Subgroups — C₁₉×#22707 #18028666

Order 4864.

910 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2354	C₂	1	2	yes	maximal
3	C₁₉×#1947	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1854	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1808	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#1735	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#117	C₂²	1	4	yes	—
12	C₈×D₄×C₁₉	C₄	1	8	yes	complemented
13	C₂×C₁₉×#107	C₂²	1	4	yes	—
14	C₂×C₁₉×#106	C₂²	1	2	yes	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#390	—	2	2	no	—
19	C₁₉×#388	—	2	4	no	—
20	C₁₉×#384	C₂²	1	2	yes	—
21	C₁₉×#380	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₁₉×#340	C₂²	1	4	yes	—
25	C₁₉×#340	C₂²	1	1	yes	characteristic
26	C₁₉×#340	C₂²	1	2	yes	—
27	C₁₉×#340	C₂²	1	1	yes	characteristic
28	C₁₉×#340	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₉	—	4	4	no	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
36	C₄×D₄×C₁₉	C₂×C₄	1	4	yes	—
37	C₄×D₄×C₁₉	Q₈	1	4	yes	—
38	C₁₉×#117	—	2	8	no	—
39	C₁₉×#117	—	4	4	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
49	C₂×C₄²×C₁₉	—	2	2	no	—
50	C₂×C₄²×C₁₉	—	2	4	no	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₁₉×#107	—	4	4	no	—
53	C₁₉×#106	—	2	4	no	—
54	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₁₉×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
57	C₁₉×#95	—	2	4	no	—
58	C₁₉×#95	—	2	2	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	2	yes	—
60	C₂×D₄×C₁₉	—	2	4	no	—
61	C₂×D₄×C₁₉	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	4	8	no	—
77	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	4	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	4	no	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	4	4	no	—
86	C₄⋊C₄×C₁₉	—	8	2	no	—
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₁₉	#41	1	4	yes	—
89	C₂²⋊C₄×C₁₉	—	4	8	no	—
90	C₂²⋊C₄×C₁₉	—	8	4	no	—
91	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	8	yes	—
92	C₄²×C₁₉	—	4	8	no	—
93	C₄²×C₁₉	—	4	4	no	—
94	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
96	#22707	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	—	2	4	no	—
99	C₂³×C₁₉	—	2	2	no	—
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
102	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
103	D₄×C₁₉	—	2	8	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	8	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	—	4	2	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	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₉	#135	1	4	yes	—
123	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
127	C₈×C₁₉	—	2	2	no	—
128	C₈×C₁₉	—	2	2	no	—
129	C₈×C₁₉	—	2	4	no	—
130	C₂×C₈×D₄	C₂×C₁₉	1	1	yes	characteristic
131	#2354	C₂×C₁₉	1	2	yes	—
132	#1947	C₂×C₁₉	1	2	yes	complemented
133	#1854	C₂×C₁₉	1	4	yes	complemented
134	#1834	C₂×C₁₉	1	1	yes	characteristic, complemented
135	#1808	C₂×C₁₉	1	2	yes	complemented
136	C₄×#106	C₂×C₁₉	1	1	yes	characteristic, complemented
137	#1735	C₂×C₁₉	1	2	yes	complemented
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	—	2	4	no	—
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	4	4	no	—
142	C₂²×C₁₉	—	4	4	no	—
143	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₁₉	#546	1	2	yes	—
145	C₂²×C₁₉	C₂×#135	1	2	yes	—
146	C₂²×C₁₉	#576	1	1	yes	characteristic
147	C₂²×C₁₉	#578	1	1	yes	characteristic
148	C₄×C₁₉	—	2	4	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	4	4	no	—
151	C₄×C₁₉	—	8	4	no	—
152	C₄×C₁₉	—	8	2	no	—
153	C₄×C₁₉	#427	1	4	yes	—
154	C₄×C₁₉	C₄×#41	1	1	yes	characteristic
155	C₄×C₁₉	#518	1	1	yes	characteristic
156	C₄×C₁₉	#554	1	2	yes	—
