Subgroups — C₂×C₁₉×#1960 #18027156

Order 4864.

1598 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#103	C₂	1	4	yes	maximal
3	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#382	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1960	C₂	1	8	yes	maximal, direct product factor
6	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
8	C₂×C₁₉×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₁₉×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₁₉×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#103	—	2	16	no	—
16	C₂×C₁₉×#103	C₂²	1	16	yes	—
17	C₂×C₁₉×#97	—	2	4	no	—
18	C₂×C₁₉×#97	C₂²	1	4	yes	—
19	C₁₉×#382	—	2	2	no	—
20	C₁₉×#382	C₂²	1	4	yes	—
21	C₂×C₁₉×#95	—	2	2	no	—
22	C₂²×Q₈×C₁₉	—	2	4	no	—
23	C₂²×Q₈×C₁₉	—	2	2	no	—
24	C₂²×Q₈×C₁₉	—	2	8	no	—
25	C₂²×Q₈×C₁₉	—	2	8	no	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₉	—	2	2	no	—
29	C₂³×C₄×C₁₉	—	2	2	no	—
30	C₂³×C₄×C₁₉	D₄	1	2	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	—	2	4	no	—
33	C₂²×C₈×C₁₉	—	2	4	no	—
34	C₂²×C₈×C₁₉	D₄	1	8	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
36	C₁₉×C₄⋊Q₈	—	2	16	no	—
37	C₁₉×C₄⋊Q₈	—	2	8	no	—
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
39	C₂×C₄⋊C₄×C₁₉	—	2	16	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₁₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₉	D₄	1	4	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
48	C₁₉×#103	—	2	32	no	—
49	C₁₉×#97	—	2	8	no	—
50	C₁₉×#95	—	2	8	no	—
51	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈×C₁₉	—	2	8	no	—
53	C₂×Q₈×C₁₉	—	2	4	no	—
54	C₂×Q₈×C₁₉	—	2	32	no	—
55	C₂×Q₈×C₁₉	—	2	8	no	—
56	C₂×Q₈×C₁₉	—	2	8	no	—
57	C₂×Q₈×C₁₉	—	4	16	no	—
58	C₂×Q₈×C₁₉	—	4	4	no	—
59	C₂×Q₈×C₁₉	—	4	4	no	—
60	C₂×Q₈×C₁₉	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₁₉	—	2	8	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	4	4	no	—
66	C₂²×C₄×C₁₉	—	4	4	no	—
67	C₂²×C₄×C₁₉	—	4	4	no	—
68	C₂²×C₄×C₁₉	—	4	8	no	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₉	#41	1	4	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₉	—	2	16	no	—
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	4	no	—
80	C₂×C₈×C₁₉	C₂×D₄	1	8	yes	—
81	C₄⋊C₄×C₁₉	—	2	32	no	—
82	C₄⋊C₄×C₁₉	—	2	8	no	—
83	C₄⋊C₄×C₁₉	—	4	4	no	—
84	C₄²×C₁₉	C₂²⋊C₄	1	8	yes	—
85	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
86	C₄²×C₁₉	C₂×D₄	1	4	yes	—
87	C₂×#1960	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₁₉	#124	1	2	yes	—
90	C₂³×C₁₉	#127	1	2	yes	—
91	C₂³×C₁₉	#137	1	8	yes	—
92	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₁₉	—	2	16	no	—
94	Q₈×C₁₉	—	4	8	no	—
95	Q₈×C₁₉	—	4	16	no	—
96	C₂×C₄×C₁₉	—	2	8	no	—
97	C₂×C₄×C₁₉	—	2	16	no	—
98	C₂×C₄×C₁₉	—	2	8	no	—
99	C₂×C₄×C₁₉	—	4	4	no	—
100	C₂×C₄×C₁₉	—	4	4	no	—
101	C₂×C₄×C₁₉	—	4	16	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	4	8	no	—
104	C₂×C₄×C₁₉	—	8	8	no	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
106	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
108	C₂×C₄×C₁₉	#121	1	8	yes	—
109	C₂×C₄×C₁₉	#124	1	2	yes	—
110	C₂×C₄×C₁₉	#127	1	2	yes	—
111	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
115	C₈×C₁₉	—	2	8	no	—
116	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
117	C₂²×#103	C₂×C₁₉	1	4	yes	—
118	C₂²×#97	C₂×C₁₉	1	1	yes	characteristic
119	C₂×#382	C₂×C₁₉	1	1	yes	characteristic
120	#1960	C₂×C₁₉	1	8	yes	direct product factor
121	C₂²×C₁₉	#390	1	4	yes	—
122	C₂²×C₁₉	#419	1	8	yes	—
123	C₂²×C₁₉	#491	1	8	yes	—
124	C₂²×C₁₉	#497	1	8	yes	—
125	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
127	C₂²×C₁₉	C₂×#127	1	1	yes	characteristic
128	C₂²×C₁₉	C₂×#137	1	4	yes	—
129	C₄×C₁₉	—	2	8	no	—
130	C₄×C₁₉	—	4	8	no	—
131	C₄×C₁₉	—	8	4	no	—
132	C₄×C₁₉	#390	1	4	yes	—
133	C₄×C₁₉	C₂×C₄×D₄	1	2	yes	—
134	C₄×C₁₉	C₂×#121	1	2	yes	—
135	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
136	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
137	C₂×C₄⋊Q₈	—	2	2	no	—
138	C₂×C₄⋊Q₈	—	2	4	no	—
