Subgroups — C₁₃×#23812 #15710151

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#440	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2326	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2095	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2093	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2015	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2012	C₂	1	2	yes	maximal
9	C₁₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×SD₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#124	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#440	—	2	8	no	—
17	C₄×C₁₃×Q₁₆	—	2	4	no	—
18	C₄×C₁₃×D₈	—	2	4	no	—
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#107	C₂²	1	2	yes	—
21	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#103	C₂²	1	2	yes	—
23	C₂×C₁₃×#103	C₂²	1	2	yes	—
24	C₂×C₁₃×#102	C₂²	1	2	yes	—
25	C₂×C₁₃×#102	C₂²	1	2	yes	—
26	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#391	C₂²	1	2	yes	—
29	C₁₃×#390	C₂²	1	2	yes	—
30	C₁₃×#388	—	2	4	no	—
31	C₁₃×#385	—	2	4	no	—
32	C₁₃×#384	C₂²	1	2	yes	—
33	C₁₃×#376	C₂²	1	1	yes	characteristic
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₁₃×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃	—	2	2	no	—
39	C₂×C₁₃×Q₁₆	—	2	4	no	—
40	C₂×C₁₃×Q₁₆	—	2	2	no	—
41	C₂×C₁₃×SD₁₆	—	2	4	no	—
42	C₂×C₁₃×SD₁₆	—	2	4	no	—
43	C₂×C₁₃×SD₁₆	—	2	4	no	—
44	C₂×C₁₃×D₈	—	2	4	no	—
45	C₂×C₁₃×D₈	—	2	2	no	—
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
49	C₁₃×#124	—	2	4	no	—
50	C₁₃×#124	—	2	4	no	—
51	C₄×Q₈×C₁₃	—	2	4	no	—
52	C₄×Q₈×C₁₃	—	2	4	no	—
53	C₄×D₄×C₁₃	—	2	4	no	—
54	C₄×D₄×C₁₃	—	2	4	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
57	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
59	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
60	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₂×C₄²×C₁₃	—	2	2	no	—
63	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
65	C₁₃×#107	—	2	2	no	—
66	C₁₃×#107	—	2	2	no	—
67	C₁₃×#107	—	2	4	no	—
68	C₁₃×#103	—	2	4	no	—
69	C₁₃×#103	—	2	4	no	—
70	C₁₃×#102	—	2	4	no	—
71	C₁₃×#102	—	2	4	no	—
72	C₁₃×#97	D₄	1	4	yes	—
73	C₄×C₈×C₁₃	D₄	1	4	yes	—
74	C₁₃×#95	—	2	4	no	—
75	C₁₃×#95	D₄	1	2	yes	—
76	C₁₃×#95	D₄	1	2	yes	—
77	#23812	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂⁴×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	#41	1	4	yes	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	2	2	no	—
90	C₂×D₄×C₁₃	#41	1	4	yes	—
91	C₂²×C₄×C₁₃	—	2	2	no	—
92	C₂²×C₄×C₁₃	—	2	2	no	—
93	C₂²×C₄×C₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₃	—	4	4	no	—
95	C₂²×C₄×C₁₃	—	8	4	no	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
97	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×Q₁₆	—	2	8	no	—
101	C₁₃×SD₁₆	—	2	16	no	—
102	C₁₃×D₈	—	2	8	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	4	4	no	—
115	C₁₃×C₄⋊C₄	—	8	4	no	—
116	C₁₃×C₂²⋊C₄	—	2	16	no	—
117	C₁₃×C₂²⋊C₄	—	4	4	no	—
118	C₁₃×C₂²⋊C₄	—	8	4	no	—
119	C₄²×C₁₃	—	2	2	no	—
120	C₄²×C₁₃	—	2	2	no	—
121	C₄²×C₁₃	—	4	4	no	—
122	C₄²×C₁₃	C₂×D₄	1	2	yes	—
123	C₄²×C₁₃	C₂×D₄	1	2	yes	—
124	C₂×#440	C₂×C₁₃	1	2	yes	complemented
125	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
127	#2326	C₂×C₁₃	1	2	yes	complemented
128	#2095	C₂×C₁₃	1	2	yes	complemented
129	#2093	C₂×C₁₃	1	2	yes	complemented
