Subgroups — C₃₁×#33027 #21489786

Order 7936.

854 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×#477	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2862	C₂	1	1	yes	maximal, characteristic
4	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1987	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1928	C₂	1	4	yes	maximal
8	C₃₁×#1878	C₂	1	4	yes	maximal
9	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#477	—	2	4	no	—
16	C₃₁×#462	—	2	4	no	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	4	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₃₁×#391	—	2	2	no	—
24	C₃₁×#391	C₂²	1	4	yes	—
25	C₃₁×#382	—	2	1	no	—
26	C₃₁×#382	—	2	1	no	—
27	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄³×C₃₁	—	2	1	no	—
32	C₄³×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#340	C₂²	1	4	yes	—
34	C₃₁×#336	C₂²	1	4	yes	—
35	C₂²×Q₈×C₃₁	—	2	1	no	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×Q₁₆×C₃₁	—	2	4	no	—
39	C₂×Q₁₆×C₃₁	—	2	8	no	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	2	no	—
46	C₄×Q₈×C₃₁	—	2	4	no	—
47	C₄×Q₈×C₃₁	—	2	4	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	D₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	—	2	1	no	—
60	C₂×C₄²×C₃₁	—	2	2	no	—
61	C₂×C₄²×C₃₁	—	2	1	no	—
62	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	D₄	1	2	yes	—
64	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#107	—	2	4	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×#103	—	2	4	no	—
71	C₃₁×#103	—	2	4	no	—
72	C₃₁×#103	—	2	8	no	—
73	C₃₁×#95	—	2	2	no	—
74	C₃₁×#95	—	2	1	no	—
75	C₃₁×#95	—	2	4	no	—
76	C₃₁×#95	—	2	4	no	—
77	C₂×Q₈×C₃₁	—	2	1	no	—
78	C₂×Q₈×C₃₁	—	2	2	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	4	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	4	no	—
83	C₂×Q₈×C₃₁	—	4	4	no	—
84	C₂×Q₈×C₃₁	—	4	2	no	—
85	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×Q₈×C₃₁	#41	1	4	yes	—
87	C₂²×C₄×C₃₁	—	2	1	no	—
88	C₂²×C₄×C₃₁	—	2	1	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₃₁	—	4	8	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	4	no	—
101	C₂×C₈×C₃₁	—	2	4	no	—
102	C₄⋊C₄×C₃₁	—	2	8	no	—
103	C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₃₁	#41	1	4	yes	—
113	C₄²×C₃₁	—	2	8	no	—
114	C₄²×C₃₁	—	2	4	no	—
115	C₄²×C₃₁	—	2	2	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	2	2	no	—
118	C₄²×C₃₁	—	4	2	no	—
119	C₄²×C₃₁	—	4	4	no	—
120	C₄²×C₃₁	—	4	2	no	—
121	C₄²×C₃₁	Q₁₆	1	4	yes	complemented
122	C₄²×C₃₁	C₂×D₄	1	2	yes	—
123	C₄²×C₃₁	#41	1	2	yes	—
124	#33027	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₃₁	—	2	4	no	—
127	Q₈×C₃₁	—	2	2	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	4	4	no	—
130	Q₈×C₃₁	—	4	2	no	—
131	Q₈×C₃₁	—	8	4	no	—
132	Q₈×C₃₁	#124	1	4	yes	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	4	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	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	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	4	2	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	#121	1	2	yes	—
151	C₂×C₄×C₃₁	#121	1	2	yes	—
152	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
153	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
154	C₂×C₄×C₃₁	#137	1	2	yes	—
155	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
157	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
159	C₈×C₃₁	—	4	2	no	—
160	C₈×C₃₁	—	4	2	no	—
161	C₂×#477	C₂×C₃₁	1	1	yes	characteristic
162	C₂×#462	C₂×C₃₁	1	1	yes	characteristic
163	#2862	C₂×C₃₁	1	1	yes	characteristic
164	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
165	#1987	C₂×C₃₁	1	1	yes	characteristic
166	#1981	C₂×C₃₁	1	1	yes	characteristic
167	#1928	C₂×C₃₁	1	4	yes	—
168	#1878	C₂×C₃₁	1	4	yes	—
169	#1831	C₂×C₃₁	1	1	yes	characteristic
170	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
171	C₂²×C₃₁	#525	1	2	yes	—
172	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
173	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
174	C₂²×C₃₁	#575	1	2	yes	—
175	C₄×C₃₁	—	2	4	no	—
176	C₄×C₃₁	—	2	4	no	—
177	C₄×C₃₁	—	2	2	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	4	4	no	—
180	C₄×C₃₁	—	4	2	no	—
181	C₄×C₃₁	—	8	2	no	—
182	C₄×C₃₁	—	8	2	no	—
183	C₄×C₃₁	#462	1	4	yes	—
