Subgroups — C₂×C₁₉×#1533 #18023615

Order 4864.

1310 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#432	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1533	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₉×#329	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₁₉	C₄	1	1	yes	characteristic
6	C₂³×D₄×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₁₉×#124	—	2	2	no	—
8	C₂×C₁₉×#124	—	2	4	no	—
9	C₂×C₁₉×#124	C₄	1	2	yes	—
10	C₂×C₁₉×#124	C₄	1	4	yes	—
11	C₂×C₁₉×#124	C₂²	1	2	yes	—
12	C₂×C₁₉×#124	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₁₉	—	2	2	no	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#432	—	2	4	no	—
16	C₁₉×#432	C₂²	1	8	yes	complemented
17	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
18	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
19	C₂×C₁₉×#97	C₂²	1	4	yes	complemented
20	C₁₉×#329	C₂²	1	16	yes	complemented
21	C₂⁵×C₁₉	—	2	1	no	—
22	C₂²×Q₈×C₁₉	—	2	2	no	—
23	C₂²×Q₈×C₁₉	—	2	1	no	—
24	C₂²×Q₈×C₁₉	—	4	4	no	—
25	C₂²×Q₈×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₁₉	—	2	2	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	—	4	4	no	—
30	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
31	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₁₉	—	2	1	no	—
33	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₉	D₄	1	2	yes	—
35	C₂²×C₈×C₁₉	—	2	4	no	—
36	C₁₉×#124	—	2	8	no	—
37	C₁₉×#124	—	2	4	no	—
38	C₁₉×#124	—	2	4	no	—
39	C₁₉×#124	—	2	8	no	—
40	C₁₉×#124	C₂×C₄	1	4	yes	—
41	C₁₉×#124	C₂×C₄	1	8	yes	—
42	C₁₉×#124	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
46	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
47	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₉	D₄	1	2	yes	—
50	C₂×C₄²×C₁₉	D₄	1	4	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
53	C₁₉×C₄⋊C₈	—	2	8	no	—
54	C₁₉×C₂²⋊C₈	—	2	8	no	—
55	C₁₉×#97	—	2	4	no	—
56	C₁₉×#97	C₂³	1	8	yes	complemented
57	C₂⁴×C₁₉	—	2	4	no	—
58	C₂⁴×C₁₉	—	2	2	no	—
59	C₂⁴×C₁₉	—	4	8	no	—
60	C₂⁴×C₁₉	—	4	4	no	—
61	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	2	4	no	—
64	C₂×Q₈×C₁₉	—	4	8	no	—
65	C₂×Q₈×C₁₉	—	4	4	no	—
66	C₂×Q₈×C₁₉	—	4	8	no	—
67	C₂×Q₈×C₁₉	—	4	4	no	—
68	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
69	C₂×D₄×C₁₉	—	2	2	no	—
70	C₂×D₄×C₁₉	—	2	4	no	—
71	C₂×D₄×C₁₉	—	4	8	no	—
72	C₂×D₄×C₁₉	—	4	4	no	—
73	C₂×D₄×C₁₉	—	4	8	no	—
74	C₂×D₄×C₁₉	—	4	4	no	—
75	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₉	—	2	4	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	4	4	no	—
80	C₂²×C₄×C₁₉	—	4	4	no	—
81	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₁₉	—	2	8	no	—
90	C₂×C₈×C₁₉	—	4	8	no	—
91	C₂²⋊C₄×C₁₉	—	2	8	no	—
92	C₂²⋊C₄×C₁₉	—	2	16	no	—
93	C₂²⋊C₄×C₁₉	—	2	8	no	—
94	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
95	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₉	C₂×D₄	1	2	yes	—
99	C₄²×C₁₉	C₂×D₄	1	4	yes	—
100	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
101	C₂×#1533	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₁₉	—	2	4	no	—
103	C₂³×C₁₉	—	2	4	no	—
104	C₂³×C₁₉	—	4	16	no	—
105	C₂³×C₁₉	—	4	8	no	—
106	C₂³×C₁₉	—	4	4	no	—
107	C₂³×C₁₉	—	4	8	no	—
108	C₂³×C₁₉	—	8	16	no	—
109	C₂³×C₁₉	#100	1	4	yes	—
110	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	Q₈×C₁₉	—	4	4	no	—
113	Q₈×C₁₉	—	4	4	no	—
114	Q₈×C₁₉	—	4	8	no	—
115	D₄×C₁₉	—	4	4	no	—
116	D₄×C₁₉	—	4	4	no	—
117	D₄×C₁₉	—	4	8	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	4	no	—
125	C₂×C₄×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	8	no	—
128	C₂×C₄×C₁₉	—	4	4	no	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
131	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
135	C₈×C₁₉	—	4	8	no	—
136	C₂²×#124	C₂×C₁₉	1	1	yes	characteristic
137	C₂×#432	C₂×C₁₉	1	2	yes	complemented
138	#1533	C₂×C₁₉	1	8	yes	direct product factor
139	C₂×#329	C₂×C₁₉	1	4	yes	complemented
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	2	8	no	—
142	C₂²×C₁₉	—	4	4	no	—
143	C₂²×C₁₉	—	4	8	no	—
144	C₂²×C₁₉	—	8	16	no	—
145	C₂²×C₁₉	—	8	8	no	—
146	C₂²×C₁₉	C₂×#100	1	4	yes	—
147	C₂²×C₁₉	C₂×#100	1	2	yes	—
148	C₂²×C₁₉	#421	1	4	yes	—
149	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	4	no	—
156	C₄×C₁₉	—	4	4	no	—
