Subgroups — C₄×C₁₉×#443 #18026468

Order 4864.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#443	C₂	1	2	yes	maximal
2	C₄×C₁₉×#135	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×#131	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1772	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1764	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₉×#104	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#131	C₂²	1	4	yes	—
11	C₄×#41×C₁₉	C₂²	1	2	yes	—
12	C₁₉×#443	C₄	1	16	yes	direct product factor
13	C₄×Q₁₆×C₁₉	C₄	1	2	yes	complemented
14	C₄×SD₁₆×C₁₉	C₄	1	4	yes	complemented
15	C₄×D₈×C₁₉	C₄	1	2	yes	complemented
16	C₂×C₁₉×#108	C₂²	1	2	yes	—
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#104	C₂²	1	4	yes	—
19	C₁₉×#416	C₂²	1	2	yes	—
20	C₁₉×#405	—	2	4	no	—
21	C₁₉×#405	C₂²	1	4	yes	—
22	C₄×M₁₆×C₁₉	—	2	2	no	—
23	C₄×M₁₆×C₁₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₉	—	2	2	no	—
25	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
29	C₄³×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#341	C₂²	1	2	yes	—
31	C₁₉×#339	C₂²	1	4	yes	—
32	C₂×#41×C₁₉	C₂³	1	2	yes	—
33	C₁₉×#135	C₂×C₄	1	8	yes	—
34	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×SD₁₆×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₁₉×#131	—	2	8	no	—
38	C₁₉×#131	C₂×C₄	1	16	yes	complemented
39	C₂×M₁₆×C₁₉	—	2	4	no	—
40	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
41	C₂²×C₈×C₁₉	—	2	4	no	—
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
45	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₁₉	—	2	4	no	—
47	C₄×D₄×C₁₉	C₂×C₄	1	4	yes	—
48	C₁₉×#117	—	2	2	no	—
49	C₁₉×#117	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₉	—	2	2	no	—
51	C₂×C₄²×C₁₉	—	2	2	no	—
52	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₁₉×#108	C₂×C₄	1	8	yes	complemented
55	C₁₉×#107	C₂×C₄	1	2	yes	—
56	C₁₉×#106	C₂×C₄	1	2	yes	—
57	C₁₉×#104	C₂×C₄	1	16	yes	complemented
58	C₁₉×#103	C₂×C₄	1	4	yes	—
59	C₁₉×#102	C₂×C₄	1	4	yes	—
60	C₁₉×#97	—	2	4	no	—
61	C₁₉×#97	—	2	2	no	—
62	C₄×C₈×C₁₉	—	2	4	no	—
63	C₄×C₈×C₁₉	—	2	4	no	—
64	C₄×C₈×C₁₉	—	2	2	no	—
65	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	complemented
68	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₉	D₄	1	2	yes	—
70	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
71	#41×C₁₉	—	2	4	no	—
72	#41×C₁₉	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₁₉	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	4	no	—
79	C₂²×C₄×C₁₉	#41	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₁₉	C₄²	1	4	yes	complemented
82	SD₁₆×C₁₉	C₄²	1	8	yes	complemented
83	D₈×C₁₉	C₄²	1	4	yes	complemented
84	M₁₆×C₁₉	—	2	4	no	—
85	M₁₆×C₁₉	—	2	16	no	—
86	M₁₆×C₁₉	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₁₉	—	2	4	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	16	no	—
90	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₉	#41	1	4	yes	—
98	C₄⋊C₄×C₁₉	—	2	4	no	—
99	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₁₉	—	2	2	no	—
101	C₂²⋊C₄×C₁₉	—	2	4	no	—
102	C₄²×C₁₉	—	2	8	no	—
103	C₄²×C₁₉	—	2	4	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	C₄²×C₁₉	—	2	2	no	—
106	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
107	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄×#443	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₁₉	—	2	2	no	—
