Subgroups — C₇₁×#17720 #31838592

Order 18176.

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	C₇₁×#3498	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1547	C₂	1	3	yes	maximal, complemented
5	C₇₁×#1545	C₂	1	3	yes	maximal, complemented
6	C₇₁×#1540	C₂	1	3	yes	maximal, complemented
7	C₂×#331×C₇₁	C₂	1	3	yes	maximal, complemented
8	C₂×H(5,𝔽₂)×C₇₁	—	2	1	no	—
9	C₂×#131×C₇₁	—	2	1	no	—
10	#550×C₇₁	—	2	1	no	—
11	#550×C₇₁	C₄	1	2	yes	—
12	D₄²×C₇₁	—	2	3	no	—
13	D₄²×C₇₁	C₄	1	6	yes	—
14	#535×C₇₁	—	2	3	no	—
15	#532×C₇₁	C₂²	1	3	yes	—
16	C₂×#127×C₇₁	C₂²	1	3	yes	—
17	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
18	C₈×Q₈×C₇₁	—	2	1	no	—
19	C₈×D₄×C₇₁	—	2	3	no	—
20	#431×C₇₁	C₂²	1	3	yes	complemented
21	#424×C₇₁	—	2	3	no	—
22	#423×C₇₁	C₂²	1	3	yes	complemented
23	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	complemented
24	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	complemented
25	#332×C₇₁	C₂²	1	1	yes	characteristic, complemented
26	#332×C₇₁	C₂²	1	3	yes	complemented
27	#331×C₇₁	C₂²	1	3	yes	complemented
28	#331×C₇₁	C₂²	1	6	yes	complemented
29	#331×C₇₁	C₂²	1	3	yes	complemented
30	H(5,𝔽₂)×C₇₁	—	4	4	no	—
31	H(5,𝔽₂)×C₇₁	—	4	4	no	—
32	C₂×#41×C₇₁	—	2	2	no	—
33	C₂×#41×C₇₁	—	2	3	no	—
34	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₇₁	—	2	6	no	—
36	C₂²×D₄×C₇₁	—	2	3	no	—
37	C₂²×D₄×C₇₁	—	2	3	no	—
38	C₂²×D₄×C₇₁	C₂×C₄	1	6	yes	—
39	#131×C₇₁	—	4	4	no	—
40	C₂×M₁₆×C₇₁	—	2	3	no	—
41	C₂²×C₈×C₇₁	—	2	3	no	—
42	C₂²×C₈×C₇₁	—	2	3	no	—
43	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
44	#127×C₇₁	—	2	3	no	—
45	#127×C₇₁	—	2	3	no	—
46	#127×C₇₁	C₂×C₄	1	6	yes	—
47	#127×C₇₁	C₂³	1	3	yes	—
48	#124×C₇₁	—	2	3	no	—
49	#121×C₇₁	—	2	3	no	—
50	#121×C₇₁	—	2	6	no	—
51	#121×C₇₁	—	2	6	no	—
52	#121×C₇₁	C₂×C₄	1	6	yes	—
53	#120×C₇₁	—	2	12	no	—
54	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
56	C₄×D₄×C₇₁	—	2	3	no	—
57	C₄×D₄×C₇₁	C₂×C₄	1	6	yes	—
58	C₄×D₄×C₇₁	C₂×C₄	1	3	yes	—
59	C₄×D₄×C₇₁	D₄	1	3	yes	—
60	#117×C₇₁	D₄	1	3	yes	—
61	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
62	C₄⋊C₈×C₇₁	—	2	3	no	—
63	C₄⋊C₈×C₇₁	—	2	3	no	—
64	C₄⋊C₈×C₇₁	—	2	3	no	—
65	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₇₁	C₂³	1	3	yes	—
67	C₄⋊C₈×C₇₁	C₂³	1	3	yes	—
68	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
69	C₂²⋊C₈×C₇₁	—	2	3	no	—
70	C₄×C₈×C₇₁	—	2	3	no	—
71	C₂⁴×C₇₁	—	2	6	no	—
72	#41×C₇₁	—	4	8	no	—
73	#41×C₇₁	—	4	6	no	—
74	#41×C₇₁	—	4	6	no	—
75	#41×C₇₁	D₈	1	4	yes	—
76	#41×C₇₁	SD₁₆	1	4	yes	—
77	C₂×Q₈×C₇₁	—	2	1	no	—
78	C₂×Q₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₇₁	—	2	6	no	—
80	C₂×D₄×C₇₁	—	2	6	no	—
81	C₂×D₄×C₇₁	—	2	6	no	—
82	C₂×D₄×C₇₁	—	2	6	no	—
83	C₂×D₄×C₇₁	—	2	3	no	—
84	C₂×D₄×C₇₁	—	2	3	no	—
85	C₂×D₄×C₇₁	—	4	24	no	—
86	C₂×D₄×C₇₁	—	4	6	no	—
87	C₂×D₄×C₇₁	—	4	6	no	—
