Subgroups — C₁₉×#60284 #18066243

Order 4864.

2038 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×C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2991	C₂	1	8	yes	maximal, complemented
3	C₁₉×#2982	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2975	C₂	1	6	yes	maximal, complemented
5	C₁₉×#2971	C₂	1	8	yes	maximal, complemented
6	C₁₉×#2920	C₂	1	6	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₄	1	1	yes	characteristic
8	C₄×#41×C₁₉	C₄	1	4	yes	—
9	C₄×#41×C₁₉	C₄	1	4	yes	—
10	C₄×#41×C₁₉	C₂²	1	4	yes	—
11	C₄×#41×C₁₉	C₂²	1	4	yes	—
12	C₂×C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₉	C₄	1	3	yes	—
15	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
16	C₂×C₁₉×#117	C₄	1	3	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
18	C₈×Q₈×C₁₉	C₂²	1	16	yes	complemented
19	C₁₉×#436	C₂²	1	48	yes	complemented
20	C₁₉×#432	C₂²	1	4	yes	—
21	C₁₉×#432	C₂²	1	12	yes	complemented
22	C₁₉×#431	C₂²	1	12	yes	—
23	C₁₉×#431	C₂²	1	12	yes	complemented
24	C₁₉×#423	C₂²	1	12	yes	complemented
25	C₁₉×#407	C₂²	1	12	yes	—
26	C₄×M₁₆×C₁₉	C₂²	1	12	yes	complemented
27	C₂×#41×C₁₉	—	2	8	no	—
28	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
30	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₉	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₁₉	C₂×C₄	1	3	yes	—
33	C₂³×C₄×C₁₉	C₂×C₄	1	3	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
35	C₂×M₁₆×C₁₉	C₂³	1	24	yes	—
36	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
40	C₄×D₄×C₁₉	C₂×C₄	1	12	yes	—
41	C₄×D₄×C₁₉	C₂×C₄	1	12	yes	—
42	C₄×D₄×C₁₉	C₂×C₄	1	12	yes	—
43	C₄×D₄×C₁₉	C₂³	1	12	yes	—
44	C₁₉×#117	C₂×C₄	1	12	yes	—
45	C₁₉×#117	C₂×C₄	1	12	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
49	C₂×C₄²×C₁₉	C₂×C₄	1	3	yes	—
50	C₂×C₄²×C₁₉	C₂×C₄	1	3	yes	—
51	C₂×C₄²×C₁₉	C₂×C₄	1	6	yes	—
52	C₂×C₄²×C₁₉	C₂×C₄	1	6	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
56	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
58	C₁₉×C₄⋊C₈	C₂³	1	24	yes	—
59	C₁₉×C₂²⋊C₈	C₂³	1	8	yes	—
60	C₁₉×C₂²⋊C₈	C₂³	1	24	yes	—
61	C₁₉×#97	C₂³	1	8	yes	—
62	C₄×C₈×C₁₉	C₂³	1	24	yes	—
63	C₂⁴×C₁₉	C₂²×C₄	1	3	yes	—
64	#41×C₁₉	—	2	16	no	—
65	#41×C₁₉	—	2	16	no	—
66	C₂×Q₈×C₁₉	M₁₆	1	8	yes	complemented
67	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
68	C₂×D₄×C₁₉	—	2	12	no	—
69	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
70	C₂²×C₄×C₁₉	—	2	12	no	—
71	C₂²×C₄×C₁₉	—	4	6	no	—
72	C₂²×C₄×C₁₉	—	4	6	no	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
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	3	yes	—
79	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
80	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
81	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
85	M₁₆×C₁₉	—	4	24	no	—
86	C₂×C₈×C₁₉	C₂⁴	1	16	yes	—
87	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
88	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
89	C₂²⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
90	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
93	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
94	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
95	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
96	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
97	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
98	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
99	C₄²×C₁₉	C₂⁴	1	6	yes	—
100	#60284	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₉	—	2	6	no	—
102	C₂³×C₁₉	—	4	3	no	—
103	C₂³×C₁₉	—	4	3	no	—
104	C₂³×C₁₉	C₂³×C₄	1	6	yes	—
105	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₁₉	C₂×M₁₆	1	16	yes	—
107	D₄×C₁₉	—	2	24	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	12	no	—
110	C₂×C₄×C₁₉	—	2	12	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	12	no	—
