Subgroups — C₇₁×#23164 #31844036

Order 18176.

706 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2370	C₂	1	2	yes	maximal
3	C₇₁×#1989	C₂	1	2	yes	maximal
4	C₇₁×#1943	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1943	C₂	1	1	yes	maximal, characteristic
6	C₇₁×#1920	C₂	1	2	yes	maximal
7	C₇₁×#1919	C₂	1	2	yes	maximal
8	C₇₁×#1776	C₂	1	1	yes	maximal, characteristic
9	C₇₁×#1776	C₂	1	1	yes	maximal, characteristic
10	C₄×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
11	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₇₁	C₂²	1	2	yes	—
13	#499×C₇₁	C₄	1	8	yes	complemented
14	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#106×C₇₁	C₂²	1	2	yes	—
18	C₂×#106×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#389×C₇₁	—	2	2	no	—
24	#389×C₇₁	C₂²	1	2	yes	—
25	#389×C₇₁	C₂²	1	2	yes	—
26	#387×C₇₁	—	2	2	no	—
27	#387×C₇₁	—	2	2	no	—
28	C₄×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
33	#340×C₇₁	C₂²	1	2	yes	—
34	#340×C₇₁	C₂²	1	2	yes	—
35	#340×C₇₁	C₂²	1	2	yes	—
36	#340×C₇₁	C₂²	1	2	yes	—
37	#340×C₇₁	C₂²	1	2	yes	—
38	#336×C₇₁	C₂²	1	1	yes	characteristic
39	#336×C₇₁	C₂²	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	#125×C₇₁	—	2	4	no	—
45	#125×C₇₁	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₇₁	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₁	—	2	2	no	—
56	C₂×C₄²×C₇₁	—	2	2	no	—
57	C₂×C₄²×C₇₁	—	2	2	no	—
58	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	#107×C₇₁	C₂×C₄	1	4	yes	—
62	#107×C₇₁	C₂×C₄	1	4	yes	—
63	#106×C₇₁	—	2	2	no	—
64	#106×C₇₁	C₂×C₄	1	8	yes	—
65	#106×C₇₁	Q₈	1	4	yes	—
66	C₄⋊C₈×C₇₁	—	2	2	no	—
67	#97×C₇₁	—	2	2	no	—
68	#97×C₇₁	—	2	2	no	—
69	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
70	#95×C₇₁	—	2	2	no	—
71	#95×C₇₁	—	2	2	no	—
72	#95×C₇₁	—	2	2	no	—
73	#95×C₇₁	—	2	2	no	—
74	#95×C₇₁	—	2	2	no	—
75	#95×C₇₁	D₄	1	2	yes	—
76	C₂²×C₄×C₇₁	—	2	2	no	—
77	C₂²×C₄×C₇₁	—	2	2	no	—
78	C₂²×C₄×C₇₁	—	2	2	no	—
79	C₂²×C₄×C₇₁	—	2	2	no	—
80	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₇₁	—	2	1	no	—
86	C₂×C₈×C₇₁	—	2	1	no	—
87	C₂×C₈×C₇₁	—	2	1	no	—
88	C₂×C₈×C₇₁	—	2	1	no	—
89	C₂×C₈×C₇₁	—	2	1	no	—
90	C₂×C₈×C₇₁	—	2	1	no	—
91	C₂×C₈×C₇₁	—	2	1	no	—
92	C₂×C₈×C₇₁	—	2	1	no	—
93	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₇₁	#41	1	2	yes	—
97	C₄⋊C₄×C₇₁	—	2	8	no	—
98	C₄⋊C₄×C₇₁	—	2	8	no	—
99	C₄⋊C₄×C₇₁	—	2	2	no	—
100	C₄⋊C₄×C₇₁	—	2	2	no	—
101	C₄⋊C₄×C₇₁	—	2	2	no	—
102	C₄⋊C₄×C₇₁	—	2	2	no	—
103	C₄⋊C₄×C₇₁	—	2	2	no	—
104	C₄⋊C₄×C₇₁	—	4	2	no	—
105	C₄⋊C₄×C₇₁	—	4	4	no	—
106	C₄⋊C₄×C₇₁	—	8	2	no	—
107	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
108	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
109	C₄⋊C₄×C₇₁	C₂×Q₈	1	4	yes	—
110	C₄²×C₇₁	—	2	1	no	—
111	C₄²×C₇₁	—	2	1	no	—
112	C₄²×C₇₁	—	2	1	no	—
113	C₄²×C₇₁	—	2	1	no	—
114	C₄²×C₇₁	—	4	4	no	—
115	C₄²×C₇₁	—	8	2	no	—
116	C₄²×C₇₁	—	8	2	no	—
