Subgroups — C₂₃×#17720 #19165346

Order 5888.

1308 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3498	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1547	C₂	1	3	yes	maximal, complemented
5	C₂₃×#1545	C₂	1	3	yes	maximal, complemented
6	C₂₃×#1540	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#331	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
9	C₂×C₂₃×#131	—	2	1	no	—
10	C₂₃×#550	—	2	1	no	—
11	C₂₃×#550	C₄	1	2	yes	—
12	D₄²×C₂₃	—	2	3	no	—
13	D₄²×C₂₃	C₄	1	6	yes	—
14	C₂₃×#535	—	2	3	no	—
15	C₂₃×#532	C₂²	1	3	yes	—
16	C₂×C₂₃×#127	C₂²	1	3	yes	—
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₈×Q₈×C₂₃	—	2	1	no	—
19	C₈×D₄×C₂₃	—	2	3	no	—
20	C₂₃×#431	C₂²	1	3	yes	complemented
21	C₂₃×#424	—	2	3	no	—
22	C₂₃×#423	C₂²	1	3	yes	complemented
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	complemented
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	complemented
25	C₂₃×#332	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×#332	C₂²	1	3	yes	complemented
27	C₂₃×#331	C₂²	1	3	yes	complemented
28	C₂₃×#331	C₂²	1	6	yes	complemented
29	C₂₃×#331	C₂²	1	3	yes	complemented
30	C₂₃×H(5,𝔽₂)	—	4	4	no	—
31	C₂₃×H(5,𝔽₂)	—	4	4	no	—
32	C₂×#41×C₂₃	—	2	2	no	—
33	C₂×#41×C₂₃	—	2	3	no	—
34	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₂₃	—	2	6	no	—
36	C₂²×D₄×C₂₃	—	2	3	no	—
37	C₂²×D₄×C₂₃	—	2	3	no	—
38	C₂²×D₄×C₂₃	C₂×C₄	1	6	yes	—
39	C₂₃×#131	—	4	4	no	—
40	C₂×M₁₆×C₂₃	—	2	3	no	—
41	C₂²×C₈×C₂₃	—	2	3	no	—
42	C₂²×C₈×C₂₃	—	2	3	no	—
43	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
44	C₂₃×#127	—	2	3	no	—
45	C₂₃×#127	—	2	3	no	—
46	C₂₃×#127	C₂×C₄	1	6	yes	—
47	C₂₃×#127	C₂³	1	3	yes	—
48	C₂₃×#124	—	2	3	no	—
49	C₂₃×#121	—	2	3	no	—
50	C₂₃×#121	—	2	6	no	—
51	C₂₃×#121	—	2	6	no	—
52	C₂₃×#121	C₂×C₄	1	6	yes	—
53	C₂₃×#120	—	2	12	no	—
54	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
56	C₄×D₄×C₂₃	—	2	3	no	—
57	C₄×D₄×C₂₃	C₂×C₄	1	6	yes	—
58	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
59	C₄×D₄×C₂₃	D₄	1	3	yes	—
60	C₂₃×#117	D₄	1	3	yes	—
61	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
62	C₂₃×C₄⋊C₈	—	2	3	no	—
63	C₂₃×C₄⋊C₈	—	2	3	no	—
64	C₂₃×C₄⋊C₈	—	2	3	no	—
65	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
66	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
67	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
68	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₂₃×C₂²⋊C₈	—	2	3	no	—
70	C₄×C₈×C₂₃	—	2	3	no	—
71	C₂⁴×C₂₃	—	2	6	no	—
72	#41×C₂₃	—	4	8	no	—
73	#41×C₂₃	—	4	6	no	—
74	#41×C₂₃	—	4	6	no	—
75	#41×C₂₃	D₈	1	4	yes	—
76	#41×C₂₃	SD₁₆	1	4	yes	—
77	C₂×Q₈×C₂₃	—	2	1	no	—
78	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₂₃	—	2	6	no	—
80	C₂×D₄×C₂₃	—	2	6	no	—
81	C₂×D₄×C₂₃	—	2	6	no	—
82	C₂×D₄×C₂₃	—	2	6	no	—
83	C₂×D₄×C₂₃	—	2	3	no	—
84	C₂×D₄×C₂₃	—	2	3	no	—
85	C₂×D₄×C₂₃	—	4	24	no	—
86	C₂×D₄×C₂₃	—	4	6	no	—
87	C₂×D₄×C₂₃	—	4	6	no	—
88	C₂×D₄×C₂₃	—	8	3	no	—
89	C₂×D₄×C₂₃	—	8	3	no	—
90	C₂×D₄×C₂₃	—	8	6	no	—
91	C₂×D₄×C₂₃	C₂²⋊C₄	1	3	yes	—
92	C₂×D₄×C₂₃	C₂²×C₄	1	6	yes	—
