Subgroups — C₂×C₁₉×#2660 #18040582

Order 4864.

2462 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#125	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2660	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₉×#398	C₂	1	4	yes	maximal
5	C₂×C₁₉×#395	C₂	1	4	yes	maximal
6	C₂×C₁₉×#384	C₂	1	2	yes	maximal
7	C₂×C₁₉×#382	C₂	1	2	yes	maximal
8	C₂×C₁₉×#376	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
10	C₂×C₁₉×C₄⋊Q₈	—	2	8	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	8	yes	—
12	C₂×C₁₉×#125	—	2	8	no	—
13	C₂×C₁₉×#125	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
21	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
23	C₁₉×#398	C₂²	1	32	yes	—
24	C₁₉×#395	C₂²	1	32	yes	—
25	C₁₉×#384	C₂²	1	16	yes	—
26	C₁₉×#382	C₂²	1	16	yes	—
27	C₁₉×#376	C₂²	1	8	yes	—
28	C₂×C₁₉×#95	C₂²	1	2	yes	—
29	C₂×C₁₉×#95	C₂²	1	2	yes	—
30	C₂×C₁₉×#95	C₂²	1	8	yes	—
31	C₂²×Q₈×C₁₉	—	2	8	no	—
32	C₂²×Q₈×C₁₉	—	4	8	no	—
33	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
40	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	—	2	32	no	—
42	C₁₉×#125	—	2	32	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
49	C₂×C₄⋊C₄×C₁₉	—	4	16	no	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
52	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
54	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₁₉	C₂³	1	32	yes	—
57	C₂×C₄²×C₁₉	D₄	1	8	yes	—
58	C₂×C₄²×C₁₉	Q₈	1	16	yes	complemented
59	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
60	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
61	C₁₉×#95	C₂³	1	8	yes	—
62	C₁₉×#95	C₂³	1	8	yes	—
63	C₁₉×#95	C₂³	1	32	yes	—
64	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
65	C₂×Q₈×C₁₉	—	2	16	no	—
66	C₂×Q₈×C₁₉	—	4	16	no	—
67	C₂×Q₈×C₁₉	—	4	32	no	—
68	C₂²×C₄×C₁₉	—	2	8	no	—
69	C₂²×C₄×C₁₉	—	2	4	no	—
70	C₂²×C₄×C₁₉	—	2	4	no	—
71	C₂²×C₄×C₁₉	—	2	8	no	—
72	C₂²×C₄×C₁₉	—	2	4	no	—
73	C₂²×C₄×C₁₉	—	2	4	no	—
74	C₂²×C₄×C₁₉	—	2	8	no	—
75	C₂²×C₄×C₁₉	—	2	8	no	—
76	C₂²×C₄×C₁₉	—	2	16	no	—
77	C₂²×C₄×C₁₉	—	2	8	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₉	C₂×Q₈	1	16	yes	—
82	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
86	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
87	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
90	C₄⋊C₄×C₁₉	—	2	16	no	—
91	C₄⋊C₄×C₁₉	—	2	16	no	—
92	C₄⋊C₄×C₁₉	—	2	8	no	—
93	C₄⋊C₄×C₁₉	—	2	8	no	—
94	C₄⋊C₄×C₁₉	—	2	16	no	—
95	C₄⋊C₄×C₁₉	—	2	16	no	—
96	C₄⋊C₄×C₁₉	—	4	32	no	—
97	C₄²×C₁₉	C₂×D₄	1	16	yes	—
98	C₄²×C₁₉	C₂×Q₈	1	32	yes	complemented
99	C₂×#2660	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
101	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
102	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
103	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
104	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
105	Q₈×C₁₉	—	4	32	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	2	8	no	—
108	C₂×C₄×C₁₉	—	2	32	no	—
109	C₂×C₄×C₁₉	—	2	8	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	8	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	8	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	16	no	—
118	C₂×C₄×C₁₉	—	2	8	no	—
119	C₂×C₄×C₁₉	—	4	8	no	—
120	C₂×C₄×C₁₉	—	4	16	no	—
121	C₂×C₄×C₁₉	—	4	8	no	—
122	C₂×C₄×C₁₉	C₄⋊Q₈	1	16	yes	—
123	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
124	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
125	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
126	C₂×C₄×C₁₉	C₂²×Q₈	1	16	yes	—
127	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
128	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
129	C₂²×#125	C₂×C₁₉	1	1	yes	characteristic
130	#2660	C₂×C₁₉	1	16	yes	direct product factor
131	C₂×#398	C₂×C₁₉	1	4	yes	—
132	C₂×#395	C₂×C₁₉	1	4	yes	—
133	C₂×#384	C₂×C₁₉	1	2	yes	—
134	C₂×#382	C₂×C₁₉	1	2	yes	—
135	C₂×#376	C₂×C₁₉	1	1	yes	characteristic
136	C₂²×C₁₉	C₂×C₄⋊Q₈	1	4	yes	—
137	C₂²×C₁₉	#535	1	4	yes	—
138	C₂²×C₁₉	#536	1	4	yes	—
139	C₂²×C₁₉	#544	1	16	yes	—
140	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
141	C₂²×C₁₉	C₂³×Q₈	1	2	yes	—
142	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
143	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
144	C₄×C₁₉	—	2	4	no	—
145	C₄×C₁₉	—	2	4	no	—
146	C₄×C₁₉	—	2	16	no	—
147	C₄×C₁₉	—	4	4	no	—
148	C₄×C₁₉	—	4	8	no	—
149	C₄×C₁₉	—	4	4	no	—
150	C₄×C₁₉	C₂×C₄⋊Q₈	1	8	yes	—
151	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
152	C₂×C₄⋊Q₈	—	2	8	no	—
153	C₂×C₄⋊Q₈	C₂²×C₁₉	1	8	yes	—
154	C₂×#125	—	2	8	no	—
155	C₂×#125	C₂²×C₁₉	1	8	yes	—
