Subgroups — C₃₁×#23812 #21480571

Order 7936.

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×C₃₁×#440	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2326	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2095	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2093	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2015	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2012	C₂	1	2	yes	maximal
9	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#440	—	2	8	no	—
17	C₄×Q₁₆×C₃₁	—	2	4	no	—
18	C₄×D₈×C₃₁	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#391	C₂²	1	2	yes	—
29	C₃₁×#390	C₂²	1	2	yes	—
30	C₃₁×#388	—	2	4	no	—
31	C₃₁×#385	—	2	4	no	—
32	C₃₁×#384	C₂²	1	2	yes	—
33	C₃₁×#376	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂×Q₁₆×C₃₁	—	2	4	no	—
40	C₂×Q₁₆×C₃₁	—	2	2	no	—
41	C₂×SD₁₆×C₃₁	—	2	4	no	—
42	C₂×SD₁₆×C₃₁	—	2	4	no	—
43	C₂×SD₁₆×C₃₁	—	2	4	no	—
44	C₂×D₈×C₃₁	—	2	4	no	—
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#124	—	2	4	no	—
50	C₃₁×#124	—	2	4	no	—
51	C₄×Q₈×C₃₁	—	2	4	no	—
52	C₄×Q₈×C₃₁	—	2	4	no	—
53	C₄×D₄×C₃₁	—	2	4	no	—
54	C₄×D₄×C₃₁	—	2	4	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
62	C₂×C₄²×C₃₁	—	2	2	no	—
63	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	—	2	2	no	—
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#107	—	2	4	no	—
68	C₃₁×#103	—	2	4	no	—
69	C₃₁×#103	—	2	4	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#97	D₄	1	4	yes	—
73	C₄×C₈×C₃₁	D₄	1	4	yes	—
74	C₃₁×#95	—	2	4	no	—
75	C₃₁×#95	D₄	1	2	yes	—
76	C₃₁×#95	D₄	1	2	yes	—
77	C₂⁴×C₃₁	—	2	2	no	—
78	C₂×Q₈×C₃₁	—	2	2	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	#41	1	4	yes	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	#41	1	4	yes	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	4	4	no	—
94	C₂²×C₄×C₃₁	—	8	4	no	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	Q₁₆×C₃₁	—	2	8	no	—
100	SD₁₆×C₃₁	—	2	16	no	—
101	D₈×C₃₁	—	2	8	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	4	4	no	—
114	C₄⋊C₄×C₃₁	—	8	4	no	—
115	C₂²⋊C₄×C₃₁	—	2	16	no	—
116	C₂²⋊C₄×C₃₁	—	4	4	no	—
117	C₂²⋊C₄×C₃₁	—	8	4	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	4	4	no	—
121	C₄²×C₃₁	C₂×D₄	1	2	yes	—
122	C₄²×C₃₁	C₂×D₄	1	2	yes	—
123	#23812	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	—	2	4	no	—
125	C₂³×C₃₁	—	4	2	no	—
126	C₂³×C₃₁	—	4	2	no	—
127	C₂³×C₃₁	—	4	4	no	—
128	C₂³×C₃₁	—	4	4	no	—
129	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	Q₈×C₃₁	—	2	4	no	—
131	Q₈×C₃₁	—	2	4	no	—
132	Q₈×C₃₁	—	4	2	no	—
133	Q₈×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	2	4	no	—
135	D₄×C₃₁	—	2	4	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	4	no	—
142	C₂×C₄×C₃₁	—	2	4	no	—
143	C₂×C₄×C₃₁	—	2	4	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	—	4	4	no	—
150	C₂×C₄×C₃₁	—	8	8	no	—
151	C₂×C₄×C₃₁	#127	1	2	yes	—
152	C₂×C₄×C₃₁	#127	1	2	yes	—
153	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
156	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
157	C₈×C₃₁	—	2	2	no	—
158	C₈×C₃₁	—	2	4	no	—
159	C₈×C₃₁	—	2	2	no	—
160	C₂×#440	C₂×C₃₁	1	2	yes	complemented
161	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
163	#2326	C₂×C₃₁	1	2	yes	complemented
164	#2095	C₂×C₃₁	1	2	yes	complemented
165	#2093	C₂×C₃₁	1	2	yes	complemented
166	#2015	C₂×C₃₁	1	2	yes	complemented
167	#2012	C₂×C₃₁	1	2	yes	—
168	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
169	C₂²×C₃₁	—	4	8	no	—
170	C₂²×C₃₁	—	4	4	no	—
171	C₂²×C₃₁	—	8	4	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	—	8	2	no	—
174	C₂²×C₃₁	#525	1	2	yes	—
175	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
176	C₂²×C₃₁	#576	1	2	yes	—
177	C₂²×C₃₁	#578	1	2	yes	—
178	C₄×C₃₁	—	2	2	no	—
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	4	4	no	—
181	C₄×C₃₁	—	4	4	no	—
182	C₄×C₃₁	—	8	2	no	—
183	C₄×C₃₁	—	8	2	no	—
184	C₄×C₃₁	#525	1	1	yes	characteristic
185	C₄×C₃₁	#525	1	2	yes	—
186	C₄×C₃₁	#525	1	1	yes	characteristic
187	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
188	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
189	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×#124	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
192	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
193	#440	—	2	8	no	—
194	C₄×Q₁₆	—	2	4	no	—
195	C₄×D₈	—	2	4	no	—
