Subgroups — C₂×C₁₉×#2100 #18029702

Order 4864.

998 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#389	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2100	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₉×#340	C₂	1	2	yes	maximal
5	C₂×C₁₉×#340	C₂	1	2	yes	maximal
6	C₂×C₁₉×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#389	—	2	2	no	—
16	C₁₉×#389	C₂²	1	4	yes	—
17	C₁₉×#384	—	2	2	no	—
18	C₁₉×#384	C₂²	1	4	yes	—
19	C₂×C₁₉×#95	—	2	1	no	—
20	C₁₉×#340	C₂²	1	8	yes	—
21	C₁₉×#340	C₂²	1	8	yes	—
22	C₁₉×#336	C₂²	1	4	yes	—
23	C₂³×C₄×C₁₉	—	2	1	no	—
24	C₂³×C₄×C₁₉	—	2	1	no	—
25	C₂³×C₄×C₁₉	—	2	2	no	—
26	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	—	2	4	no	—
30	C₂²×C₈×C₁₉	—	2	4	no	—
31	C₂²×C₈×C₁₉	—	2	4	no	—
32	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
43	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
44	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
45	C₂×C₄⋊C₄×C₁₉	Q₈	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₉	—	2	4	no	—
50	C₂×C₄²×C₁₉	D₄	1	2	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
52	C₁₉×#95	—	2	4	no	—
53	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	—	4	4	no	—
66	C₂²×C₄×C₁₉	—	4	2	no	—
67	C₂²×C₄×C₁₉	—	4	4	no	—
68	C₂²×C₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	—	4	2	no	—
70	C₂²×C₄×C₁₉	SD₁₆	1	4	yes	—
71	C₂²×C₄×C₁₉	Q₁₆	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×C₄×C₁₉	#41	1	4	yes	—
78	C₂²×C₄×C₁₉	#41	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	2	4	no	—
83	C₂×C₈×C₁₉	—	4	8	no	—
84	C₄⋊C₄×C₁₉	—	2	4	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	—	2	8	no	—
88	C₄⋊C₄×C₁₉	—	4	8	no	—
89	C₄⋊C₄×C₁₉	—	4	8	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	4	no	—
92	C₄⋊C₄×C₁₉	—	4	8	no	—
93	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
94	C₄²×C₁₉	—	2	4	no	—
95	C₄²×C₁₉	—	2	4	no	—
96	C₂×#2100	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₁₉	#123	1	2	yes	—
99	C₂³×C₁₉	#124	1	2	yes	—
100	C₂³×C₁₉	C₂×SD₁₆	1	2	yes	—
101	C₂³×C₁₉	C₂×Q₁₆	1	2	yes	—
102	C₂³×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂³×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
104	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	8	no	—
110	C₂×C₄×C₁₉	—	2	8	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	8	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	8	4	no	—
122	C₂×C₄×C₁₉	—	8	8	no	—
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₉	#123	1	2	yes	—
125	C₂×C₄×C₁₉	#124	1	2	yes	—
126	C₂×C₄×C₁₉	#125	1	8	yes	—
127	C₂×C₄×C₁₉	#126	1	8	yes	—
128	C₂×C₄×C₁₉	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
130	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₉	—	4	4	no	—
135	C₂×#389	C₂×C₁₉	1	1	yes	characteristic
136	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
137	#2100	C₂×C₁₉	1	8	yes	direct product factor
138	C₂×#340	C₂×C₁₉	1	2	yes	—
139	C₂×#340	C₂×C₁₉	1	2	yes	—
140	C₂×#336	C₂×C₁₉	1	1	yes	characteristic
141	C₂²×C₁₉	#400	1	4	yes	—
142	C₂²×C₁₉	#476	1	4	yes	—
143	C₂²×C₁₉	#477	1	4	yes	—
144	C₂²×C₁₉	#481	1	4	yes	—
145	C₂²×C₁₉	#484	1	4	yes	—
146	C₂²×C₁₉	#487	1	4	yes	—
147	C₂²×C₁₉	#490	1	4	yes	—
148	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#123	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
151	C₂²×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂²×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄×C₁₉	—	2	4	no	—
156	C₄×C₁₉	—	2	4	no	—
157	C₄×C₁₉	—	4	4	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	—	8	4	no	—
160	C₄×C₁₉	#400	1	4	yes	—
161	C₄×C₁₉	C₂×#125	1	2	yes	—
162	C₄×C₁₉	C₂×#126	1	2	yes	—
163	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
164	C₂²×C₄⋊C₄	—	2	1	no	—
165	C₂²×C₄⋊C₄	—	2	1	no	—
166	C₂²×C₄⋊C₄	—	2	1	no	—
167	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
168	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