156	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
157	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
162	C₂²×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
163	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
164	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
165	#398	C₂²×C₁₉	1	32	yes	—
166	#395	C₂²×C₁₉	1	32	yes	—
167	#384	C₂²×C₁₉	1	16	yes	—
168	#382	C₂²×C₁₉	1	16	yes	—
169	#376	C₂²×C₁₉	1	8	yes	—
170	C₂×#95	C₂²×C₁₉	1	2	yes	—
171	C₂×#95	C₂²×C₁₉	1	8	yes	—
172	C₂×#95	C₂²×C₁₉	1	2	yes	—
173	C₂×C₁₉	#2660	1	8	yes	direct product factor
174	C₂×C₁₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₁₉	C₂×#535	1	1	yes	characteristic
176	C₂×C₁₉	C₂×#536	1	1	yes	characteristic
177	C₂×C₁₉	C₂×#544	1	4	yes	—
178	C₂²×Q₈	—	2	8	no	—
179	C₂²×Q₈	—	4	8	no	—
180	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
181	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
183	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
184	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
187	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	32	no	—
189	#125	—	2	32	no	—
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	8	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	—	2	8	no	—
196	C₂×C₄⋊C₄	—	4	16	no	—
197	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₉	1	32	yes	—
204	C₂×C₄²	D₄×C₁₉	1	8	yes	—
205	C₂×C₄²	Q₈×C₁₉	1	16	yes	complemented
206	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
207	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
208	#95	C₂³×C₁₉	1	8	yes	—
209	#95	C₂³×C₁₉	1	8	yes	—
210	#95	C₂³×C₁₉	1	32	yes	—
211	C₁₉	C₂×#2660	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
213	C₂×Q₈	—	2	16	no	—
214	C₂×Q₈	—	4	16	no	—
215	C₂×Q₈	—	4	32	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	16	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
228	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
229	C₂²×C₄	C₂×Q₈×C₁₉	1	16	yes	—
230	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
231	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
233	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
236	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
237	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	2	16	no	—
243	C₄⋊C₄	—	2	16	no	—
244	C₄⋊C₄	—	4	32	no	—
245	C₄²	C₂×D₄×C₁₉	1	16	yes	—
246	C₄²	C₂×Q₈×C₁₉	1	32	yes	complemented
247	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
248	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
249	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
250	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
251	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
252	Q₈	—	4	32	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	32	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	16	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	16	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	C₁₉×C₄⋊Q₈	1	16	yes	—
270	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
271	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
272	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
273	C₂×C₄	C₂²×Q₈×C₁₉	1	16	yes	—
274	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
275	C₂²	C₂×C₁₉×C₄⋊Q₈	1	4	yes	—
276	C₂²	C₁₉×#535	1	4	yes	—
277	C₂²	C₁₉×#536	1	4	yes	—
278	C₂²	C₁₉×#544	1	16	yes	—
279	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
280	C₂²	C₂³×Q₈×C₁₉	1	2	yes	—
281	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
282	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	16	no	—
286	C₄	—	4	8	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	C₂×C₁₉×C₄⋊Q₈	1	8	yes	—
290	C₂	C₁₉×#2660	1	8	yes	direct product factor
291	C₂	C₂²×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
292	C₂	C₂×C₁₉×#535	1	1	yes	characteristic
293	C₂	C₂×C₁₉×#536	1	1	yes	characteristic
294	C₂	C₂×C₁₉×#544	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2660	C₂	direct (N × H)
58. C₂×C₄²×C₁₉	Q₈	semidirect
98. C₄²×C₁₉	C₂×Q₈	semidirect
99. C₂×#2660	C₁₉	direct (N × H)
130. #2660	C₂×C₁₉	direct (N × H)
173. C₂×C₁₉	#2660	direct (N × H)
205. C₂×C₄²	Q₈×C₁₉	semidirect
211. C₁₉	C₂×#2660	direct (N × H)
246. C₄²	C₂×Q₈×C₁₉	semidirect
290. C₂	C₁₉×#2660	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×C₄⋊Q₈	2
C₂²×C₁₉×#125	2
C₁₉×#2660	2
C₂×C₁₉×#398	2
C₂×C₁₉×#395	2
C₂×C₁₉×#384	2
C₂×C₁₉×#382	2
C₂×C₁₉×#376	2
C₂×#2660	19

Subgroups — C2×C19×#2660 #18040582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2660 #18040582