113	C₂×C₄×C₁₉	—	2	12	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	4	6	no	—
116	C₂×C₄×C₁₉	—	4	3	no	—
117	C₂×C₄×C₁₉	—	4	6	no	—
118	C₂×C₄×C₁₉	—	4	3	no	—
119	C₂×C₄×C₁₉	C₂×M₁₆	1	12	yes	—
120	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
125	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
126	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
127	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
129	C₈×C₁₉	—	2	16	no	—
130	C₂×C₄×#41	C₂×C₁₉	1	1	yes	characteristic
131	#2991	C₂×C₁₉	1	8	yes	complemented
132	#2982	C₂×C₁₉	1	2	yes	complemented
133	#2975	C₂×C₁₉	1	6	yes	complemented
134	#2971	C₂×C₁₉	1	8	yes	complemented
135	#2920	C₂×C₁₉	1	6	yes	complemented
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	6	no	—
139	C₂²×C₁₉	—	2	6	no	—
140	C₂²×C₁₉	—	2	6	no	—
141	C₂²×C₁₉	—	4	12	no	—
142	C₂²×C₁₉	C₂²×M₁₆	1	2	yes	—
143	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	6	no	—
146	C₄×C₁₉	—	2	1	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	6	no	—
150	C₄×C₁₉	C₂²×M₁₆	1	12	yes	—
151	C₄×C₁₉	#568	1	2	yes	—
152	C₄×C₁₉	#585	1	2	yes	—
153	C₂²×#41	C₄×C₁₉	1	1	yes	characteristic
154	C₄×#41	C₄×C₁₉	1	4	yes	—
155	C₄×#41	C₄×C₁₉	1	4	yes	—
156	C₄×#41	C₂²×C₁₉	1	4	yes	—
157	C₄×#41	C₂²×C₁₉	1	4	yes	—
158	C₂×C₄×Q₈	C₄×C₁₉	1	1	yes	characteristic
159	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
160	C₂×C₄×D₄	C₄×C₁₉	1	3	yes	—
161	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
162	C₂×#117	C₄×C₁₉	1	3	yes	—
163	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
164	C₈×Q₈	C₂²×C₁₉	1	16	yes	complemented
165	#436	C₂²×C₁₉	1	48	yes	complemented
166	#432	C₂²×C₁₉	1	4	yes	—
167	#432	C₂²×C₁₉	1	12	yes	complemented
168	#431	C₂²×C₁₉	1	12	yes	—
169	#431	C₂²×C₁₉	1	12	yes	complemented
170	#423	C₂²×C₁₉	1	12	yes	complemented
171	#407	C₂²×C₁₉	1	12	yes	—
172	C₄×M₁₆	C₂²×C₁₉	1	12	yes	complemented
173	C₂×C₁₉	—	2	2	no	—
174	C₂×C₁₉	—	4	6	no	—
175	C₂×C₁₉	#3428	1	1	yes	characteristic
176	C₂×C₁₉	C₂³×M₁₆	1	1	yes	characteristic
177	C₂×C₁₉	C₂×#568	1	1	yes	characteristic
178	C₂×#41	—	2	8	no	—
179	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
180	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
181	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
182	C₂²×D₄	C₂×C₄×C₁₉	1	3	yes	—
183	C₂³×C₄	C₂×C₄×C₁₉	1	3	yes	—
184	C₂³×C₄	C₂×C₄×C₁₉	1	3	yes	—
185	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
186	C₂×M₁₆	C₂³×C₁₉	1	24	yes	—
187	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
188	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
189	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
190	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
191	C₄×D₄	C₂×C₄×C₁₉	1	12	yes	—
192	C₄×D₄	C₂×C₄×C₁₉	1	12	yes	—
193	C₄×D₄	C₂×C₄×C₁₉	1	12	yes	—
194	C₄×D₄	C₂³×C₁₉	1	12	yes	—
195	#117	C₂×C₄×C₁₉	1	12	yes	—
196	#117	C₂×C₄×C₁₉	1	12	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	3	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	3	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	6	yes	—
200	C₂×C₄²	C₂×C₄×C₁₉	1	3	yes	—
201	C₂×C₄²	C₂×C₄×C₁₉	1	3	yes	—
202	C₂×C₄²	C₂×C₄×C₁₉	1	6	yes	—
203	C₂×C₄²	C₂×C₄×C₁₉	1	6	yes	—
204	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
205	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
206	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
207	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
208	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
209	C₄⋊C₈	C₂³×C₁₉	1	24	yes	—
210	C₂²⋊C₈	C₂³×C₁₉	1	8	yes	—
211	C₂²⋊C₈	C₂³×C₁₉	1	24	yes	—
212	#97	C₂³×C₁₉	1	8	yes	—
213	C₄×C₈	C₂³×C₁₉	1	24	yes	—
214	C₁₉	#60284	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂²×C₄×C₁₉	1	3	yes	—
216	#41	—	2	16	no	—
217	#41	—	2	16	no	—
218	C₂×Q₈	M₁₆×C₁₉	1	8	yes	complemented
219	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
220	C₂×D₄	—	2	12	no	—