117	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
118	C₄²×C₇₁	#41	1	2	yes	—
119	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	—	2	2	no	—
121	C₂×C₄×C₇₁	—	2	2	no	—
122	C₂×C₄×C₇₁	—	2	4	no	—
123	C₂×C₄×C₇₁	—	2	4	no	—
124	C₂×C₄×C₇₁	—	2	2	no	—
125	C₂×C₄×C₇₁	—	2	2	no	—
126	C₂×C₄×C₇₁	—	2	4	no	—
127	C₂×C₄×C₇₁	—	2	2	no	—
128	C₂×C₄×C₇₁	—	2	2	no	—
129	C₂×C₄×C₇₁	—	2	2	no	—
130	C₂×C₄×C₇₁	—	2	2	no	—
131	C₂×C₄×C₇₁	—	4	2	no	—
132	C₂×C₄×C₇₁	—	4	2	no	—
133	C₂×C₄×C₇₁	—	4	2	no	—
134	C₂×C₄×C₇₁	—	4	2	no	—
135	C₂×C₄×C₇₁	—	4	2	no	—
136	C₂×C₄×C₇₁	—	4	2	no	—
137	C₂×C₄×C₇₁	—	4	2	no	—
138	C₂×C₄×C₇₁	—	4	2	no	—
139	C₂×C₄×C₇₁	—	4	2	no	—
140	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₇₁	—	2	2	no	—
149	C₈×C₇₁	—	4	1	no	—
150	C₈×C₇₁	—	4	1	no	—
151	C₈×C₇₁	C₄×Q₈	1	4	yes	—
152	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₇₁	#547	1	1	yes	characteristic
154	C₂²×C₇₁	D₄×Q₈	1	1	yes	characteristic
155	C₂²×C₇₁	#575	1	2	yes	—
156	C₂²×C₇₁	#577	1	2	yes	—
157	C₄×C₇₁	—	2	2	no	—
158	C₄×C₇₁	—	2	2	no	—
159	C₄×C₇₁	—	4	2	no	—
160	C₄×C₇₁	—	4	2	no	—
161	C₄×C₇₁	—	4	4	no	—
162	C₄×C₇₁	—	4	4	no	—
163	C₄×C₇₁	—	4	4	no	—
164	C₄×C₇₁	—	8	2	no	—
165	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₄×C₇₁	#519	1	1	yes	characteristic
167	C₄×C₇₁	#556	1	1	yes	characteristic
168	C₄×C₇₁	#557	1	1	yes	characteristic
169	#23164	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₇₁	#2365	1	1	yes	characteristic
171	C₂×C₇₁	#2943	1	1	yes	characteristic
172	C₂×C₇₁	#2952	1	1	yes	characteristic
173	C₂×C₇₁	#3351	1	2	yes	—
174	C₂×C₇₁	#3388	1	2	yes	—
175	C₂×#499	C₂×C₇₁	1	1	yes	characteristic
176	#2370	C₂×C₇₁	1	2	yes	—
177	#1989	C₂×C₇₁	1	2	yes	—
178	#1943	C₂×C₇₁	1	1	yes	characteristic
179	#1943	C₂×C₇₁	1	1	yes	characteristic
180	#1920	C₂×C₇₁	1	2	yes	—
181	#1919	C₂×C₇₁	1	2	yes	—
182	#1776	C₂×C₇₁	1	1	yes	characteristic
183	#1776	C₂×C₇₁	1	1	yes	characteristic
184	C₄×#106	C₂×C₇₁	1	1	yes	characteristic
185	#1770	C₂×C₇₁	1	1	yes	characteristic
186	C₇₁	#23164	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#125	C₂²×C₇₁	1	2	yes	—
188	#499	C₄×C₇₁	1	8	yes	complemented
189	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
190	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
191	C₂×#106	C₂²×C₇₁	1	1	yes	characteristic
192	C₂×#106	C₂²×C₇₁	1	2	yes	—
193	C₂×#106	C₂²×C₇₁	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
195	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
196	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
198	#389	—	2	2	no	—
199	#389	C₂²×C₇₁	1	2	yes	—
200	#389	C₂²×C₇₁	1	2	yes	—
201	#387	—	2	2	no	—
202	#387	—	2	2	no	—
203	C₄×C₄⋊C₄	—	2	2	no	—
204	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
205	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
206	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
207	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