139	C₂×C₄⋊Q₈	C₄×C₁₉	1	8	yes	complemented
140	C₂×C₄⋊Q₈	C₂²×C₁₉	1	4	yes	—
141	C₂²×C₄⋊C₄	—	2	2	no	—
142	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
143	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
144	C₂×#103	—	2	16	no	—
145	C₂×#103	C₂²×C₁₉	1	16	yes	—
146	C₂×#97	—	2	4	no	—
147	C₂×#97	C₂²×C₁₉	1	4	yes	—
148	#382	—	2	2	no	—
149	#382	C₂²×C₁₉	1	4	yes	—
150	C₂×#95	—	2	2	no	—
151	C₂×C₁₉	#1960	1	8	yes	direct product factor
152	C₂×C₁₉	C₂×#390	1	1	yes	characteristic
153	C₂×C₁₉	C₂×#419	1	2	yes	—
154	C₂×C₁₉	C₂×#491	1	2	yes	—
155	C₂×C₁₉	C₂×#497	1	2	yes	—
156	C₂²×Q₈	—	2	4	no	—
157	C₂²×Q₈	—	2	2	no	—
158	C₂²×Q₈	—	2	8	no	—
159	C₂²×Q₈	—	2	8	no	—
160	C₂²×Q₈	C₂×C₄×C₁₉	1	8	yes	—
161	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	D₄×C₁₉	1	2	yes	—
165	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	D₄×C₁₉	1	8	yes	—
169	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
170	C₄⋊Q₈	—	2	16	no	—
171	C₄⋊Q₈	—	2	8	no	—
172	C₄⋊Q₈	C₂×C₄×C₁₉	1	16	yes	complemented
173	C₂×C₄⋊C₄	—	2	16	no	—
174	C₂×C₄⋊C₄	—	2	8	no	—
175	C₂×C₄⋊C₄	—	2	8	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
179	C₂×C₄²	D₄×C₁₉	1	2	yes	—
180	C₂×C₄²	D₄×C₁₉	1	4	yes	—
181	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
182	#103	—	2	32	no	—
183	#97	—	2	8	no	—
184	#95	—	2	8	no	—
185	C₁₉	C₂×#1960	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
187	C₂×Q₈	—	2	8	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	32	no	—
190	C₂×Q₈	—	2	8	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	4	16	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	C₂²×C₄×C₁₉	1	8	yes	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
210	C₂²×C₄	#41×C₁₉	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
212	C₂×C₈	—	2	16	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂×D₄×C₁₉	1	8	yes	—
216	C₄⋊C₄	—	2	32	no	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄²	C₂²⋊C₄×C₁₉	1	8	yes	—
220	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
221	C₄²	C₂×D₄×C₁₉	1	4	yes	—
222	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
223	C₂³	C₁₉×#124	1	2	yes	—
224	C₂³	C₁₉×#127	1	2	yes	—
225	C₂³	C₁₉×#137	1	8	yes	—
226	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
227	Q₈	—	2	16	no	—
228	Q₈	—	4	16	no	—
229	Q₈	—	4	8	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	4	16	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	8	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
241	C₂×C₄	C₄×D₄×C₁₉	1	8	yes	—
242	C₂×C₄	C₁₉×#121	1	8	yes	—
243	C₂×C₄	C₁₉×#124	1	2	yes	—
244	C₂×C₄	C₁₉×#127	1	2	yes	—
245	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
247	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₂²	C₁₉×#390	1	4	yes	—
251	C₂²	C₁₉×#419	1	8	yes	—
252	C₂²	C₁₉×#491	1	8	yes	—
253	C₂²	C₁₉×#497	1	8	yes	—
254	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
255	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
256	C₂²	C₂×C₁₉×#127	1	1	yes	characteristic
257	C₂²	C₂×C₁₉×#137	1	4	yes	—
258	C₄	—	2	8	no	—
259	C₄	—	4	8	no	—
260	C₄	—	8	4	no	—
261	C₄	C₁₉×#390	1	4	yes	—
262	C₄	C₂×C₄×D₄×C₁₉	1	2	yes	—
263	C₄	C₂×C₁₉×#121	1	2	yes	—
264	C₂	C₁₉×#1960	1	8	yes	direct product factor
265	C₂	C₂×C₁₉×#390	1	1	yes	characteristic
266	C₂	C₂×C₁₉×#419	1	2	yes	—
267	C₂	C₂×C₁₉×#491	1	2	yes	—
268	C₂	C₂×C₁₉×#497	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₁₉×#1960	C₂	direct (N × H)
10. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
38. C₁₉×C₄⋊Q₈	C₂×C₄	semidirect
87. C₂×#1960	C₁₉	direct (N × H)
120. #1960	C₂×C₁₉	direct (N × H)
139. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
151. C₂×C₁₉	#1960	direct (N × H)
172. C₄⋊Q₈	C₂×C₄×C₁₉	semidirect
185. C₁₉	C₂×#1960	direct (N × H)
264. C₂	C₁₉×#1960	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×#103	2
C₂²×C₁₉×#97	2
C₂×C₁₉×#382	2
C₁₉×#1960	2
C₂×#1960	19

Subgroups — C2×C19×#1960 #18027156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1960 #18027156