88	C₂×D₄×C₇₁	—	8	3	no	—
89	C₂×D₄×C₇₁	—	8	3	no	—
90	C₂×D₄×C₇₁	—	8	6	no	—
91	C₂×D₄×C₇₁	C₂²⋊C₄	1	3	yes	—
92	C₂×D₄×C₇₁	C₂²×C₄	1	6	yes	—
93	C₂²×C₄×C₇₁	—	2	6	no	—
94	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
95	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
96	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
97	M₁₆×C₇₁	—	4	6	no	—
98	C₂×C₈×C₇₁	—	2	1	no	—
99	C₂×C₈×C₇₁	—	2	3	no	—
100	C₂×C₈×C₇₁	—	2	1	no	—
101	C₂×C₈×C₇₁	—	2	3	no	—
102	C₂×C₈×C₇₁	—	4	6	no	—
103	C₂×C₈×C₇₁	—	8	3	no	—
104	C₄⋊C₄×C₇₁	—	2	1	no	—
105	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	3	yes	—
106	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	3	yes	—
107	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
108	C₂²⋊C₄×C₇₁	—	2	6	no	—
109	C₂²⋊C₄×C₇₁	—	2	6	no	—
110	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	6	yes	—
111	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
112	C₄²×C₇₁	C₂×D₄	1	3	yes	—
113	C₄²×C₇₁	C₂×D₄	1	3	yes	—
114	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
115	C₂³×C₇₁	—	2	6	no	—
116	C₂³×C₇₁	—	4	12	no	—
117	C₂³×C₇₁	—	8	6	no	—
118	C₂³×C₇₁	—	8	6	no	—
119	C₂³×C₇₁	C₂×C₂²⋊C₄	1	3	yes	—
120	Q₈×C₇₁	—	4	1	no	—
121	Q₈×C₇₁	—	4	1	no	—
122	Q₈×C₇₁	#102	1	4	yes	—
123	D₄×C₇₁	—	4	6	no	—
124	D₄×C₇₁	—	4	12	no	—
125	D₄×C₇₁	—	4	3	no	—
126	D₄×C₇₁	—	4	3	no	—
127	D₄×C₇₁	—	4	3	no	—
128	D₄×C₇₁	—	4	3	no	—
129	D₄×C₇₁	—	4	12	no	—
130	D₄×C₇₁	—	8	6	no	—
131	D₄×C₇₁	—	8	6	no	—
132	D₄×C₇₁	#102	1	12	yes	—
133	C₂×C₄×C₇₁	—	2	3	no	—
134	C₂×C₄×C₇₁	—	2	3	no	—
135	C₂×C₄×C₇₁	—	2	2	no	—
136	C₂×C₄×C₇₁	—	4	3	no	—
137	C₂×C₄×C₇₁	—	4	12	no	—
138	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	3	yes	—
139	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	3	yes	—
140	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	3	yes	—
141	C₂×C₄×C₇₁	C₂×D₈	1	6	yes	—
142	C₂×C₄×C₇₁	C₂×SD₁₆	1	6	yes	—
143	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₇₁	—	4	1	no	—
147	C₈×C₇₁	—	4	3	no	—
148	C₈×C₇₁	—	4	3	no	—
149	C₈×C₇₁	—	4	1	no	—
150	C₂²×C₇₁	—	2	3	no	—
151	C₂²×C₇₁	—	2	3	no	—
152	C₂²×C₇₁	—	4	6	no	—
153	C₂²×C₇₁	—	4	6	no	—
154	C₂²×C₇₁	—	4	6	no	—
155	C₂²×C₇₁	—	16	12	no	—
156	C₂²×C₇₁	C₂×#102	1	6	yes	—
157	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₇₁	—	2	1	no	—
159	C₄×C₇₁	—	2	3	no	—
160	C₄×C₇₁	—	2	3	no	—
161	C₄×C₇₁	—	4	2	no	—
162	C₄×C₇₁	C₂×#102	1	6	yes	—
163	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
165	C₄×C₇₁	#576	1	1	yes	characteristic
166	C₄×C₇₁	#577	1	1	yes	characteristic
167	#17720	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₇₁	—	2	6	no	—
169	C₂×C₇₁	—	8	6	no	—
170	C₂×C₇₁	#2888	1	1	yes	characteristic
171	C₂×C₇₁	C₂²×#102	1	1	yes	characteristic
172	C₂×C₇₁	#2897	1	1	yes	characteristic
173	#3498	C₂×C₇₁	1	1	yes	characteristic
174	#2988	C₂×C₇₁	1	1	yes	characteristic, complemented