184	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
185	C₄×C₃₁	#529	1	2	yes	—
186	C₄×C₃₁	#532	1	1	yes	characteristic
187	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
188	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
192	#477	—	2	4	no	—
193	#462	—	2	4	no	—
194	C₂×#107	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×#103	C₂²×C₃₁	1	2	yes	—
198	C₂×#103	C₂²×C₃₁	1	4	yes	—
199	C₂×#103	C₂²×C₃₁	1	2	yes	—
200	#391	—	2	2	no	—
201	#391	C₂²×C₃₁	1	4	yes	—
202	#382	—	2	1	no	—
203	#382	—	2	1	no	—
204	C₄×C₄⋊C₄	—	2	1	no	—
205	C₄×C₄⋊C₄	—	2	2	no	—
206	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
207	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
208	C₄³	—	2	1	no	—
209	C₄³	C₂²×C₃₁	1	1	yes	characteristic
210	#340	C₂²×C₃₁	1	4	yes	—
211	#336	C₂²×C₃₁	1	4	yes	—
212	C₂×C₃₁	#2850	1	1	yes	characteristic
213	C₂×C₃₁	C₂×#462	1	1	yes	characteristic
214	C₂×C₃₁	#3040	1	1	yes	characteristic
215	C₂×C₃₁	#3104	1	2	yes	—
216	C₂×C₃₁	#3116	1	2	yes	—
217	C₂²×Q₈	—	2	1	no	—
218	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
219	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×Q₁₆	—	2	4	no	—
221	C₂×Q₁₆	—	2	8	no	—
222	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
223	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
224	C₄⋊Q₈	—	2	2	no	—
225	C₄⋊Q₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	4	no	—
229	C₄×Q₈	—	2	4	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	D₄×C₃₁	1	4	yes	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
239	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×C₄²	—	2	2	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	—	2	2	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄²	D₄×C₃₁	1	2	yes	—
246	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	#107	—	2	4	no	—
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	2	2	no	—
252	#103	—	2	4	no	—
253	#103	—	2	4	no	—
254	#103	—	2	8	no	—
255	#95	—	2	2	no	—
256	#95	—	2	1	no	—
257	#95	—	2	4	no	—
258	#95	—	2	4	no	—
259	C₃₁	#33027	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	4	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	4	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
269	C₂×Q₈	#41×C₃₁	1	4	yes	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
280	Q₁₆	—	4	8	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
295	C₄⋊C₄	#41×C₃₁	1	4	yes	—
296	C₄²	—	2	8	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	—	4	4	no	—
303	C₄²	—	4	2	no	—
304	C₄²	Q₁₆×C₃₁	1	4	yes	complemented
305	C₄²	C₂×D₄×C₃₁	1	2	yes	—
306	C₄²	#41×C₃₁	1	2	yes	—
307	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	Q₈	—	2	4	no	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	4	no	—
314	Q₈	C₃₁×#124	1	4	yes	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	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	C₃₁×#121	1	2	yes	—
333	C₂×C₄	C₃₁×#121	1	2	yes	—
334	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
335	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
336	C₂×C₄	C₃₁×#137	1	2	yes	—
337	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
339	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
340	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
344	C₂²	C₃₁×#525	1	2	yes	—
345	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
346	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
347	C₂²	C₃₁×#575	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	2	4	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₃₁×#462	1	4	yes	—
357	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
358	C₄	C₃₁×#529	1	2	yes	—
359	C₄	C₃₁×#532	1	1	yes	characteristic
360	C₂	C₃₁×#2850	1	1	yes	characteristic
361	C₂	C₂×C₃₁×#462	1	1	yes	characteristic
362	C₂	C₃₁×#3040	1	1	yes	characteristic
363	C₂	C₃₁×#3104	1	2	yes	—
364	C₂	C₃₁×#3116	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
121. C₄²×C₃₁	Q₁₆	semidirect
124. #33027	C₃₁	direct (N × H)
259. C₃₁	#33027	direct (N × H)
304. C₄²	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#477	2
C₂×C₃₁×#462	2
C₃₁×#2862	2
C₄²×Q₈×C₃₁	2
C₃₁×#1987	2
C₃₁×#1981	2
C₃₁×#1928	2
C₃₁×#1878	2
C₃₁×#1831	2
#33027	31

Subgroups — C31×#33027 #21489786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33027 #21489786