Subgroups — C₁₉×#17934 #18023893

Order 4864.

582 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3104	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2960	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#446	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1593	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1587	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1570	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1487	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#1481	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#1452	C₂	1	1	yes	maximal, characteristic
10	C₁₉×#533	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#484	C₄	1	2	yes	—
13	C₁₉×#477	C₄	1	2	yes	—
14	C₁₉×#462	C₄	1	2	yes	—
15	C₁₉×#451	C₄	1	2	yes	—
16	C₁₉×#446	—	2	1	no	—
17	C₁₉×#446	—	2	1	no	—
18	C₁₉×#446	—	2	1	no	—
19	C₁₉×#446	C₂²	1	2	yes	—
20	C₁₉×#446	C₂²	1	2	yes	—
21	C₁₉×#446	C₂²	1	2	yes	—
22	C₈×Q₈×C₁₉	—	2	1	no	—
23	C₈×Q₈×C₁₉	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₁₉	C₂²	1	4	yes	—
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₁₉×#431	C₂²	1	1	yes	characteristic
27	C₁₉×#424	—	2	1	no	—
28	C₁₉×#423	C₂²	1	1	yes	characteristic
29	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
30	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₁₉×#405	—	2	1	no	—
32	C₂×C₁₉×#97	—	2	1	no	—
33	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
34	C₁₉×#334	C₂²	1	4	yes	—
35	C₁₉×Q₈⋊C₈	C₂²	1	4	yes	—
36	C₁₉×Q₈⋊C₈	C₂²	1	4	yes	—
37	C₁₉×#322	C₂²	1	2	yes	—
38	C₁₉×#322	C₂²	1	2	yes	—
39	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×Q₁₆×C₁₉	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₁₉	—	2	1	no	—
42	C₂²×C₈×C₁₉	—	2	1	no	—
43	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
47	C₁₉×#125	—	2	1	no	—
48	C₁₉×C₂²⋊Q₈	—	2	1	no	—
49	C₁₉×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₁₉	—	2	1	no	—
51	C₄×Q₈×C₁₉	—	2	1	no	—
52	C₄×Q₈×C₁₉	—	2	1	no	—
53	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
54	C₄×Q₈×C₁₉	C₂³	1	2	yes	—
55	C₄×Q₈×C₁₉	C₂³	1	2	yes	—
56	C₁₉×#117	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	—	2	1	no	—
60	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₁₉×#107	C₂×C₄	1	4	yes	—
62	C₁₉×C₄⋊C₈	—	2	2	no	—
63	C₁₉×C₄⋊C₈	—	2	2	no	—
64	C₁₉×C₄⋊C₈	—	2	2	no	—
65	C₁₉×C₄⋊C₈	—	2	2	no	—
66	C₁₉×C₄⋊C₈	—	2	1	no	—
67	C₁₉×C₄⋊C₈	—	2	1	no	—
68	C₁₉×C₄⋊C₈	—	2	1	no	—
69	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
70	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
71	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
72	C₁₉×#103	C₂×C₄	1	4	yes	—
73	C₁₉×#103	C₂×C₄	1	4	yes	—
74	C₁₉×C₂²⋊C₈	—	2	1	no	—
75	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
76	C₁₉×#97	—	2	2	no	—
77	C₁₉×#97	—	2	2	no	—
78	C₁₉×#97	—	2	2	no	—
79	C₄×C₈×C₁₉	—	2	2	no	—
80	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
81	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
82	C₄×C₈×C₁₉	C₂³	1	2	yes	—
83	C₄×C₈×C₁₉	C₂³	1	2	yes	—
84	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₁₉	—	2	1	no	—
87	C₂×Q₈×C₁₉	—	2	1	no	—
88	C₂×Q₈×C₁₉	—	4	2	no	—
89	C₂×Q₈×C₁₉	C₂²×C₄	1	2	yes	—
90	C₂×Q₈×C₁₉	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₉	—	2	1	no	—