175	#2906	C₂×C₇₁	1	1	yes	characteristic, complemented
176	#1547	C₂×C₇₁	1	3	yes	complemented
177	#1545	C₂×C₇₁	1	3	yes	complemented
178	#1540	C₂×C₇₁	1	3	yes	complemented
179	C₂×#331	C₂×C₇₁	1	3	yes	complemented
180	C₇₁	#17720	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×H(5,𝔽₂)	—	2	1	no	—
182	C₂×#131	—	2	1	no	—
183	#550	—	2	1	no	—
184	#550	C₄×C₇₁	1	2	yes	—
185	D₄²	—	2	3	no	—
186	D₄²	C₄×C₇₁	1	6	yes	—
187	#535	—	2	3	no	—
188	#532	C₂²×C₇₁	1	3	yes	—
189	C₂×#127	C₂²×C₇₁	1	3	yes	—
190	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
191	C₈×Q₈	—	2	1	no	—
192	C₈×D₄	—	2	3	no	—
193	#431	C₂²×C₇₁	1	3	yes	complemented
194	#424	—	2	3	no	—
195	#423	C₂²×C₇₁	1	3	yes	complemented
196	C₂×C₄⋊C₈	C₂²×C₇₁	1	3	yes	complemented
197	C₂×C₄⋊C₈	C₂²×C₇₁	1	3	yes	complemented
198	#332	C₂²×C₇₁	1	1	yes	characteristic, complemented
199	#332	C₂²×C₇₁	1	3	yes	complemented
200	#331	C₂²×C₇₁	1	3	yes	complemented
201	#331	C₂²×C₇₁	1	6	yes	complemented
202	#331	C₂²×C₇₁	1	3	yes	complemented
203	H(5,𝔽₂)	—	4	4	no	—
204	H(5,𝔽₂)	—	4	4	no	—
205	C₂×#41	—	2	2	no	—
206	C₂×#41	—	2	3	no	—
207	C₂×#41	D₄×C₇₁	1	1	yes	characteristic
208	C₂²×D₄	—	2	6	no	—
209	C₂²×D₄	—	2	3	no	—
210	C₂²×D₄	—	2	3	no	—
211	C₂²×D₄	C₂×C₄×C₇₁	1	6	yes	—
212	#131	—	4	4	no	—
213	C₂×M₁₆	—	2	3	no	—
214	C₂²×C₈	—	2	3	no	—
215	C₂²×C₈	—	2	3	no	—
216	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
217	#127	—	2	3	no	—
218	#127	—	2	3	no	—
219	#127	C₂×C₄×C₇₁	1	6	yes	—
220	#127	C₂³×C₇₁	1	3	yes	—
221	#124	—	2	3	no	—
222	#121	—	2	3	no	—
223	#121	—	2	6	no	—
224	#121	—	2	6	no	—
225	#121	C₂×C₄×C₇₁	1	6	yes	—
226	#120	—	2	12	no	—
227	C₄×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
228	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
229	C₄×D₄	—	2	3	no	—
230	C₄×D₄	C₂×C₄×C₇₁	1	6	yes	—
231	C₄×D₄	C₂×C₄×C₇₁	1	3	yes	—
232	C₄×D₄	D₄×C₇₁	1	3	yes	—
233	#117	D₄×C₇₁	1	3	yes	—
234	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
235	C₄⋊C₈	—	2	3	no	—
236	C₄⋊C₈	—	2	3	no	—
237	C₄⋊C₈	—	2	3	no	—
238	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
239	C₄⋊C₈	C₂³×C₇₁	1	3	yes	—
240	C₄⋊C₈	C₂³×C₇₁	1	3	yes	—
241	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
242	C₂²⋊C₈	—	2	3	no	—
243	C₄×C₈	—	2	3	no	—
244	C₂⁴	—	2	6	no	—
245	#41	—	4	8	no	—
246	#41	—	4	6	no	—
247	#41	—	4	6	no	—
248	#41	D₈×C₇₁	1	4	yes	—
249	#41	SD₁₆×C₇₁	1	4	yes	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	C₂²⋊C₄×C₇₁	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×C₇₁	1	3	yes	—
265	C₂×D₄	C₂²×C₄×C₇₁	1	6	yes	—
266	C₂²×C₄	—	2	6	no	—
267	C₂²×C₄	C₂²×C₄×C₇₁	1	3	yes	—
268	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
269	C₂²×C₄	C₂×D₄×C₇₁	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×C₇₁	1	3	yes	—
279	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	3	yes	—