112	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
113	Q₈×C₁₉	—	2	2	no	—
114	Q₈×C₁₉	C₂×C₄²	1	4	yes	—
115	D₄×C₁₉	—	2	8	no	—
116	D₄×C₁₉	—	2	2	no	—
117	D₄×C₁₉	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₁₉	—	2	8	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	4	8	no	—
125	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
131	C₈×C₁₉	—	2	8	no	—
132	C₈×C₁₉	—	2	2	no	—
133	C₈×C₁₉	C₂×C₄²	1	2	yes	—
134	C₈×C₁₉	C₂×C₄²	1	2	yes	—
135	C₈×C₁₉	C₄×D₄	1	4	yes	—
136	C₈×C₁₉	C₄×D₄	1	4	yes	—
137	C₂×#443	C₂×C₁₉	1	2	yes	—
138	C₄×#135	C₂×C₁₉	1	1	yes	characteristic
139	C₄×#131	C₂×C₁₉	1	2	yes	complemented
140	C₄×#108	C₂×C₁₉	1	1	yes	characteristic, complemented
141	#1772	C₂×C₁₉	1	2	yes	complemented
142	#1764	C₂×C₁₉	1	4	yes	complemented
143	C₄×#104	C₂×C₁₉	1	2	yes	complemented
144	C₄²×C₈	C₂×C₁₉	1	1	yes	characteristic, complemented
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	—	4	4	no	—
148	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₁₉	C₄×#41	1	2	yes	—
150	C₄×C₁₉	—	2	8	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	2	4	no	—
153	C₄×C₁₉	—	4	4	no	—
154	C₄×C₁₉	#443	1	4	yes	direct product factor
155	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
156	C₄×C₁₉	C₂×C₄×D₄	1	2	yes	—
157	C₄×C₁₉	C₄×#41	1	1	yes	characteristic
158	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
159	C₂×#131	C₂²×C₁₉	1	4	yes	—
160	C₄×#41	C₂²×C₁₉	1	2	yes	—
161	#443	C₄×C₁₉	1	16	yes	direct product factor
162	C₄×Q₁₆	C₄×C₁₉	1	2	yes	complemented
163	C₄×SD₁₆	C₄×C₁₉	1	4	yes	complemented
164	C₄×D₈	C₄×C₁₉	1	2	yes	complemented
165	C₂×#108	C₂²×C₁₉	1	2	yes	—
166	#427	C₂²×C₁₉	1	1	yes	characteristic
167	C₂×#104	C₂²×C₁₉	1	4	yes	—
168	#416	C₂²×C₁₉	1	2	yes	—
169	#405	—	2	4	no	—
170	#405	C₂²×C₁₉	1	4	yes	—
171	C₄×M₁₆	—	2	2	no	—
172	C₄×M₁₆	C₂²×C₁₉	1	2	yes	—
173	C₂×C₄×C₈	—	2	2	no	—
174	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
178	C₄³	C₂²×C₁₉	1	1	yes	characteristic
179	#341	C₂²×C₁₉	1	2	yes	—
180	#339	C₂²×C₁₉	1	4	yes	—
181	C₂×C₁₉	—	2	2	no	—
182	C₂×C₁₉	—	4	4	no	—
183	C₂×C₁₉	C₄²×D₄	1	1	yes	characteristic
184	C₂×C₁₉	C₂×#443	1	2	yes	—
185	C₂×#41	C₂³×C₁₉	1	2	yes	—
186	#135	C₂×C₄×C₁₉	1	8	yes	—
187	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
188	C₂×SD₁₆	C₂×C₄×C₁₉	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
190	#131	—	2	8	no	—
191	#131	C₂×C₄×C₁₉	1	16	yes	complemented
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
198	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	C₂×C₄×C₁₉	1	4	yes	—
201	#117	—	2	2	no	—
202	#117	C₂³×C₁₉	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
206	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
207	#108	C₂×C₄×C₁₉	1	8	yes	complemented
208	#107	C₂×C₄×C₁₉	1	2	yes	—
209	#106	C₂×C₄×C₁₉	1	2	yes	—
210	#104	C₂×C₄×C₁₉	1	16	yes	complemented
211	#103	C₂×C₄×C₁₉	1	4	yes	—
212	#102	C₂×C₄×C₁₉	1	4	yes	—
213	#97	—	2	4	no	—
214	#97	—	2	2	no	—
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
219	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
220	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	complemented
221	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
222	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