92	C₂²×C₄×C₁₉	—	2	1	no	—
93	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
96	Q₁₆×C₁₉	—	4	4	no	—
97	M₁₆×C₁₉	—	4	2	no	—
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	—	2	2	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₂×C₈×C₁₉	—	2	2	no	—
103	C₂×C₈×C₁₉	—	4	2	no	—
104	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	2	1	no	—
111	C₄⋊C₄×C₁₉	—	2	1	no	—
112	C₄⋊C₄×C₁₉	—	2	1	no	—
113	C₄⋊C₄×C₁₉	—	2	1	no	—
114	C₄⋊C₄×C₁₉	—	2	2	no	—
115	C₄⋊C₄×C₁₉	—	2	2	no	—
116	C₄⋊C₄×C₁₉	—	2	2	no	—
117	C₄⋊C₄×C₁₉	C₂²×C₄	1	2	yes	—
118	C₄⋊C₄×C₁₉	C₂²×C₄	1	2	yes	—
119	C₂²⋊C₄×C₁₉	—	2	2	no	—
120	C₄²×C₁₉	—	2	2	no	—
121	C₄²×C₁₉	—	2	2	no	—
122	C₄²×C₁₉	—	2	2	no	—
123	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	#41	1	1	yes	characteristic
126	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
127	#17934	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
129	Q₈×C₁₉	—	2	2	no	—
130	Q₈×C₁₉	—	2	4	no	—
131	Q₈×C₁₉	—	2	2	no	—
132	Q₈×C₁₉	—	4	2	no	—
133	Q₈×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#137	1	2	yes	—
146	C₂×C₄×C₁₉	#137	1	2	yes	—
147	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
150	C₈×C₁₉	—	2	4	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₈×C₁₉	—	2	1	no	—
153	C₈×C₁₉	—	2	1	no	—
154	C₈×C₁₉	—	2	1	no	—
155	C₈×C₁₉	—	4	2	no	—
156	C₈×C₁₉	—	4	2	no	—
157	C₈×C₁₉	—	4	2	no	—
158	#3104	C₂×C₁₉	1	1	yes	characteristic
159	#2960	C₂×C₁₉	1	1	yes	characteristic
160	C₂×#446	C₂×C₁₉	1	1	yes	characteristic
161	#1593	C₂×C₁₉	1	4	yes	complemented
162	#1587	C₂×C₁₉	1	4	yes	complemented
163	#1570	C₂×C₁₉	1	1	yes	characteristic
164	#1487	C₂×C₁₉	1	1	yes	characteristic
165	#1481	C₂×C₁₉	1	1	yes	characteristic
166	#1452	C₂×C₁₉	1	1	yes	characteristic
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	#441	1	2	yes	—
170	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	4	2	no	—
175	C₄×C₁₉	—	4	2	no	—
176	C₄×C₁₉	—	4	2	no	—
177	C₄×C₁₉	—	4	2	no	—
178	C₄×C₁₉	#441	1	2	yes	—
179	C₄×C₁₉	#568	1	2	yes	—
180	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
181	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
182	#533	C₂²×C₁₉	1	1	yes	characteristic
183	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
184	#484	C₄×C₁₉	1	2	yes	—
185	#477	C₄×C₁₉	1	2	yes	—
186	#462	C₄×C₁₉	1	2	yes	—
187	#451	C₄×C₁₉	1	2	yes	—
188	#446	—	2	1	no	—
189	#446	—	2	1	no	—
190	#446	—	2	1	no	—
191	#446	C₂²×C₁₉	1	2	yes	—
192	#446	C₂²×C₁₉	1	2	yes	—
193	#446	C₂²×C₁₉	1	2	yes	—
194	C₈×Q₈	—	2	1	no	—
195	C₈×Q₈	C₂²×C₁₉	1	2	yes	—
196	C₄×Q₁₆	C₂²×C₁₉	1	4	yes	—
197	#431	C₂²×C₁₉	1	1	yes	characteristic
198	#431	C₂²×C₁₉	1	1	yes	characteristic
199	#424	—	2	1	no	—
200	#423	C₂²×C₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	—	2	1	no	—
202	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
203	#405	—	2	1	no	—
204	C₂×#97	—	2	1	no	—
205	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
206	#334	C₂²×C₁₉	1	4	yes	—
207	Q₈⋊C₈	C₂²×C₁₉	1	4	yes	—
208	Q₈⋊C₈	C₂²×C₁₉	1	4	yes	—
209	#322	C₂²×C₁₉	1	2	yes	—
210	#322	C₂²×C₁₉	1	2	yes	—
211	C₂×C₁₉	—	2	2	no	—
212	C₂×C₁₉	#2930	1	1	yes	characteristic
213	C₂×C₁₉	C₂×#441	1	1	yes	characteristic
214	C₂×C₁₉	#2955	1	1	yes	characteristic