196	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×#107	C₂²×C₃₁	1	2	yes	—
198	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
199	C₂×#103	C₂²×C₃₁	1	2	yes	—
200	C₂×#103	C₂²×C₃₁	1	2	yes	—
201	C₂×#102	C₂²×C₃₁	1	2	yes	—
202	C₂×#102	C₂²×C₃₁	1	2	yes	—
203	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
205	#391	C₂²×C₃₁	1	2	yes	—
206	#390	C₂²×C₃₁	1	2	yes	—
207	#388	—	2	4	no	—
208	#385	—	2	4	no	—
209	#384	C₂²×C₃₁	1	2	yes	—
210	#376	C₂²×C₃₁	1	1	yes	characteristic
211	#336	C₂²×C₃₁	1	2	yes	—
212	#336	C₂²×C₃₁	1	2	yes	—
213	C₂×C₃₁	—	8	4	no	—
214	C₂×C₃₁	#2397	1	1	yes	characteristic
215	C₂×C₃₁	#3124	1	4	yes	—
216	C₂×C₃₁	#3155	1	1	yes	characteristic
217	C₂×C₃₁	#3156	1	1	yes	characteristic
218	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
219	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
220	C₂³×C₄	—	2	2	no	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	4	no	—
224	C₂×SD₁₆	—	2	4	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×D₈	—	2	4	no	—
227	C₂×D₈	—	2	2	no	—
228	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
230	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	#124	—	2	4	no	—
232	#124	—	2	4	no	—
233	C₄×Q₈	—	2	4	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×D₄	—	2	4	no	—
236	C₄×D₄	—	2	4	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	4	no	—
239	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
241	C₂×C₂²⋊C₄	—	2	4	no	—
242	C₂×C₂²⋊C₄	—	2	2	no	—
243	C₂×C₂²⋊C₄	—	2	4	no	—
244	C₂×C₄²	—	2	2	no	—
245	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
246	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
247	#107	—	2	2	no	—
248	#107	—	2	2	no	—
249	#107	—	2	4	no	—
250	#103	—	2	4	no	—
251	#103	—	2	4	no	—
252	#102	—	2	4	no	—
253	#102	—	2	4	no	—
254	#97	D₄×C₃₁	1	4	yes	—
255	C₄×C₈	D₄×C₃₁	1	4	yes	—
256	#95	—	2	4	no	—
257	#95	D₄×C₃₁	1	2	yes	—
258	#95	D₄×C₃₁	1	2	yes	—
259	C₃₁	#23812	1	1	yes	characteristic, direct product factor, Sylow
260	C₂⁴	—	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	—	2	2	no	—
266	C₂×Q₈	#41×C₃₁	1	4	yes	—
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	—	2	2	no	—
272	C₂×D₄	#41×C₃₁	1	4	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	4	4	no	—
277	C₂²×C₄	—	8	4	no	—
278	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
279	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
282	Q₁₆	—	2	8	no	—
283	SD₁₆	—	2	16	no	—
284	D₈	—	2	8	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
290	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
291	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₄⋊C₄	—	8	4	no	—
298	C₂²⋊C₄	—	2	16	no	—
299	C₂²⋊C₄	—	4	4	no	—
300	C₂²⋊C₄	—	8	4	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	4	4	no	—
304	C₄²	C₂×D₄×C₃₁	1	2	yes	—
305	C₄²	C₂×D₄×C₃₁	1	2	yes	—
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×C₃₁	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	C₃₁×#127	1	2	yes	—
334	C₂×C₄	C₃₁×#127	1	2	yes	—
335	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
338	C₂×C₄	C₂×#41×C₃₁	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	C₃₁×#525	1	2	yes	—
348	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
349	C₂²	C₃₁×#576	1	2	yes	—
350	C₂²	C₃₁×#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	C₃₁×#525	1	1	yes	characteristic
358	C₄	C₃₁×#525	1	2	yes	—
359	C₄	C₃₁×#525	1	1	yes	characteristic
360	C₂	—	8	4	no	—
361	C₂	C₃₁×#2397	1	1	yes	characteristic
362	C₂	C₃₁×#3124	1	4	yes	—
363	C₂	C₃₁×#3155	1	1	yes	characteristic
364	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#440	C₂	semidirect
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2326	C₂	semidirect
5. C₃₁×#2095	C₂	semidirect
6. C₃₁×#2093	C₂	semidirect
7. C₃₁×#2015	C₂	semidirect
9. C₃₁×#1746	C₂	semidirect
123. #23812	C₃₁	direct (N × H)
160. C₂×#440	C₂×C₃₁	semidirect
161. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
163. #2326	C₂×C₃₁	semidirect
164. #2095	C₂×C₃₁	semidirect
165. #2093	C₂×C₃₁	semidirect
166. #2015	C₂×C₃₁	semidirect
168. #1746	C₂×C₃₁	semidirect
259. C₃₁	#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×C₃₁×#440	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2326	2
C₃₁×#2095	2
C₃₁×#2093	2
C₃₁×#2015	2
C₃₁×#2012	2
C₃₁×#1746	2
#23812	31

Subgroups — C31×#23812 #21480571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23812 #21480571