208	#340	C₂²×C₇₁	1	2	yes	—
209	#340	C₂²×C₇₁	1	2	yes	—
210	#340	C₂²×C₇₁	1	2	yes	—
211	#340	C₂²×C₇₁	1	2	yes	—
212	#340	C₂²×C₇₁	1	2	yes	—
213	#336	C₂²×C₇₁	1	1	yes	characteristic
214	#336	C₂²×C₇₁	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
219	#125	—	2	4	no	—
220	#125	C₂×C₄×C₇₁	1	8	yes	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	D₄×C₇₁	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
236	#107	C₂×C₄×C₇₁	1	4	yes	—
237	#107	C₂×C₄×C₇₁	1	4	yes	—
238	#106	—	2	2	no	—
239	#106	C₂×C₄×C₇₁	1	8	yes	—
240	#106	Q₈×C₇₁	1	4	yes	—
241	C₄⋊C₈	—	2	2	no	—
242	#97	—	2	2	no	—
243	#97	—	2	2	no	—
244	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	D₄×C₇₁	1	2	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
259	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
269	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
270	C₂×C₈	C₂×Q₈×C₇₁	1	2	yes	—
271	C₂×C₈	#41×C₇₁	1	2	yes	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	8	2	no	—
282	C₄⋊C₄	C₂²×C₄×C₇₁	1	4	yes	—
283	C₄⋊C₄	C₂²×C₄×C₇₁	1	4	yes	—
284	C₄⋊C₄	C₂×Q₈×C₇₁	1	4	yes	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	4	4	no	—
290	C₄²	—	8	2	no	—
291	C₄²	—	8	2	no	—
292	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
293	C₄²	#41×C₇₁	1	2	yes	—
294	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
316	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
317	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
318	C₂×C₄	C₂²×Q₈×C₇₁	1	1	yes	characteristic
319	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
321	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
322	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	4	1	no	—
325	C₈	—	4	1	no	—
326	C₈	C₄×Q₈×C₇₁	1	4	yes	—
327	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
328	C₂²	#547×C₇₁	1	1	yes	characteristic
329	C₂²	D₄×Q₈×C₇₁	1	1	yes	characteristic
330	C₂²	#575×C₇₁	1	2	yes	—
331	C₂²	#577×C₇₁	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	4	no	—
337	C₄	—	4	4	no	—
338	C₄	—	4	4	no	—
339	C₄	—	8	2	no	—
340	C₄	C₂×C₄×Q₈×C₇₁	1	1	yes	characteristic
341	C₄	#519×C₇₁	1	1	yes	characteristic
342	C₄	#556×C₇₁	1	1	yes	characteristic
343	C₄	#557×C₇₁	1	1	yes	characteristic
344	C₂	C₇₁×#2365	1	1	yes	characteristic
345	C₂	C₇₁×#2943	1	1	yes	characteristic
346	C₂	C₇₁×#2952	1	1	yes	characteristic
347	C₂	C₇₁×#3351	1	2	yes	—
348	C₂	C₇₁×#3388	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄×C₄⋊C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. #499×C₇₁	C₄	semidirect
169. #23164	C₇₁	direct (N × H)
186. C₇₁	#23164	direct (N × H)
188. #499	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₇₁	2
C₇₁×#2370	2
C₇₁×#1989	2
C₇₁×#1943	2
C₇₁×#1920	2
C₇₁×#1919	2
C₇₁×#1776	2
C₄×#106×C₇₁	2
C₇₁×#1770	2
#23164	71

Subgroups — C71×#23164 #31844036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23164 #31844036