93	C₂²×C₄×C₂₃	—	2	6	no	—
94	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
96	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
97	M₁₆×C₂₃	—	4	6	no	—
98	C₂×C₈×C₂₃	—	2	1	no	—
99	C₂×C₈×C₂₃	—	2	3	no	—
100	C₂×C₈×C₂₃	—	2	3	no	—
101	C₂×C₈×C₂₃	—	2	1	no	—
102	C₂×C₈×C₂₃	—	4	6	no	—
103	C₂×C₈×C₂₃	—	8	3	no	—
104	C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
106	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
107	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
108	C₂²⋊C₄×C₂₃	—	2	6	no	—
109	C₂²⋊C₄×C₂₃	—	2	6	no	—
110	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
111	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
112	C₄²×C₂₃	C₂×D₄	1	3	yes	—
113	C₄²×C₂₃	C₂×D₄	1	3	yes	—
114	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
115	#17720	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₃	—	2	6	no	—
117	C₂³×C₂₃	—	4	12	no	—
118	C₂³×C₂₃	—	8	6	no	—
119	C₂³×C₂₃	—	8	6	no	—
120	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
121	Q₈×C₂₃	—	4	1	no	—
122	Q₈×C₂₃	—	4	1	no	—
123	Q₈×C₂₃	#102	1	4	yes	—
124	D₄×C₂₃	—	4	6	no	—
125	D₄×C₂₃	—	4	12	no	—
126	D₄×C₂₃	—	4	3	no	—
127	D₄×C₂₃	—	4	3	no	—
128	D₄×C₂₃	—	4	3	no	—
129	D₄×C₂₃	—	4	3	no	—
130	D₄×C₂₃	—	4	12	no	—
131	D₄×C₂₃	—	8	6	no	—
132	D₄×C₂₃	—	8	6	no	—
133	D₄×C₂₃	#102	1	12	yes	—
134	C₂×C₄×C₂₃	—	2	3	no	—
135	C₂×C₄×C₂₃	—	2	3	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	—	4	3	no	—
138	C₂×C₄×C₂₃	—	4	12	no	—
139	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
140	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
141	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
142	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
143	C₂×C₄×C₂₃	C₂×SD₁₆	1	6	yes	—
144	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₂₃	—	4	1	no	—
148	C₈×C₂₃	—	4	3	no	—
149	C₈×C₂₃	—	4	3	no	—
150	C₈×C₂₃	—	4	1	no	—
151	#3498	C₂×C₂₃	1	1	yes	characteristic
152	#2988	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#1547	C₂×C₂₃	1	3	yes	complemented
155	#1545	C₂×C₂₃	1	3	yes	complemented
156	#1540	C₂×C₂₃	1	3	yes	complemented
157	C₂×#331	C₂×C₂₃	1	3	yes	complemented
158	C₂²×C₂₃	—	2	3	no	—
159	C₂²×C₂₃	—	2	3	no	—
160	C₂²×C₂₃	—	4	6	no	—
161	C₂²×C₂₃	—	4	6	no	—
162	C₂²×C₂₃	—	4	6	no	—
163	C₂²×C₂₃	—	16	12	no	—
164	C₂²×C₂₃	C₂×#102	1	6	yes	—
165	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	3	no	—
168	C₄×C₂₃	—	2	3	no	—
169	C₄×C₂₃	—	4	2	no	—
170	C₄×C₂₃	C₂×#102	1	6	yes	—
171	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
173	C₄×C₂₃	#576	1	1	yes	characteristic
174	C₄×C₂₃	#577	1	1	yes	characteristic
175	C₂×H(5,𝔽₂)	—	2	1	no	—
176	C₂×#131	—	2	1	no	—
177	#550	—	2	1	no	—
178	#550	C₄×C₂₃	1	2	yes	—
179	D₄²	—	2	3	no	—
180	D₄²	C₄×C₂₃	1	6	yes	—
181	#535	—	2	3	no	—
182	#532	C₂²×C₂₃	1	3	yes	—
183	C₂×#127	C₂²×C₂₃	1	3	yes	—
184	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
185	C₈×Q₈	—	2	1	no	—
186	C₈×D₄	—	2	3	no	—
187	#431	C₂²×C₂₃	1	3	yes	complemented
188	#424	—	2	3	no	—
189	#423	C₂²×C₂₃	1	3	yes	complemented