280	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
281	C₂²⋊C₄	—	2	6	no	—
282	C₂²⋊C₄	—	2	6	no	—
283	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	6	yes	—
284	C₄²	C₂²×C₄×C₇₁	1	3	yes	—
285	C₄²	C₂×D₄×C₇₁	1	3	yes	—
286	C₄²	C₂×D₄×C₇₁	1	3	yes	—
287	C₄²	C₂⁴×C₇₁	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×C₇₁	1	3	yes	—
293	Q₈	—	4	1	no	—
294	Q₈	—	4	1	no	—
295	Q₈	#102×C₇₁	1	4	yes	—
296	D₄	—	4	6	no	—
297	D₄	—	4	12	no	—
298	D₄	—	4	12	no	—
299	D₄	—	4	3	no	—
300	D₄	—	4	3	no	—
301	D₄	—	4	3	no	—
302	D₄	—	4	3	no	—
303	D₄	—	8	6	no	—
304	D₄	—	8	6	no	—
305	D₄	#102×C₇₁	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×C₇₁	1	3	yes	—
312	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	3	yes	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	3	yes	—
314	C₂×C₄	C₂×D₈×C₇₁	1	6	yes	—
315	C₂×C₄	C₂×SD₁₆×C₇₁	1	6	yes	—
316	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₇₁	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×#102×C₇₁	1	6	yes	—
330	C₂²	C₂²×C₂²⋊C₄×C₇₁	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×#102×C₇₁	1	6	yes	—
336	C₄	C₂²×D₈×C₇₁	1	1	yes	characteristic
337	C₄	C₂²×SD₁₆×C₇₁	1	1	yes	characteristic
338	C₄	#576×C₇₁	1	1	yes	characteristic
339	C₄	#577×C₇₁	1	1	yes	characteristic
340	C₂	—	2	6	no	—
341	C₂	—	8	6	no	—
342	C₂	C₇₁×#2888	1	1	yes	characteristic
343	C₂	C₂²×#102×C₇₁	1	1	yes	characteristic
344	C₂	C₇₁×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₄²	C₂⁴×C₇₁
prime radical	C₇₁×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄×#41×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2988	C₂	semidirect
3. C₇₁×#2906	C₂	semidirect
4. C₇₁×#1547	C₂	semidirect
5. C₇₁×#1545	C₂	semidirect
6. C₇₁×#1540	C₂	semidirect
7. C₂×#331×C₇₁	C₂	semidirect
20. #431×C₇₁	C₂²	semidirect
22. #423×C₇₁	C₂²	semidirect
23. C₂×C₄⋊C₈×C₇₁	C₂²	semidirect
24. C₂×C₄⋊C₈×C₇₁	C₂²	semidirect
25. #332×C₇₁	C₂²	semidirect
26. #332×C₇₁	C₂²	semidirect
27. #331×C₇₁	C₂²	semidirect
28. #331×C₇₁	C₂²	semidirect
29. #331×C₇₁	C₂²	semidirect
167. #17720	C₇₁	direct (N × H)
174. #2988	C₂×C₇₁	semidirect
175. #2906	C₂×C₇₁	semidirect
176. #1547	C₂×C₇₁	semidirect
177. #1545	C₂×C₇₁	semidirect
178. #1540	C₂×C₇₁	semidirect
179. C₂×#331	C₂×C₇₁	semidirect
180. C₇₁	#17720	direct (N × H)
193. #431	C₂²×C₇₁	semidirect
195. #423	C₂²×C₇₁	semidirect
196. C₂×C₄⋊C₈	C₂²×C₇₁	semidirect
197. C₂×C₄⋊C₈	C₂²×C₇₁	semidirect
198. #332	C₂²×C₇₁	semidirect
199. #332	C₂²×C₇₁	semidirect
200. #331	C₂²×C₇₁	semidirect
201. #331	C₂²×C₇₁	semidirect
202. #331	C₂²×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3498	2
C₇₁×#2988	2
C₇₁×#2906	2
C₇₁×#1547	2
C₇₁×#1545	2
C₇₁×#1540	2
C₂×#331×C₇₁	2
#17720	71

Subgroups — C71×#17720 #31838592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#17720 #31838592