215	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
216	C₂×Q₁₆	C₂×C₄×C₁₉	1	4	yes	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
221	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
222	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
223	#125	—	2	1	no	—
224	C₂²⋊Q₈	—	2	1	no	—
225	C₂²⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	C₂×C₄×C₁₉	1	2	yes	—
230	C₄×Q₈	C₂³×C₁₉	1	2	yes	—
231	C₄×Q₈	C₂³×C₁₉	1	2	yes	—
232	#117	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
237	#107	C₂×C₄×C₁₉	1	4	yes	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
246	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
247	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
248	#103	C₂×C₄×C₁₉	1	4	yes	—
249	#103	C₂×C₄×C₁₉	1	4	yes	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
252	#97	—	2	2	no	—
253	#97	—	2	2	no	—
254	#97	—	2	2	no	—
255	C₄×C₈	—	2	2	no	—
256	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
257	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₁₉	1	2	yes	—
259	C₄×C₈	C₂³×C₁₉	1	2	yes	—
260	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
262	C₁₉	#17934	1	1	yes	characteristic, direct product factor, Sylow
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	C₂²×C₄×C₁₉	1	2	yes	—
267	C₂×Q₈	C₂²×C₄×C₁₉	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
272	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
273	Q₁₆	—	4	4	no	—
274	M₁₆	—	4	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
282	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	C₂²×C₄×C₁₉	1	2	yes	—
295	C₄⋊C₄	C₂²×C₄×C₁₉	1	2	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
302	C₄²	#41×C₁₉	1	1	yes	characteristic
303	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
304	C₂³	C₄×D₄×C₁₉	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	4	no	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
319	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
320	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
321	C₂×C₄	C₁₉×#137	1	2	yes	—
322	C₂×C₄	C₁₉×#137	1	2	yes	—
323	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
325	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
326	C₈	—	2	4	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	C₁₉×#441	1	2	yes	—
337	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	C₁₉×#441	1	2	yes	—
346	C₄	C₁₉×#568	1	2	yes	—
347	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
348	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	C₁₉×#2930	1	1	yes	characteristic
351	C₂	C₂×C₁₉×#441	1	1	yes	characteristic
352	C₂	C₁₉×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×C₄×D₄
prime radical	C₂³×C₁₉	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1593	C₂	semidirect
5. C₁₉×#1587	C₂	semidirect
127. #17934	C₁₉	direct (N × H)
161. #1593	C₂×C₁₉	semidirect
162. #1587	C₂×C₁₉	semidirect
262. C₁₉	#17934	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3104	2
C₁₉×#2960	2
C₂×C₁₉×#446	2
C₁₉×#1593	2
C₁₉×#1587	2
C₁₉×#1570	2
C₁₉×#1487	2
C₁₉×#1481	2
C₁₉×#1452	2
#17934	19

Subgroups — C19×#17934 #18023893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17934 #18023893