130	#2015	C₂×C₁₃	1	2	yes	complemented
131	#2012	C₂×C₁₃	1	2	yes	—
132	#1746	C₂×C₁₃	1	1	yes	characteristic, complemented
133	C₂³×C₁₃	—	2	4	no	—
134	C₂³×C₁₃	—	4	2	no	—
135	C₂³×C₁₃	—	4	2	no	—
136	C₂³×C₁₃	—	4	4	no	—
137	C₂³×C₁₃	—	4	4	no	—
138	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₁₃	—	2	4	no	—
140	Q₈×C₁₃	—	2	4	no	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	2	4	no	—
144	D₄×C₁₃	—	2	4	no	—
145	D₄×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	4	no	—
150	C₂×C₄×C₁₃	—	2	4	no	—
151	C₂×C₄×C₁₃	—	2	4	no	—
152	C₂×C₄×C₁₃	—	2	4	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	4	no	—
158	C₂×C₄×C₁₃	—	4	4	no	—
159	C₂×C₄×C₁₃	—	8	8	no	—
160	C₂×C₄×C₁₃	#127	1	2	yes	—
161	C₂×C₄×C₁₃	#127	1	2	yes	—
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
165	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
166	C₈×C₁₃	—	2	2	no	—
167	C₈×C₁₃	—	2	4	no	—
168	C₈×C₁₃	—	2	2	no	—
169	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
170	C₂²×SD₁₆	C₂²×C₁₃	1	2	yes	—
171	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×#124	C₂²×C₁₃	1	2	yes	—
173	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
174	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
175	#440	—	2	8	no	—
176	C₄×Q₁₆	—	2	4	no	—
177	C₄×D₈	—	2	4	no	—
178	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×#107	C₂²×C₁₃	1	2	yes	—
180	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
181	C₂×#103	C₂²×C₁₃	1	2	yes	—
182	C₂×#103	C₂²×C₁₃	1	2	yes	—
183	C₂×#102	C₂²×C₁₃	1	2	yes	—
184	C₂×#102	C₂²×C₁₃	1	2	yes	—
185	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
187	#391	C₂²×C₁₃	1	2	yes	—
188	#390	C₂²×C₁₃	1	2	yes	—
189	#388	—	2	4	no	—
190	#385	—	2	4	no	—
191	#384	C₂²×C₁₃	1	2	yes	—
192	#376	C₂²×C₁₃	1	1	yes	characteristic
193	#336	C₂²×C₁₃	1	2	yes	—
194	#336	C₂²×C₁₃	1	2	yes	—
195	C₂²×C₁₃	—	4	8	no	—
196	C₂²×C₁₃	—	4	4	no	—
197	C₂²×C₁₃	—	8	4	no	—
198	C₂²×C₁₃	—	8	2	no	—
199	C₂²×C₁₃	—	8	2	no	—
200	C₂²×C₁₃	#525	1	2	yes	—
201	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
202	C₂²×C₁₃	#576	1	2	yes	—
203	C₂²×C₁₃	#578	1	2	yes	—
204	C₄×C₁₃	—	2	2	no	—
205	C₄×C₁₃	—	2	2	no	—
206	C₄×C₁₃	—	4	4	no	—
207	C₄×C₁₃	—	4	4	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	—	8	2	no	—
210	C₄×C₁₃	#525	1	1	yes	characteristic
211	C₄×C₁₃	#525	1	2	yes	—
212	C₄×C₁₃	#525	1	1	yes	characteristic
213	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
214	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
215	C₂³×C₄	—	2	2	no	—
216	C₂×Q₁₆	—	2	4	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×SD₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×D₈	—	2	4	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
225	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
226	#124	—	2	4	no	—
227	#124	—	2	4	no	—
228	C₄×Q₈	—	2	4	no	—
229	C₄×Q₈	—	2	4	no	—
230	C₄×D₄	—	2	4	no	—
231	C₄×D₄	—	2	4	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	4	no	—
234	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
235	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
236	C₂×C₂²⋊C₄	—	2	4	no	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₂²⋊C₄	—	2	4	no	—
239	C₂×C₄²	—	2	2	no	—
240	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
241	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	2	2	no	—
244	#107	—	2	4	no	—
