Subgroups — C₁₉×#20454 #18026413

Order 4864.

1126 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#405	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2274	C₂	1	2	yes	maximal
5	C₁₉×#1776	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₉×#106	C₂	1	4	yes	maximal, complemented
7	C₁₉×#1736	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#117	C₂²	1	2	yes	—
11	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₁₉×#427	C₄	1	8	yes	complemented
14	C₂×C₁₉×#107	C₂²	1	2	yes	—
15	C₂×C₁₉×#106	C₄	1	8	yes	complemented
16	C₂×C₁₉×#106	C₂²	1	2	yes	—
17	C₂×C₁₉×#106	C₂²	1	2	yes	—
18	C₂×C₁₉×#106	C₂²	1	2	yes	—
19	C₁₉×#405	C₄	1	8	yes	complemented
20	C₂×C₁₉×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₁₉	—	2	4	no	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
24	C₄×C₂²⋊C₄×C₁₉	—	2	4	no	—
25	C₁₉×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₉	—	2	2	no	—
27	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
37	C₁₉×#117	—	2	4	no	—
38	C₁₉×#117	C₂×C₄	1	8	yes	—
39	C₁₉×#117	C₂×C₄	1	4	yes	—
40	C₁₉×#117	Q₈	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
49	C₂×C₄²×C₁₉	—	2	2	no	—
50	C₂×C₄²×C₁₉	—	2	8	no	—
51	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
52	C₁₉×#107	C₂×C₄	1	8	yes	complemented
53	C₁₉×#106	C₂×C₄	1	8	yes	—
54	C₁₉×#106	C₂×C₄	1	8	yes	—
55	C₁₉×#106	C₂×C₄	1	8	yes	complemented
56	C₁₉×#97	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	complemented
58	C₁₉×#95	—	2	8	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	8	4	no	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₉	—	2	8	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	C₄²	1	16	yes	complemented
77	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
84	C₄⋊C₄×C₁₉	—	2	8	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	8	yes	—
88	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₁₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	8	yes	—
92	C₄²×C₁₉	—	2	8	no	—
93	C₄²×C₁₉	—	8	8	no	—
94	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
95	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
96	#20454	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₁₉	C₄×D₄	1	2	yes	—
100	C₂³×C₁₉	C₄×D₄	1	2	yes	—
101	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	—	2	8	no	—
103	C₂×C₄×C₁₉	—	2	8	no	—
104	C₂×C₄×C₁₉	—	2	8	no	—
105	C₂×C₄×C₁₉	—	4	4	no	—
106	C₂×C₄×C₁₉	—	4	8	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	—	8	8	no	—
109	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₉	#117	1	4	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
116	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
117	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
123	C₈×C₁₉	—	2	4	no	—
124	C₈×C₁₉	C₂×C₄²	1	4	yes	—
125	C₈×C₁₉	C₂×C₄²	1	4	yes	—
126	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
127	C₂²×#106	C₂×C₁₉	1	1	yes	characteristic
128	C₂×#405	C₂×C₁₉	1	1	yes	characteristic
129	#2274	C₂×C₁₉	1	2	yes	—
130	#1776	C₂×C₁₉	1	4	yes	complemented
131	C₄×#106	C₂×C₁₉	1	4	yes	complemented
132	#1736	C₂×C₁₉	1	2	yes	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	—	2	4	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
137	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₁₉	#577	1	4	yes	—
141	C₄×C₁₉	—	2	8	no	—
142	C₄×C₁₉	—	4	8	no	—
143	C₄×C₁₉	—	4	8	no	—
144	C₄×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
145	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₁₉	C₂×#117	1	1	yes	characteristic
147	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
151	C₂×#117	C₂²×C₁₉	1	2	yes	—
152	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
154	#427	C₄×C₁₉	1	8	yes	complemented
155	C₂×#107	C₂²×C₁₉	1	2	yes	—
156	C₂×#106	C₄×C₁₉	1	8	yes	complemented
157	C₂×#106	C₂²×C₁₉	1	2	yes	—
158	C₂×#106	C₂²×C₁₉	1	2	yes	—
159	C₂×#106	C₂²×C₁₉	1	2	yes	—
160	#405	C₄×C₁₉	1	8	yes	complemented
161	C₂×#97	C₂²×C₁₉	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
163	C₄×C₄⋊C₄	—	2	4	no	—
164	C₄×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
165	C₄×C₂²⋊C₄	—	2	4	no	—
166	#340	C₂²×C₁₉	1	8	yes	—