190	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	complemented
191	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	complemented
192	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
193	#332	C₂²×C₂₃	1	3	yes	complemented
194	#331	C₂²×C₂₃	1	3	yes	complemented
195	#331	C₂²×C₂₃	1	6	yes	complemented
196	#331	C₂²×C₂₃	1	3	yes	complemented
197	C₂×C₂₃	—	2	6	no	—
198	C₂×C₂₃	—	8	6	no	—
199	C₂×C₂₃	#2888	1	1	yes	characteristic
200	C₂×C₂₃	C₂²×#102	1	1	yes	characteristic
201	C₂×C₂₃	#2897	1	1	yes	characteristic
202	H(5,𝔽₂)	—	4	4	no	—
203	H(5,𝔽₂)	—	4	4	no	—
204	C₂×#41	—	2	2	no	—
205	C₂×#41	—	2	3	no	—
206	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
207	C₂²×D₄	—	2	6	no	—
208	C₂²×D₄	—	2	3	no	—
209	C₂²×D₄	—	2	3	no	—
210	C₂²×D₄	C₂×C₄×C₂₃	1	6	yes	—
211	#131	—	4	4	no	—
212	C₂×M₁₆	—	2	3	no	—
213	C₂²×C₈	—	2	3	no	—
214	C₂²×C₈	—	2	3	no	—
215	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
216	#127	—	2	3	no	—
217	#127	—	2	3	no	—
218	#127	C₂×C₄×C₂₃	1	6	yes	—
219	#127	C₂³×C₂₃	1	3	yes	—
220	#124	—	2	3	no	—
221	#121	—	2	3	no	—
222	#121	—	2	6	no	—
223	#121	—	2	6	no	—
224	#121	C₂×C₄×C₂₃	1	6	yes	—
225	#120	—	2	12	no	—
226	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
227	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
228	C₄×D₄	—	2	3	no	—
229	C₄×D₄	C₂×C₄×C₂₃	1	6	yes	—
230	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
231	C₄×D₄	D₄×C₂₃	1	3	yes	—
232	#117	D₄×C₂₃	1	3	yes	—
233	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
234	C₄⋊C₈	—	2	3	no	—
235	C₄⋊C₈	—	2	3	no	—
236	C₄⋊C₈	—	2	3	no	—
237	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
239	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
240	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
241	C₂²⋊C₈	—	2	3	no	—
242	C₄×C₈	—	2	3	no	—
243	C₂₃	#17720	1	1	yes	characteristic, direct product factor, Sylow
244	C₂⁴	—	2	6	no	—
245	#41	—	4	8	no	—
246	#41	—	4	6	no	—
247	#41	—	4	6	no	—
248	#41	D₈×C₂₃	1	4	yes	—
249	#41	SD₁₆×C₂₃	1	4	yes	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×D₄	—	2	6	no	—
253	C₂×D₄	—	2	6	no	—
254	C₂×D₄	—	2	6	no	—
255	C₂×D₄	—	2	6	no	—
256	C₂×D₄	—	2	3	no	—
257	C₂×D₄	—	2	3	no	—
258	C₂×D₄	—	4	24	no	—
259	C₂×D₄	—	4	6	no	—
260	C₂×D₄	—	4	6	no	—
261	C₂×D₄	—	8	3	no	—
262	C₂×D₄	—	8	3	no	—
263	C₂×D₄	—	8	6	no	—
264	C₂×D₄	C₂²⋊C₄×C₂₃	1	3	yes	—
265	C₂×D₄	C₂²×C₄×C₂₃	1	6	yes	—
266	C₂²×C₄	—	2	6	no	—
267	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
268	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
269	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
270	M₁₆	—	4	6	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	3	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	3	no	—
275	C₂×C₈	—	4	6	no	—
276	C₂×C₈	—	8	3	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
279	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
280	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
281	C₂²⋊C₄	—	2	6	no	—
282	C₂²⋊C₄	—	2	6	no	—
283	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