170	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
171	#389	—	2	2	no	—
172	#389	C₂²×C₁₉	1	4	yes	—
173	#384	—	2	2	no	—
174	#384	C₂²×C₁₉	1	4	yes	—
175	C₂×#95	—	2	1	no	—
176	#340	C₂²×C₁₉	1	8	yes	—
177	#340	C₂²×C₁₉	1	8	yes	—
178	#336	C₂²×C₁₉	1	4	yes	—
179	C₂×C₁₉	#2100	1	8	yes	direct product factor
180	C₂×C₁₉	C₂×#400	1	1	yes	characteristic
181	C₂×C₁₉	C₂×#476	1	1	yes	characteristic
182	C₂×C₁₉	C₂×#477	1	1	yes	characteristic
183	C₂×C₁₉	C₂×#481	1	1	yes	characteristic
184	C₂×C₁₉	C₂×#484	1	1	yes	characteristic
185	C₂×C₁₉	C₂×#487	1	1	yes	characteristic
186	C₂×C₁₉	C₂×#490	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂³×C₄	—	2	2	no	—
189	C₂³×C₄	—	2	1	no	—
190	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
192	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	8	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	4	4	no	—
207	C₂×C₄⋊C₄	—	4	4	no	—
208	C₂×C₄⋊C₄	—	4	4	no	—
209	C₂×C₄⋊C₄	Q₈×C₁₉	1	4	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	D₄×C₁₉	1	2	yes	—
215	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
216	#95	—	2	4	no	—
217	C₁₉	C₂×#2100	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	SD₁₆×C₁₉	1	4	yes	—
236	C₂²×C₄	Q₁₆×C₁₉	1	4	yes	—
237	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
241	C₂²×C₄	#41×C₁₉	1	2	yes	—
242	C₂²×C₄	#41×C₁₉	1	4	yes	—
243	C₂²×C₄	#41×C₁₉	1	4	yes	—
244	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	8	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	4	8	no	—
254	C₄⋊C₄	—	4	8	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	8	no	—
258	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₂³	C₁₉×C₂²⋊Q₈	1	2	yes	—
262	C₂³	C₁₉×#123	1	2	yes	—
263	C₂³	C₁₉×#124	1	2	yes	—
264	C₂³	C₂×SD₁₆×C₁₉	1	2	yes	—
265	C₂³	C₂×Q₁₆×C₁₉	1	2	yes	—
266	C₂³	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
267	C₂³	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
268	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	—	8	8	no	—
287	C₂×C₄	C₁₉×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₁₉×#123	1	2	yes	—
289	C₂×C₄	C₁₉×#124	1	2	yes	—
290	C₂×C₄	C₁₉×#125	1	8	yes	—
291	C₂×C₄	C₁₉×#126	1	8	yes	—
292	C₂×C₄	C₂×SD₁₆×C₁₉	1	4	yes	—
293	C₂×C₄	C₂×Q₁₆×C₁₉	1	4	yes	—
294	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
296	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
297	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
298	C₈	—	4	4	no	—
299	C₂²	C₁₉×#400	1	4	yes	—
300	C₂²	C₁₉×#476	1	4	yes	—
301	C₂²	C₁₉×#477	1	4	yes	—
302	C₂²	C₁₉×#481	1	4	yes	—
303	C₂²	C₁₉×#484	1	4	yes	—
304	C₂²	C₁₉×#487	1	4	yes	—
305	C₂²	C₁₉×#490	1	4	yes	—
306	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
307	C₂²	C₂×C₁₉×#123	1	1	yes	characteristic
308	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
309	C₂²	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
310	C₂²	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
311	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	4	no	—
318	C₄	C₁₉×#400	1	4	yes	—
319	C₄	C₂×C₁₉×#125	1	2	yes	—
320	C₄	C₂×C₁₉×#126	1	2	yes	—
321	C₂	C₁₉×#2100	1	8	yes	direct product factor
322	C₂	C₂×C₁₉×#400	1	1	yes	characteristic
323	C₂	C₂×C₁₉×#476	1	1	yes	characteristic
324	C₂	C₂×C₁₉×#477	1	1	yes	characteristic
325	C₂	C₂×C₁₉×#481	1	1	yes	characteristic
326	C₂	C₂×C₁₉×#484	1	1	yes	characteristic
327	C₂	C₂×C₁₉×#487	1	1	yes	characteristic
328	C₂	C₂×C₁₉×#490	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×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
3. C₁₉×#2100	C₂	direct (N × H)
96. C₂×#2100	C₁₉	direct (N × H)
137. #2100	C₂×C₁₉	direct (N × H)
179. C₂×C₁₉	#2100	direct (N × H)
217. C₁₉	C₂×#2100	direct (N × H)
321. C₂	C₁₉×#2100	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#389	2
C₂×C₁₉×#384	2
C₁₉×#2100	2
C₂×C₁₉×#340	2
C₂×C₁₉×#336	2
C₂×#2100	19

Subgroups — C2×C19×#2100 #18029702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2100 #18029702