Subgroups — C₁₉×#21882 #18027841

Order 4864.

1082 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#540	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2921	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2905	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1998	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1965	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1900	C₂	1	4	yes	maximal, complemented
7	C₁₉×#1800	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
9	C₁₉×#540	—	2	4	no	—
10	C₁₉×#540	—	2	4	no	—
11	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#123	C₄	1	4	yes	—
14	C₂×C₁₉×#121	C₄	1	4	yes	—
15	C₂×C₁₉×#121	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₉	C₂²	1	2	yes	—
17	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
18	C₁₉×#433	—	2	4	no	—
19	C₁₉×#424	—	2	4	no	—
20	C₁₉×#423	C₂²	1	2	yes	—
21	C₁₉×#423	C₂²	1	2	yes	—
22	C₁₉×#423	C₂²	1	2	yes	—
23	C₂×C₁₉×#100	C₂²	1	8	yes	—
24	C₁₉×#407	C₂²	1	2	yes	—
25	C₄×M₁₆×C₁₉	C₂²	1	2	yes	—
26	C₁₉×#343	C₂²	1	8	yes	—
27	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
28	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
29	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂×M₁₆×C₁₉	—	4	4	no	—
34	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
35	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₉	—	2	2	no	—
37	C₁₉×#127	—	2	2	no	—
38	C₁₉×#127	—	2	2	no	—
39	C₁₉×#125	—	2	2	no	—
40	C₁₉×#125	—	2	2	no	—
41	C₁₉×#123	—	2	16	no	—
42	C₁₉×#121	—	2	4	no	—
43	C₁₉×#121	—	2	4	no	—
44	C₁₉×#121	—	2	16	no	—
45	C₄×D₄×C₁₉	—	2	4	no	—
46	C₄×D₄×C₁₉	—	2	4	no	—
47	C₁₉×#117	D₄	1	4	yes	—
48	C₁₉×#117	D₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×C₄⋊C₈	—	2	4	no	—
58	C₁₉×C₄⋊C₈	—	2	4	no	—
59	C₁₉×C₄⋊C₈	—	2	4	no	—
60	C₁₉×#100	—	4	8	no	—
61	C₁₉×C₂²⋊C₈	—	2	4	no	—
62	C₁₉×#97	—	2	2	no	—
63	C₄×C₈×C₁₉	—	2	2	no	—
64	C₂⁴×C₁₉	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×D₄×C₁₉	—	2	8	no	—
67	C₂×D₄×C₁₉	—	2	8	no	—
68	C₂×D₄×C₁₉	—	2	2	no	—
69	C₂×D₄×C₁₉	—	2	2	no	—
70	C₂×D₄×C₁₉	—	2	4	no	—
71	C₂×D₄×C₁₉	—	2	4	no	—
72	C₂×D₄×C₁₉	—	8	4	no	—
73	C₂×D₄×C₁₉	—	8	4	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	—	2	8	no	—
77	C₂²×C₄×C₁₉	—	8	4	no	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₁₉	—	2	8	no	—
86	M₁₆×C₁₉	—	2	4	no	—
87	M₁₆×C₁₉	—	2	4	no	—
88	M₁₆×C₁₉	—	4	4	no	—
89	C₂×C₈×C₁₉	—	2	4	no	—
90	C₂×C₈×C₁₉	—	2	4	no	—
91	C₂×C₈×C₁₉	—	4	4	no	—
92	C₄⋊C₄×C₁₉	—	2	8	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
96	C₂²⋊C₄×C₁₉	—	2	8	no	—
97	C₂²⋊C₄×C₁₉	—	2	8	no	—
98	C₂²⋊C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
99	C₄²×C₁₉	C₂×D₄	1	2	yes	—
100	C₄²×C₁₉	C₂×D₄	1	2	yes	—
101	C₄²×C₁₉	C₂×D₄	1	2	yes	—
102	C₄²×C₁₉	C₂×D₄	1	2	yes	—
103	#21882	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₁₉	—	2	2	no	—
105	C₂³×C₁₉	—	2	4	no	—
106	C₂³×C₁₉	—	2	8	no	—
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	—	4	4	no	—
109	C₂³×C₁₉	—	4	4	no	—
110	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
112	D₄×C₁₉	—	4	2	no	—
113	D₄×C₁₉	—	4	2	no	—
114	D₄×C₁₉	—	4	4	no	—
115	D₄×C₁₉	—	8	8	no	—
116	D₄×C₁₉	—	8	8	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	8	8	no	—
126	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
128	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
132	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₁₉	—	4	4	no	—
134	C₈×C₁₉	—	4	4	no	—
135	C₂×#540	C₂×C₁₉	1	1	yes	characteristic
136	#2921	C₂×C₁₉	1	1	yes	characteristic, complemented
137	#2905	C₂×C₁₉	1	1	yes	characteristic, complemented
138	#1998	C₂×C₁₉	1	2	yes	complemented
139	#1965	C₂×C₁₉	1	2	yes	complemented
140	#1900	C₂×C₁₉	1	4	yes	complemented
141	#1800	C₂×C₁₉	1	4	yes	complemented
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	4	no	—
145	C₂²×C₁₉	—	4	4	no	—
146	C₂²×C₁₉	—	4	4	no	—
147	C₂²×C₁₉	—	4	4	no	—
148	C₂²×C₁₉	—	8	2	no	—
149	C₂²×C₁₉	—	8	8	no	—
150	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₉	#520	1	2	yes	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	2	4	no	—
155	C₄×C₁₉	—	4	4	no	—
156	C₄×C₁₉	#535	1	1	yes	characteristic