167	C₂×C₁₉	—	2	4	no	—
168	C₂×C₁₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₁₉	#2912	1	2	yes	—
170	C₂×C₁₉	#2949	1	2	yes	—
171	C₂×C₁₉	#2952	1	2	yes	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
177	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
178	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
179	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
180	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
183	#117	—	2	4	no	—
184	#117	C₂×C₄×C₁₉	1	8	yes	—
185	#117	C₂×C₄×C₁₉	1	4	yes	—
186	#117	Q₈×C₁₉	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
190	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	8	no	—
197	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
198	#107	C₂×C₄×C₁₉	1	8	yes	complemented
199	#106	C₂×C₄×C₁₉	1	8	yes	—
200	#106	C₂×C₄×C₁₉	1	8	yes	—
201	#106	C₂×C₄×C₁₉	1	8	yes	complemented
202	#97	C₂×C₄×C₁₉	1	8	yes	complemented
203	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	complemented
204	#95	—	2	8	no	—
205	C₁₉	#20454	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	8	4	no	—
211	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
215	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
217	C₂²×C₄	#41×C₁₉	1	2	yes	—
218	C₂²×C₄	#41×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
220	M₁₆	—	2	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	C₄²×C₁₉	1	16	yes	complemented
224	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
225	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
230	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₄⋊C₄×C₁₉	1	8	yes	—
235	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
236	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	C₄⋊C₄×C₁₉	1	8	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	8	8	no	—
241	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
242	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
245	C₂³	C₄×D₄×C₁₉	1	2	yes	—
246	C₂³	C₄×D₄×C₁₉	1	2	yes	—
247	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
258	C₂×C₄	C₁₉×#117	1	4	yes	—
259	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₁₉	1	4	yes	—
262	C₂×C₄	C₄×Q₈×C₁₉	1	4	yes	—
263	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
264	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
265	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
269	C₈	—	2	4	no	—
270	C₈	C₂×C₄²×C₁₉	1	4	yes	—
271	C₈	C₂×C₄²×C₁₉	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
276	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
277	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
278	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
279	C₂²	C₁₉×#577	1	4	yes	—
280	C₄	—	2	8	no	—
281	C₄	—	4	8	no	—
282	C₄	—	4	8	no	—
283	C₄	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
284	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
285	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
286	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
287	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄×C₄⋊C₄×C₁₉	1	1	yes	characteristic
290	C₂	C₁₉×#2912	1	2	yes	—
291	C₂	C₁₉×#2949	1	2	yes	—
292	C₂	C₁₉×#2952	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₉×#1776	C₂	semidirect
6. C₄×C₁₉×#106	C₂	semidirect
13. C₁₉×#427	C₄	semidirect
15. C₂×C₁₉×#106	C₄	semidirect
19. C₁₉×#405	C₄	semidirect
52. C₁₉×#107	C₂×C₄	semidirect
55. C₁₉×#106	C₂×C₄	semidirect
56. C₁₉×#97	C₂×C₄	semidirect
57. C₄×C₈×C₁₉	C₂×C₄	semidirect
76. C₂×C₈×C₁₉	C₄²	semidirect
96. #20454	C₁₉	direct (N × H)
130. #1776	C₂×C₁₉	semidirect
131. C₄×#106	C₂×C₁₉	semidirect
154. #427	C₄×C₁₉	semidirect
156. C₂×#106	C₄×C₁₉	semidirect
160. #405	C₄×C₁₉	semidirect
198. #107	C₂×C₄×C₁₉	semidirect
201. #106	C₂×C₄×C₁₉	semidirect
202. #97	C₂×C₄×C₁₉	semidirect
203. C₄×C₈	C₂×C₄×C₁₉	semidirect
205. C₁₉	#20454	direct (N × H)
223. C₂×C₈	C₄²×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#427	2
C₂²×C₁₉×#106	2
C₂×C₁₉×#405	2
C₁₉×#2274	2
C₁₉×#1776	2
C₄×C₁₉×#106	2
C₁₉×#1736	2
#20454	19

Subgroups — C19×#20454 #18026413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20454 #18026413