2	12	no	—
227	C₂²×C₄	—	4	12	no	—
228	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
231	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
232	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
233	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
234	C₂²×C₄	C₂×Q₈×C₁₉	1	6	yes	—
235	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
236	M₁₆	—	2	24	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	C₂²×C₄×C₁₉	1	8	yes	—
239	C₂×C₈	—	2	24	no	—
240	C₂×C₈	—	2	4	no	—
241	C₄⋊C₄	—	2	12	no	—
242	C₄⋊C₄	—	2	12	no	—
243	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
244	C₂²⋊C₄	—	2	12	no	—
245	C₂²⋊C₄	—	2	12	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	—	2	12	no	—
248	C₄²	—	2	12	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	3	no	—
252	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
253	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
254	C₂³	—	4	6	no	—
255	C₂³	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
256	C₂³	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
257	C₂³	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
258	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
259	Q₈	C₁₉×#95	1	16	yes	—
260	D₄	C₁₉×#95	1	48	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	—	4	6	no	—
267	C₂×C₄	—	8	12	no	—
268	C₂×C₄	C₂×C₄²×C₁₉	1	12	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	12	yes	—
272	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
275	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
277	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
280	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
281	C₈	—	2	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	12	no	—
284	C₂²	—	4	6	no	—
285	C₂²	—	4	6	no	—
286	C₂²	C₂×C₁₉×#95	1	12	yes	—
287	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
288	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
289	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	4	6	no	—
292	C₄	—	4	8	no	—
293	C₄	C₂×C₁₉×#95	1	12	yes	—
294	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
295	C₄	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
296	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
297	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	12	no	—
301	C₂	C₂²×C₁₉×#95	1	1	yes	characteristic
302	C₂	C₁₉×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1733	C₂	semidirect
5. C₂×C₁₉×#339	C₂	semidirect
8. C₁₉×#568	C₄	semidirect
11. C₁₉×#520	C₄	semidirect
16. C₄×#41×C₁₉	C₄	semidirect
37. C₂×M₁₆×C₁₉	C₂×C₄	semidirect
47. C₁₉×#117	C₂×C₄	semidirect
48. C₁₉×#117	C₂×C₄	semidirect
51. C₂×C₄⋊C₄×C₁₉	C₂×C₄	semidirect
57. C₂×C₄²×C₁₉	C₂×C₄	semidirect
102. #20343	C₁₉	direct (N × H)
134. #1733	C₂×C₁₉	semidirect
135. C₂×#339	C₂×C₁₉	semidirect
154. #568	C₄×C₁₉	semidirect
157. #520	C₄×C₁₉	semidirect
162. C₄×#41	C₄×C₁₉	semidirect
188. C₂×M₁₆	C₂×C₄×C₁₉	semidirect
198. #117	C₂×C₄×C₁₉	semidirect
199. #117	C₂×C₄×C₁₉	semidirect
202. C₂×C₄⋊C₄	C₂×C₄×C₁₉	semidirect
208. C₂×C₄²	C₂×C₄×C₁₉	semidirect
211. C₁₉	#20343	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#568	2
C₁₉×#3428	2
C₂×C₄×#41×C₁₉	2
C₁₉×#1733	2
C₂×C₁₉×#339	2
#20343	19

Subgroups — C19×#20343 #18026302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20343 #18026302