245	#103	—	2	4	no	—
246	#103	—	2	4	no	—
247	#102	—	2	4	no	—
248	#102	—	2	4	no	—
249	#97	D₄×C₁₃	1	4	yes	—
250	C₄×C₈	D₄×C₁₃	1	4	yes	—
251	#95	—	2	4	no	—
252	#95	D₄×C₁₃	1	2	yes	—
253	#95	D₄×C₁₃	1	2	yes	—
254	C₂×C₁₃	—	8	4	no	—
255	C₂×C₁₃	#2397	1	1	yes	characteristic
256	C₂×C₁₃	#3124	1	4	yes	—
257	C₂×C₁₃	#3155	1	1	yes	characteristic
258	C₂×C₁₃	#3156	1	1	yes	characteristic
259	C₂⁴	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	C₁₃×#41	1	4	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	C₁₃×#41	1	4	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	4	4	no	—
276	C₂²×C₄	—	8	4	no	—
277	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
278	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
281	Q₁₆	—	2	8	no	—
282	SD₁₆	—	2	16	no	—
283	D₈	—	2	8	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
290	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	4	no	—
296	C₄⋊C₄	—	8	4	no	—
297	C₂²⋊C₄	—	2	16	no	—
298	C₂²⋊C₄	—	4	4	no	—
299	C₂²⋊C₄	—	8	4	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	4	no	—
303	C₄²	C₂×D₄×C₁₃	1	2	yes	—
304	C₄²	C₂×D₄×C₁₃	1	2	yes	—
305	C₁₃	#23812	1	1	yes	characteristic, direct product factor, Sylow
306	C₂³	—	2	4	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	4	no	—
310	C₂³	—	4	4	no	—
311	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
312	Q₈	—	2	4	no	—
313	Q₈	—	2	4	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	4	2	no	—
316	D₄	—	2	4	no	—
317	D₄	—	2	4	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	8	8	no	—
333	C₂×C₄	C₁₃×#127	1	2	yes	—
334	C₂×C₄	C₁₃×#127	1	2	yes	—
335	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
336	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
337	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
338	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
339	C₈	—	2	2	no	—
340	C₈	—	2	4	no	—
341	C₈	—	2	2	no	—
342	C₂²	—	4	8	no	—
343	C₂²	—	4	4	no	—
344	C₂²	—	8	4	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	C₁₃×#525	1	2	yes	—
348	C₂²	C₂×C₁₃×#127	1	1	yes	characteristic
349	C₂²	C₁₃×#576	1	2	yes	—
350	C₂²	C₁₃×#578	1	2	yes	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	4	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	C₁₃×#525	1	1	yes	characteristic
358	C₄	C₁₃×#525	1	2	yes	—
359	C₄	C₁₃×#525	1	1	yes	characteristic
360	C₂	—	8	4	no	—
361	C₂	C₁₃×#2397	1	1	yes	characteristic
362	C₂	C₁₃×#3124	1	4	yes	—
363	C₂	C₁₃×#3155	1	1	yes	characteristic
364	C₂	C₁₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#440	C₂	semidirect
2. C₂×C₄×C₁₃×Q₁₆	C₂	semidirect
4. C₁₃×#2326	C₂	semidirect
5. C₁₃×#2095	C₂	semidirect
6. C₁₃×#2093	C₂	semidirect
7. C₁₃×#2015	C₂	semidirect
9. C₁₃×#1746	C₂	semidirect
77. #23812	C₁₃	direct (N × H)
124. C₂×#440	C₂×C₁₃	semidirect
125. C₂×C₄×Q₁₆	C₂×C₁₃	semidirect
127. #2326	C₂×C₁₃	semidirect
128. #2095	C₂×C₁₃	semidirect
129. #2093	C₂×C₁₃	semidirect
130. #2015	C₂×C₁₃	semidirect
132. #1746	C₂×C₁₃	semidirect
305. C₁₃	#23812	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×C₁₃×#440	2
C₂×C₄×C₁₃×Q₁₆	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2326	2
C₁₃×#2095	2
C₁₃×#2093	2
C₁₃×#2015	2
C₁₃×#2012	2
C₁₃×#1746	2
#23812	13

Subgroups — C13×#23812 #15710151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23812 #15710151