157	C₄×C₁₉	#536	1	1	yes	characteristic
158	C₄×C₁₉	#537	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
160	#540	—	2	4	no	—
161	#540	—	2	4	no	—
162	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×#125	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×#123	C₄×C₁₉	1	4	yes	—
165	C₂×#121	C₄×C₁₉	1	4	yes	—
166	C₂×#121	C₂²×C₁₉	1	2	yes	—
167	C₂×C₄×D₄	C₂²×C₁₉	1	2	yes	—
168	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
169	#433	—	2	4	no	—
170	#424	—	2	4	no	—
171	#423	C₂²×C₁₉	1	2	yes	—
172	#423	C₂²×C₁₉	1	2	yes	—
173	#423	C₂²×C₁₉	1	2	yes	—
174	C₂×#100	C₂²×C₁₉	1	8	yes	—
175	#407	C₂²×C₁₉	1	2	yes	—
176	C₄×M₁₆	C₂²×C₁₉	1	2	yes	—
177	#343	C₂²×C₁₉	1	8	yes	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	—	4	2	no	—
180	C₂×C₁₉	—	8	4	no	—
181	C₂×C₁₉	#2311	1	1	yes	characteristic
182	C₂×C₁₉	#2888	1	2	yes	—
183	C₂²×D₄	C₂×C₄×C₁₉	1	4	yes	—
184	C₂²×D₄	C₂×C₄×C₁₉	1	4	yes	—
185	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
186	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
187	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	4	4	no	—
190	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
191	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	#127	—	2	2	no	—
194	#127	—	2	2	no	—
195	#125	—	2	2	no	—
196	#125	—	2	2	no	—
197	#123	—	2	16	no	—
198	#121	—	2	4	no	—
199	#121	—	2	4	no	—
200	#121	—	2	16	no	—
201	C₄×D₄	—	2	4	no	—
202	C₄×D₄	—	2	4	no	—
203	#117	D₄×C₁₉	1	4	yes	—
204	#117	D₄×C₁₉	1	4	yes	—
205	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
208	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
209	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
210	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
211	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	#100	—	4	8	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	#97	—	2	2	no	—
219	C₄×C₈	—	2	2	no	—
220	C₁₉	#21882	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	C₂²×C₄×C₁₉	1	4	yes	—
222	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	8	4	no	—
230	C₂×D₄	—	8	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	8	4	no	—
235	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	8	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	4	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
256	C₄²	C₂×D₄×C₁₉	1	2	yes	—
257	C₄²	C₂×D₄×C₁₉	1	2	yes	—
258	C₄²	C₂×D₄×C₁₉	1	2	yes	—
259	C₄²	C₂×D₄×C₁₉	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
267	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	8	no	—
272	D₄	—	8	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	8	8	no	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
285	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
287	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
288	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
289	C₈	—	4	4	no	—
290	C₈	—	4	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	8	no	—
299	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
300	C₂²	C₁₉×#520	1	2	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	4	4	no	—
305	C₄	C₁₉×#535	1	1	yes	characteristic
306	C₄	C₁₉×#536	1	1	yes	characteristic
307	C₄	C₁₉×#537	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	4	2	no	—
310	C₂	—	8	4	no	—
311	C₂	C₁₉×#2311	1	1	yes	characteristic
312	C₂	C₁₉×#2888	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#540	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2921	C₂	semidirect
3. C₁₉×#2905	C₂	semidirect
4. C₁₉×#1998	C₂	semidirect
5. C₁₉×#1965	C₂	semidirect
6. C₁₉×#1900	C₂	semidirect
7. C₁₉×#1800	C₂	semidirect
103. #21882	C₁₉	direct (N × H)
136. #2921	C₂×C₁₉	semidirect
137. #2905	C₂×C₁₉	semidirect
138. #1998	C₂×C₁₉	semidirect
139. #1965	C₂×C₁₉	semidirect
140. #1900	C₂×C₁₉	semidirect
141. #1800	C₂×C₁₉	semidirect
220. C₁₉	#21882	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×#540	2
C₁₉×#2921	2
C₁₉×#2905	2
C₁₉×#1998	2
C₁₉×#1965	2
C₁₉×#1900	2
C₁₉×#1800	2
#21882	19

Subgroups — C19×#21882 #18027841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21882 #18027841