284	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
285	C₄²	C₂×D₄×C₂₃	1	3	yes	—
286	C₄²	C₂×D₄×C₂₃	1	3	yes	—
287	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
288	C₂³	—	2	6	no	—
289	C₂³	—	4	12	no	—
290	C₂³	—	8	6	no	—
291	C₂³	—	8	6	no	—
292	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
293	Q₈	—	4	1	no	—
294	Q₈	—	4	1	no	—
295	Q₈	C₂₃×#102	1	4	yes	—
296	D₄	—	4	6	no	—
297	D₄	—	4	12	no	—
298	D₄	—	4	3	no	—
299	D₄	—	4	3	no	—
300	D₄	—	4	3	no	—
301	D₄	—	4	3	no	—
302	D₄	—	4	12	no	—
303	D₄	—	8	6	no	—
304	D₄	—	8	6	no	—
305	D₄	C₂₃×#102	1	12	yes	—
306	C₂×C₄	—	2	3	no	—
307	C₂×C₄	—	2	3	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	3	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
312	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
314	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
315	C₂×C₄	C₂×SD₁₆×C₂₃	1	6	yes	—
316	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
319	C₈	—	4	1	no	—
320	C₈	—	4	3	no	—
321	C₈	—	4	3	no	—
322	C₈	—	4	1	no	—
323	C₂²	—	2	3	no	—
324	C₂²	—	2	3	no	—
325	C₂²	—	4	6	no	—
326	C₂²	—	4	6	no	—
327	C₂²	—	4	6	no	—
328	C₂²	—	16	12	no	—
329	C₂²	C₂×C₂₃×#102	1	6	yes	—
330	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	3	no	—
333	C₄	—	2	3	no	—
334	C₄	—	4	2	no	—
335	C₄	C₂×C₂₃×#102	1	6	yes	—
336	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
337	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
338	C₄	C₂₃×#576	1	1	yes	characteristic
339	C₄	C₂₃×#577	1	1	yes	characteristic
340	C₂	—	2	6	no	—
341	C₂	—	8	6	no	—
342	C₂	C₂₃×#2888	1	1	yes	characteristic
343	C₂	C₂²×C₂₃×#102	1	1	yes	characteristic
344	C₂	C₂₃×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2988	C₂	semidirect
3. C₂₃×#2906	C₂	semidirect
4. C₂₃×#1547	C₂	semidirect
5. C₂₃×#1545	C₂	semidirect
6. C₂₃×#1540	C₂	semidirect
7. C₂×C₂₃×#331	C₂	semidirect
20. C₂₃×#431	C₂²	semidirect
22. C₂₃×#423	C₂²	semidirect
23. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
24. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
25. C₂₃×#332	C₂²	semidirect
26. C₂₃×#332	C₂²	semidirect
27. C₂₃×#331	C₂²	semidirect
28. C₂₃×#331	C₂²	semidirect
29. C₂₃×#331	C₂²	semidirect
115. #17720	C₂₃	direct (N × H)
152. #2988	C₂×C₂₃	semidirect
153. #2906	C₂×C₂₃	semidirect
154. #1547	C₂×C₂₃	semidirect
155. #1545	C₂×C₂₃	semidirect
156. #1540	C₂×C₂₃	semidirect
157. C₂×#331	C₂×C₂₃	semidirect
187. #431	C₂²×C₂₃	semidirect
189. #423	C₂²×C₂₃	semidirect
190. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
191. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
192. #332	C₂²×C₂₃	semidirect
193. #332	C₂²×C₂₃	semidirect
194. #331	C₂²×C₂₃	semidirect
195. #331	C₂²×C₂₃	semidirect
196. #331	C₂²×C₂₃	semidirect
243. C₂₃	#17720	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3498	2
C₂₃×#2988	2
C₂₃×#2906	2
C₂₃×#1547	2
C₂₃×#1545	2
C₂₃×#1540	2
C₂×C₂₃×#331	2
#17720	23

Subgroups — C23×#17720 #19165346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17720 #19165346