Subgroups — C₁₉×#22542 #18028501

Order 4864.

1086 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#557	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1981	C₂	1	3	yes	maximal
5	C₁₉×#1956	C₂	1	3	yes	maximal
6	C₁₉×#1803	C₂	1	3	yes	maximal
7	C₄×C₁₉×#103	C₂	1	3	yes	maximal
8	C₁₉×#557	—	2	4	no	—
9	C₁₉×#557	C₄	1	8	yes	complemented
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₂×C₁₉×#125	—	2	3	no	—
12	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
14	C₂×C₄×Q₈×C₁₉	C₂²	1	3	yes	—
15	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₁₉	—	2	4	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₁₉×#103	C₂²	1	3	yes	—
19	C₂×C₁₉×#103	C₂²	1	3	yes	—
20	C₂×C₁₉×#103	C₂²	1	6	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
22	C₁₉×#389	—	2	3	no	—
23	C₁₉×#384	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
25	C₁₉×#340	C₂²	1	1	yes	characteristic
26	C₁₉×#340	C₂²	1	3	yes	—
27	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×Q₈×C₁₉	C₂³	1	3	yes	—
29	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
30	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₁₉×C₄⋊Q₈	—	2	6	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	12	yes	—
33	C₁₉×#125	—	2	24	no	—
34	C₄×Q₈×C₁₉	—	2	8	no	—
35	C₄×Q₈×C₁₉	—	2	6	no	—
36	C₄×Q₈×C₁₉	—	4	4	no	—
37	C₄×Q₈×C₁₉	C₂×C₄	1	12	yes	—
38	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₁₉	D₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
45	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	—	2	3	no	—
50	C₂×C₄²×C₁₉	—	2	3	no	—
51	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
52	C₁₉×C₄⋊C₈	—	2	6	no	—
53	C₁₉×#103	—	2	6	no	—
54	C₁₉×#103	—	2	6	no	—
55	C₁₉×#103	—	4	6	no	—
56	C₄×C₈×C₁₉	—	2	6	no	—
57	C₁₉×#95	—	2	3	no	—
58	C₂×Q₈×C₁₉	—	2	3	no	—
59	C₂×Q₈×C₁₉	—	2	3	no	—
60	C₂×Q₈×C₁₉	—	2	3	no	—
61	C₂×Q₈×C₁₉	—	2	3	no	—
62	C₂×Q₈×C₁₉	—	2	3	no	—
63	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
64	C₂×Q₈×C₁₉	SD₁₆	1	4	yes	—
65	C₂×Q₈×C₁₉	Q₁₆	1	4	yes	—
66	C₂×Q₈×C₁₉	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	—	2	3	no	—
70	C₂²×C₄×C₁₉	—	2	3	no	—
71	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	—	2	3	no	—
77	C₂×C₈×C₁₉	—	2	3	no	—
78	C₂×C₈×C₁₉	—	2	3	no	—
79	C₂×C₈×C₁₉	#41	1	2	yes	—
80	C₄⋊C₄×C₁₉	—	2	12	no	—
81	C₄⋊C₄×C₁₉	—	2	12	no	—
82	C₄⋊C₄×C₁₉	—	2	12	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	3	no	—
86	C₄⋊C₄×C₁₉	—	2	3	no	—
87	C₄⋊C₄×C₁₉	—	4	6	no	—
88	C₄⋊C₄×C₁₉	—	8	3	no	—
89	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	12	yes	—
90	C₄⋊C₄×C₁₉	C₂²×C₄	1	2	yes	—
91	C₄⋊C₄×C₁₉	#41	1	2	yes	—
92	C₄²×C₁₉	—	2	12	no	—
93	C₄²×C₁₉	—	4	6	no	—
94	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
95	C₄²×C₁₉	C₂×D₄	1	6	yes	—
96	#22542	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₉	—	2	6	no	—
99	Q₈×C₁₉	—	2	6	no	—
100	Q₈×C₁₉	—	4	3	no	—
101	Q₈×C₁₉	—	4	3	no	—
102	Q₈×C₁₉	#103	1	16	yes	complemented
103	C₂×C₄×C₁₉	—	2	3	no	—
104	C₂×C₄×C₁₉	—	2	3	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	6	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	4	3	no	—