157	C₂³×Q₈	C₄×C₁₉	1	1	yes	characteristic
158	C₂³×D₄	C₄×C₁₉	1	1	yes	characteristic
159	C₂×#124	—	2	2	no	—
160	C₂×#124	—	2	4	no	—
161	C₂×#124	C₄×C₁₉	1	2	yes	—
162	C₂×#124	C₄×C₁₉	1	4	yes	—
163	C₂×#124	C₂²×C₁₉	1	2	yes	—
164	C₂×#124	C₂²×C₁₉	1	4	yes	—
165	C₂²×C₂²⋊C₄	—	2	2	no	—
166	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
167	#432	—	2	4	no	—
168	#432	C₂²×C₁₉	1	8	yes	complemented
169	C₂×C₄⋊C₈	—	2	2	no	—
170	C₂×C₂²⋊C₈	—	2	2	no	—
171	C₂×#97	C₂²×C₁₉	1	4	yes	complemented
172	#329	C₂²×C₁₉	1	16	yes	complemented
173	C₂×C₁₉	—	2	4	no	—
174	C₂×C₁₉	—	8	4	no	—
175	C₂×C₁₉	#1533	1	4	yes	direct product factor
176	C₂×C₁₉	C₂²×#100	1	1	yes	characteristic
177	C₂×C₁₉	C₂×#421	1	2	yes	—
178	C₂⁵	—	2	1	no	—
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	4	4	no	—
182	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
183	C₂²×Q₈	C₂×C₄×C₁₉	1	4	yes	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	—	4	4	no	—
187	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
188	C₂²×D₄	C₂×C₄×C₁₉	1	4	yes	—
189	C₂³×C₄	—	2	1	no	—
190	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₁₉	1	2	yes	—
192	C₂²×C₈	—	2	4	no	—
193	#124	—	2	8	no	—
194	#124	—	2	4	no	—
195	#124	—	2	4	no	—
196	#124	—	2	8	no	—
197	#124	C₂×C₄×C₁₉	1	4	yes	—
198	#124	C₂×C₄×C₁₉	1	8	yes	—
199	#124	C₂³×C₁₉	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	—	2	8	no	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	—	2	8	no	—
204	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
206	C₂×C₄²	D₄×C₁₉	1	2	yes	—
207	C₂×C₄²	D₄×C₁₉	1	4	yes	—
208	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
210	C₄⋊C₈	—	2	8	no	—
211	C₂²⋊C₈	—	2	8	no	—
212	#97	—	2	4	no	—
213	#97	C₂³×C₁₉	1	8	yes	complemented
214	C₁₉	C₂×#1533	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	—	2	4	no	—
216	C₂⁴	—	2	2	no	—
217	C₂⁴	—	4	8	no	—
218	C₂⁴	—	4	4	no	—
219	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	8	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	4	8	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
240	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
243	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	4	8	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	2	16	no	—
251	C₂²⋊C₄	—	2	8	no	—
252	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
253	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
254	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
255	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
256	C₄²	C₂×D₄×C₁₉	1	2	yes	—
257	C₄²	C₂×D₄×C₁₉	1	4	yes	—
258	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	16	no	—
262	C₂³	—	4	8	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	8	no	—
265	C₂³	—	8	16	no	—
266	C₂³	C₁₉×#100	1	4	yes	—
267	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
268	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	8	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
290	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
292	C₈	—	4	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	8	no	—
297	C₂²	—	8	16	no	—
298	C₂²	—	8	8	no	—
299	C₂²	C₂×C₁₉×#100	1	4	yes	—
300	C₂²	C₂×C₁₉×#100	1	2	yes	—
301	C₂²	C₁₉×#421	1	4	yes	—
302	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₂	—	2	4	no	—
311	C₂	—	8	4	no	—
312	C₂	C₁₉×#1533	1	4	yes	direct product factor
313	C₂	C₂²×C₁₉×#100	1	1	yes	characteristic
314	C₂	C₂×C₁₉×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#432	C₂	semidirect
3. C₁₉×#1533	C₂	direct (N × H)
4. C₂×C₁₉×#329	C₂	semidirect
16. C₁₉×#432	C₂²	semidirect
19. C₂×C₁₉×#97	C₂²	semidirect
20. C₁₉×#329	C₂²	semidirect
56. C₁₉×#97	C₂³	semidirect
101. C₂×#1533	C₁₉	direct (N × H)
137. C₂×#432	C₂×C₁₉	semidirect
138. #1533	C₂×C₁₉	direct (N × H)
139. C₂×#329	C₂×C₁₉	semidirect
168. #432	C₂²×C₁₉	semidirect
171. C₂×#97	C₂²×C₁₉	semidirect
172. #329	C₂²×C₁₉	semidirect
175. C₂×C₁₉	#1533	direct (N × H)
213. #97	C₂³×C₁₉	semidirect
214. C₁₉	C₂×#1533	direct (N × H)
312. C₂	C₁₉×#1533	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×#124	2
C₂×C₁₉×#432	2
C₁₉×#1533	2
C₂×C₁₉×#329	2
C₂×#1533	19

Subgroups — C2×C19×#1533 #18023615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1533 #18023615