221	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
222	C₂²×C₄	—	2	12	no	—
223	C₂²×C₄	—	4	6	no	—
224	C₂²×C₄	—	4	6	no	—
225	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
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	3	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
233	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
234	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
235	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
237	M₁₆	—	4	24	no	—
238	C₂×C₈	C₂⁴×C₁₉	1	16	yes	—
239	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
240	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
241	C₂²⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
242	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
243	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
244	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
245	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
246	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
247	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
248	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
249	C₄²	C₂⁴×C₁₉	1	6	yes	—
250	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
251	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
252	C₂³	—	2	6	no	—
253	C₂³	—	4	3	no	—
254	C₂³	—	4	3	no	—
255	C₂³	C₂³×C₄×C₁₉	1	6	yes	—
256	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
257	Q₈	C₂×M₁₆×C₁₉	1	16	yes	—
258	D₄	—	2	24	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	12	no	—
261	C₂×C₄	—	2	12	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	12	no	—
264	C₂×C₄	—	2	12	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	6	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	4	6	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	C₂×M₁₆×C₁₉	1	12	yes	—
271	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
272	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
276	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
277	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
278	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
279	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
280	C₈	—	2	16	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	6	no	—
284	C₂²	—	2	6	no	—
285	C₂²	—	2	6	no	—
286	C₂²	—	4	12	no	—
287	C₂²	C₂²×M₁₆×C₁₉	1	2	yes	—
288	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	6	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	6	no	—
295	C₄	C₂²×M₁₆×C₁₉	1	12	yes	—
296	C₄	C₁₉×#568	1	2	yes	—
297	C₄	C₁₉×#585	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	6	no	—
300	C₂	C₁₉×#3428	1	1	yes	characteristic
301	C₂	C₂³×M₁₆×C₁₉	1	1	yes	characteristic
302	C₂	C₂×C₁₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
socle	C₂²×C₁₉	C₂⁴×C₄
prime radical	C₂²×#41×C₁₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2991	C₂	semidirect
3. C₁₉×#2982	C₂	semidirect
4. C₁₉×#2975	C₂	semidirect
5. C₁₉×#2971	C₂	semidirect
6. C₁₉×#2920	C₂	semidirect
18. C₈×Q₈×C₁₉	C₂²	semidirect
19. C₁₉×#436	C₂²	semidirect
21. C₁₉×#432	C₂²	semidirect
23. C₁₉×#431	C₂²	semidirect
24. C₁₉×#423	C₂²	semidirect
26. C₄×M₁₆×C₁₉	C₂²	semidirect
66. C₂×Q₈×C₁₉	M₁₆	semidirect
100. #60284	C₁₉	direct (N × H)
131. #2991	C₂×C₁₉	semidirect
132. #2982	C₂×C₁₉	semidirect
133. #2975	C₂×C₁₉	semidirect
134. #2971	C₂×C₁₉	semidirect
135. #2920	C₂×C₁₉	semidirect
164. C₈×Q₈	C₂²×C₁₉	semidirect
165. #436	C₂²×C₁₉	semidirect
167. #432	C₂²×C₁₉	semidirect
169. #431	C₂²×C₁₉	semidirect
170. #423	C₂²×C₁₉	semidirect
172. C₄×M₁₆	C₂²×C₁₉	semidirect
214. C₁₉	#60284	direct (N × H)
218. C₂×Q₈	M₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₁₉	2
C₁₉×#2991	2
C₁₉×#2982	2
C₁₉×#2975	2
C₁₉×#2971	2
C₁₉×#2920	2
#60284	19

Subgroups — C19×#60284 #18066243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60284 #18066243