109	C₂×C₄×C₁₉	—	4	3	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	3	no	—
115	C₂×C₄×C₁₉	—	4	3	no	—
116	C₂×C₄×C₁₉	—	4	3	no	—
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
119	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
120	C₂×C₄×C₁₉	C₂×Q₁₆	1	6	yes	—
121	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
127	C₈×C₁₉	—	2	2	no	—
128	C₈×C₁₉	—	4	3	no	—
129	C₂×#557	C₂×C₁₉	1	1	yes	characteristic
130	C₂×C₈×Q₈	C₂×C₁₉	1	1	yes	characteristic
131	Q₈×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
132	#1981	C₂×C₁₉	1	3	yes	—
133	#1956	C₂×C₁₉	1	3	yes	—
134	#1803	C₂×C₁₉	1	3	yes	—
135	C₄×#103	C₂×C₁₉	1	3	yes	—
136	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₉	#548	1	1	yes	characteristic
138	C₂²×C₁₉	#550	1	1	yes	characteristic
139	C₂²×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
140	C₂²×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
141	C₂²×C₁₉	#577	1	1	yes	characteristic
142	C₂²×C₁₉	#578	1	1	yes	characteristic
143	C₄×C₁₉	—	2	6	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	—	4	6	no	—
146	C₄×C₁₉	—	4	2	no	—
147	C₄×C₁₉	—	8	3	no	—
148	C₄×C₁₉	C₂×#103	1	12	yes	—
149	C₄×C₁₉	C₄×#41	1	1	yes	characteristic
150	C₄×C₁₉	#520	1	1	yes	characteristic
151	C₄×C₁₉	#548	1	1	yes	characteristic
152	C₄×C₁₉	#550	1	1	yes	characteristic
153	#557	—	2	4	no	—
154	#557	C₄×C₁₉	1	8	yes	complemented
155	C₂×C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
156	C₂×#125	—	2	3	no	—
157	C₂×C₄×Q₈	—	2	1	no	—
158	C₂×C₄×Q₈	—	2	1	no	—
159	C₂×C₄×Q₈	C₂²×C₁₉	1	3	yes	—
160	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
161	C₈×Q₈	—	2	4	no	—
162	C₂×C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
163	C₂×#103	C₂²×C₁₉	1	3	yes	—
164	C₂×#103	C₂²×C₁₉	1	3	yes	—
165	C₂×#103	C₂²×C₁₉	1	6	yes	—
166	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
167	#389	—	2	3	no	—
168	#384	C₂²×C₁₉	1	3	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
170	#340	C₂²×C₁₉	1	3	yes	—
171	#340	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×C₁₉	#2341	1	1	yes	characteristic
173	C₂×C₁₉	C₂²×#103	1	1	yes	characteristic
174	C₂×C₁₉	#2898	1	1	yes	characteristic
175	C₂×C₁₉	#3359	1	1	yes	characteristic
176	C₂×C₁₉	#3363	1	1	yes	characteristic
177	C₂×C₁₉	#3392	1	1	yes	characteristic
178	C₂×C₁₉	#3395	1	1	yes	characteristic
179	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
180	C₂²×Q₈	C₂³×C₁₉	1	3	yes	—
181	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
182	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	6	no	—
184	C₄⋊Q₈	C₂×C₄×C₁₉	1	12	yes	—
185	#125	—	2	24	no	—
186	C₄×Q₈	—	2	8	no	—
187	C₄×Q₈	—	2	6	no	—
188	C₄×Q₈	—	4	4	no	—
189	C₄×Q₈	C₂×C₄×C₁₉	1	12	yes	—
190	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
191	C₄×Q₈	D₄×C₁₉	1	4	yes	—
192	C₂×C₄⋊C₄	—	2	3	no	—
193	C₂×C₄⋊C₄	—	2	3	no	—
194	C₂×C₄⋊C₄	—	2	3	no	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
201	C₂×C₄²	—	2	3	no	—
202	C₂×C₄²	—	2	3	no	—
203	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
204	C₄⋊C₈	—	2	6	no	—