223	C₄×C₈	D₄×C₁₉	1	2	yes	—
224	C₁₉	C₄×#443	1	1	yes	characteristic, direct product factor, Sylow
225	#41	—	2	4	no	—
226	#41	C₂²×C₄×C₁₉	1	8	yes	—
227	C₂×Q₈	C₂²×C₄×C₁₉	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₁₉	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	#41×C₁₉	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
235	Q₁₆	C₄²×C₁₉	1	4	yes	complemented
236	SD₁₆	C₄²×C₁₉	1	8	yes	complemented
237	D₈	C₄²×C₁₉	1	4	yes	complemented
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	16	no	—
240	M₁₆	C₂²×C₄×C₁₉	1	8	yes	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	16	no	—
244	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
248	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
251	C₂×C₈	#41×C₁₉	1	4	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
261	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
262	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄²×C₁₉	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄²×C₁₉	1	4	yes	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
281	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
284	C₈	—	2	8	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂×C₄²×C₁₉	1	2	yes	—
287	C₈	C₂×C₄²×C₁₉	1	2	yes	—
288	C₈	C₄×D₄×C₁₉	1	4	yes	—
289	C₈	C₄×D₄×C₁₉	1	4	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
294	C₂²	C₄×#41×C₁₉	1	2	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	4	no	—
299	C₄	C₁₉×#443	1	4	yes	direct product factor
300	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
301	C₄	C₂×C₄×D₄×C₁₉	1	2	yes	—
302	C₄	C₄×#41×C₁₉	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄²×D₄×C₁₉	1	1	yes	characteristic
306	C₂	C₂×C₁₉×#443	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₉	D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×C₄×D₄
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₉×#131	C₂	semidirect
4. C₄×C₁₉×#108	C₂	semidirect
5. C₁₉×#1772	C₂	semidirect
6. C₁₉×#1764	C₂	semidirect
7. C₄×C₁₉×#104	C₂	semidirect
8. C₄²×C₈×C₁₉	C₂	semidirect
12. C₁₉×#443	C₄	direct (N × H)
13. C₄×Q₁₆×C₁₉	C₄	semidirect
14. C₄×SD₁₆×C₁₉	C₄	semidirect
15. C₄×D₈×C₁₉	C₄	semidirect
38. C₁₉×#131	C₂×C₄	semidirect
54. C₁₉×#108	C₂×C₄	semidirect
57. C₁₉×#104	C₂×C₄	semidirect
67. C₄×C₈×C₁₉	C₂×C₄	semidirect
81. Q₁₆×C₁₉	C₄²	semidirect
82. SD₁₆×C₁₉	C₄²	semidirect
83. D₈×C₁₉	C₄²	semidirect
110. C₄×#443	C₁₉	direct (N × H)
139. C₄×#131	C₂×C₁₉	semidirect
140. C₄×#108	C₂×C₁₉	semidirect
141. #1772	C₂×C₁₉	semidirect
142. #1764	C₂×C₁₉	semidirect
143. C₄×#104	C₂×C₁₉	semidirect
144. C₄²×C₈	C₂×C₁₉	semidirect
154. C₄×C₁₉	#443	direct (N × H)
161. #443	C₄×C₁₉	direct (N × H)
162. C₄×Q₁₆	C₄×C₁₉	semidirect
163. C₄×SD₁₆	C₄×C₁₉	semidirect
164. C₄×D₈	C₄×C₁₉	semidirect
191. #131	C₂×C₄×C₁₉	semidirect
207. #108	C₂×C₄×C₁₉	semidirect
210. #104	C₂×C₄×C₁₉	semidirect
220. C₄×C₈	C₂×C₄×C₁₉	semidirect
224. C₁₉	C₄×#443	direct (N × H)
235. Q₁₆	C₄²×C₁₉	semidirect
236. SD₁₆	C₄²×C₁₉	semidirect
237. D₈	C₄²×C₁₉	semidirect
299. C₄	C₁₉×#443	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×#443	2
C₄×C₁₉×#135	2
C₄×C₁₉×#131	2
C₄×C₁₉×#108	2
C₁₉×#1772	2
C₁₉×#1764	2
C₄×C₁₉×#104	2
C₄²×C₈×C₁₉	2
C₄×#443	19

Subgroups — C4×C19×#443 #18026468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₉×#443 #18026468