157	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
158	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
159	C₂×#117	C₂²×C₁₉	1	4	yes	—
160	C₈×D₄	C₄×C₁₉	1	8	yes	complemented
161	C₂×#107	C₂²×C₁₉	1	4	yes	—
162	C₂×#106	C₂²×C₁₉	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
166	#390	—	2	2	no	—
167	#388	—	2	4	no	—
168	#384	C₂²×C₁₉	1	2	yes	—
169	#380	C₂²×C₁₉	1	4	yes	—
170	C₄×C₄⋊C₄	—	2	2	no	—
171	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
172	#340	C₂²×C₁₉	1	1	yes	characteristic
173	#340	C₂²×C₁₉	1	2	yes	—
174	#340	C₂²×C₁₉	1	2	yes	—
175	#340	C₂²×C₁₉	1	4	yes	—
176	#340	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×C₁₉	—	4	4	no	—
178	C₂×C₁₉	#2349	1	1	yes	characteristic
179	C₂×C₁₉	C₂×#427	1	1	yes	characteristic
180	C₂×C₁₉	#2913	1	1	yes	characteristic
181	C₂×C₁₉	#3302	1	2	yes	—
182	C₂×C₁₉	#3305	1	2	yes	—
183	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
184	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
185	C₂²×C₈	—	4	4	no	—
186	C₂²×C₈	C₂×C₄×C₁₉	1	8	yes	—
187	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
190	C₄×D₄	C₂×C₄×C₁₉	1	4	yes	—
191	C₄×D₄	Q₈×C₁₉	1	4	yes	—
192	#117	—	2	8	no	—
193	#117	—	4	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
206	#107	—	4	4	no	—
207	#106	—	2	4	no	—
208	C₄⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
209	C₂²⋊C₈	C₂×C₄×C₁₉	1	8	yes	—
210	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	—
211	#95	—	2	4	no	—
212	#95	—	2	2	no	—
213	C₁₉	#22707	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	C₂×D₄×C₁₉	1	2	yes	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	C₄⋊C₄×C₁₉	1	4	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	8	no	—
232	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₄⋊C₄	C₄⋊C₄×C₁₉	1	4	yes	—
243	C₄⋊C₄	#41×C₁₉	1	4	yes	—
244	C₂²⋊C₄	—	4	8	no	—
245	C₂²⋊C₄	—	8	4	no	—
246	C₂²⋊C₄	C₄⋊C₄×C₁₉	1	8	yes	—
247	C₄²	—	4	8	no	—
248	C₄²	—	4	4	no	—
249	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
250	C₄²	C₂×Q₈×C₁₉	1	2	yes	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
256	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
257	D₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
276	C₂×C₄	C₁₉×#135	1	4	yes	—
277	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
280	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₈	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
290	C₂²	C₁₉×#546	1	2	yes	—
291	C₂²	C₂×C₁₉×#135	1	2	yes	—
292	C₂²	C₁₉×#576	1	1	yes	characteristic
293	C₂²	C₁₉×#578	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	—	8	2	no	—
299	C₄	C₁₉×#427	1	4	yes	—
300	C₄	C₄×#41×C₁₉	1	1	yes	characteristic
301	C₄	C₁₉×#518	1	1	yes	characteristic
302	C₄	C₁₉×#554	1	2	yes	—
303	C₂	—	4	4	no	—
304	C₂	C₁₉×#2349	1	1	yes	characteristic
305	C₂	C₂×C₁₉×#427	1	1	yes	characteristic
306	C₂	C₁₉×#2913	1	1	yes	characteristic
307	C₂	C₁₉×#3302	1	2	yes	—
308	C₂	C₁₉×#3305	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1947	C₂	semidirect
4. C₁₉×#1854	C₂	semidirect
5. C₁₉×#1834	C₂	semidirect
6. C₁₉×#1808	C₂	semidirect
7. C₄×C₁₉×#106	C₂	semidirect
8. C₁₉×#1735	C₂	semidirect
12. C₈×D₄×C₁₉	C₄	semidirect
96. #22707	C₁₉	direct (N × H)
132. #1947	C₂×C₁₉	semidirect
133. #1854	C₂×C₁₉	semidirect
134. #1834	C₂×C₁₉	semidirect
135. #1808	C₂×C₁₉	semidirect
136. C₄×#106	C₂×C₁₉	semidirect
137. #1735	C₂×C₁₉	semidirect
160. C₈×D₄	C₄×C₁₉	semidirect
213. C₁₉	#22707	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×C₈×D₄×C₁₉	2
C₁₉×#2354	2
C₁₉×#1947	2
C₁₉×#1854	2
C₁₉×#1834	2
C₁₉×#1808	2
C₄×C₁₉×#106	2
C₁₉×#1735	2
#22707	19

Subgroups — C19×#22707 #18028666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#22707 #18028666