205	#103	—	2	6	no	—
206	#103	—	2	6	no	—
207	#103	—	4	6	no	—
208	C₄×C₈	—	2	6	no	—
209	#95	—	2	3	no	—
210	C₁₉	#22542	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×Q₈	—	2	3	no	—
212	C₂×Q₈	—	2	3	no	—
213	C₂×Q₈	—	2	3	no	—
214	C₂×Q₈	—	2	3	no	—
215	C₂×Q₈	—	2	3	no	—
216	C₂×Q₈	C₂²⋊C₄×C₁₉	1	4	yes	—
217	C₂×Q₈	SD₁₆×C₁₉	1	4	yes	—
218	C₂×Q₈	Q₁₆×C₁₉	1	4	yes	—
219	C₂×Q₈	C₂²×C₄×C₁₉	1	6	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	—	2	3	no	—
224	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
225	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
226	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	3	no	—
230	C₂×C₈	—	2	3	no	—
231	C₂×C₈	—	2	3	no	—
232	C₂×C₈	#41×C₁₉	1	2	yes	—
233	C₄⋊C₄	—	2	12	no	—
234	C₄⋊C₄	—	2	12	no	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	2	3	no	—
240	C₄⋊C₄	—	4	6	no	—
241	C₄⋊C₄	—	8	3	no	—
242	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	12	yes	—
243	C₄⋊C₄	C₂²×C₄×C₁₉	1	2	yes	—
244	C₄⋊C₄	#41×C₁₉	1	2	yes	—
245	C₄²	—	2	12	no	—
246	C₄²	—	4	6	no	—
247	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
248	C₄²	C₂×D₄×C₁₉	1	6	yes	—
249	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
250	Q₈	—	2	6	no	—
251	Q₈	—	2	6	no	—
252	Q₈	—	4	3	no	—
253	Q₈	—	4	3	no	—
254	Q₈	C₁₉×#103	1	16	yes	complemented
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	3	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	4	3	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
271	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
272	C₂×C₄	C₂×Q₁₆×C₁₉	1	6	yes	—
273	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
277	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	4	3	no	—
281	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
282	C₂²	C₁₉×#548	1	1	yes	characteristic
283	C₂²	C₁₉×#550	1	1	yes	characteristic
284	C₂²	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
285	C₂²	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
286	C₂²	C₁₉×#577	1	1	yes	characteristic
287	C₂²	C₁₉×#578	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	6	no	—
291	C₄	—	4	2	no	—
292	C₄	—	8	3	no	—
293	C₄	C₂×C₁₉×#103	1	12	yes	—
294	C₄	C₄×#41×C₁₉	1	1	yes	characteristic
295	C₄	C₁₉×#520	1	1	yes	characteristic
296	C₄	C₁₉×#548	1	1	yes	characteristic
297	C₄	C₁₉×#550	1	1	yes	characteristic
298	C₂	C₁₉×#2341	1	1	yes	characteristic
299	C₂	C₂²×C₁₉×#103	1	1	yes	characteristic
300	C₂	C₁₉×#2898	1	1	yes	characteristic
301	C₂	C₁₉×#3359	1	1	yes	characteristic
302	C₂	C₁₉×#3363	1	1	yes	characteristic
303	C₂	C₁₉×#3392	1	1	yes	characteristic
304	C₂	C₁₉×#3395	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₁₉×#557	C₄	semidirect
96. #22542	C₁₉	direct (N × H)
102. Q₈×C₁₉	#103	semidirect
154. #557	C₄×C₁₉	semidirect
210. C₁₉	#22542	direct (N × H)
254. Q₈	C₁₉×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#557	2
C₂×C₈×Q₈×C₁₉	2
Q₈×C₄⋊C₄×C₁₉	2
C₁₉×#1981	2
C₁₉×#1956	2
C₁₉×#1803	2
C₄×C₁₉×#103	2
#22542	19

Subgroups — C19×#22542 #18028501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#22542 #18028501