Subgroups — C₇₁×#23165 #31844037

Order 18176.

706 conjugacy classes of subgroups.

The subgroup lattice has 340 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×#1944	C₂	1	2	yes	maximal
5	C₇₁×#1921	C₂	1	2	yes	maximal
6	C₇₁×#1918	C₂	1	2	yes	maximal
7	C₇₁×#1776	C₂	1	2	yes	maximal
8	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₇₁	C₂²	1	2	yes	—
11	#499×C₇₁	C₄	1	8	yes	complemented
12	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	#389×C₇₁	—	2	2	no	—
21	#389×C₇₁	C₂²	1	2	yes	—
22	#389×C₇₁	C₂²	1	2	yes	—
23	#387×C₇₁	—	2	2	no	—
24	#387×C₇₁	—	2	2	no	—
25	C₄×C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
30	#340×C₇₁	C₂²	1	2	yes	—
31	#340×C₇₁	C₂²	1	2	yes	—
32	#340×C₇₁	C₂²	1	2	yes	—
33	#340×C₇₁	C₂²	1	2	yes	—
34	#340×C₇₁	C₂²	1	2	yes	—
35	#336×C₇₁	C₂²	1	2	yes	—
36	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	#125×C₇₁	—	2	4	no	—
40	#125×C₇₁	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₇₁	—	2	2	no	—
52	C₂×C₄²×C₇₁	—	2	2	no	—
53	C₂×C₄²×C₇₁	—	2	2	no	—
54	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
56	#107×C₇₁	—	2	2	no	—
57	#107×C₇₁	C₂×C₄	1	4	yes	—
58	#107×C₇₁	C₂×C₄	1	4	yes	—
59	#107×C₇₁	Q₈	1	4	yes	—
60	#106×C₇₁	C₂×C₄	1	8	yes	—
61	C₄⋊C₈×C₇₁	—	2	2	no	—
62	#97×C₇₁	—	2	4	no	—
63	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
64	#95×C₇₁	—	2	2	no	—
65	#95×C₇₁	—	2	2	no	—
66	#95×C₇₁	—	2	2	no	—
67	#95×C₇₁	—	2	2	no	—
68	#95×C₇₁	—	2	2	no	—
69	#95×C₇₁	D₄	1	2	yes	—
70	C₂²×C₄×C₇₁	—	2	2	no	—
71	C₂²×C₄×C₇₁	—	2	2	no	—
72	C₂²×C₄×C₇₁	—	2	2	no	—
73	C₂²×C₄×C₇₁	—	2	2	no	—
74	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	1	no	—
81	C₂×C₈×C₇₁	—	2	2	no	—
82	C₂×C₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	—	2	2	no	—
84	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₇₁	#41	1	2	yes	—
88	C₄⋊C₄×C₇₁	—	2	8	no	—
89	C₄⋊C₄×C₇₁	—	2	8	no	—
90	C₄⋊C₄×C₇₁	—	2	2	no	—
91	C₄⋊C₄×C₇₁	—	2	2	no	—
92	C₄⋊C₄×C₇₁	—	2	2	no	—
93	C₄⋊C₄×C₇₁	—	2	2	no	—
94	C₄⋊C₄×C₇₁	—	2	2	no	—
95	C₄⋊C₄×C₇₁	—	4	1	no	—
96	C₄⋊C₄×C₇₁	—	4	1	no	—
97	C₄⋊C₄×C₇₁	—	4	4	no	—
98	C₄⋊C₄×C₇₁	—	8	2	no	—
99	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
100	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
101	C₄⋊C₄×C₇₁	C₂×Q₈	1	4	yes	—
102	C₄²×C₇₁	—	2	2	no	—
103	C₄²×C₇₁	—	2	2	no	—
104	C₄²×C₇₁	—	4	4	no	—
105	C₄²×C₇₁	—	8	2	no	—
106	C₄²×C₇₁	—	8	2	no	—
107	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
108	C₄²×C₇₁	#41	1	2	yes	—
109	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	4	no	—
115	C₂×C₄×C₇₁	—	2	4	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	2	2	no	—
118	C₂×C₄×C₇₁	—	2	4	no	—
119	C₂×C₄×C₇₁	—	2	2	no	—
120	C₂×C₄×C₇₁	—	2	2	no	—
121	C₂×C₄×C₇₁	—	2	2	no	—
122	C₂×C₄×C₇₁	—	2	2	no	—
123	C₂×C₄×C₇₁	—	4	2	no	—
124	C₂×C₄×C₇₁	—	4	2	no	—
125	C₂×C₄×C₇₁	—	4	2	no	—
126	C₂×C₄×C₇₁	—	4	2	no	—
127	C₂×C₄×C₇₁	—	4	2	no	—
128	C₂×C₄×C₇₁	—	4	2	no	—
129	C₂×C₄×C₇₁	—	4	2	no	—
130	C₂×C₄×C₇₁	—	4	2	no	—
131	C₂×C₄×C₇₁	—	4	2	no	—
132	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈×C₇₁	—	2	2	no	—
141	C₈×C₇₁	—	4	2	no	—
142	C₈×C₇₁	C₄×Q₈	1	4	yes	—
143	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₇₁	#547	1	1	yes	characteristic
145	C₂²×C₇₁	D₄×Q₈	1	1	yes	characteristic
146	C₂²×C₇₁	#575	1	1	yes	characteristic
147	C₂²×C₇₁	#575	1	1	yes	characteristic
148	C₂²×C₇₁	#576	1	1	yes	characteristic
149	C₂²×C₇₁	#578	1	1	yes	characteristic
150	C₄×C₇₁	—	2	2	no	—
151	C₄×C₇₁	—	2	2	no	—
152	C₄×C₇₁	—	4	2	no	—
153	C₄×C₇₁	—	4	2	no	—
154	C₄×C₇₁	—	4	4	no	—
155	C₄×C₇₁	—	4	4	no	—
156	C₄×C₇₁	—	4	4	no	—
157	C₄×C₇₁	—	8	2	no	—
158	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₄×C₇₁	#519	1	1	yes	characteristic
160	C₄×C₇₁	#556	1	1	yes	characteristic
161	C₄×C₇₁	#557	1	1	yes	characteristic
162	#23165	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₇₁	#2365	1	1	yes	characteristic
164	C₂×C₇₁	#2943	1	1	yes	characteristic
165	C₂×C₇₁	#2951	1	1	yes	characteristic
166	C₂×C₇₁	#3352	1	1	yes	characteristic
167	C₂×C₇₁	#3355	1	1	yes	characteristic
168	C₂×C₇₁	#3389	1	1	yes	characteristic
169	C₂×C₇₁	#3390	1	1	yes	characteristic
170	C₂×#499	C₂×C₇₁	1	1	yes	characteristic
171	#2370	C₂×C₇₁	1	2	yes	—
172	#1989	C₂×C₇₁	1	2	yes	—
173	#1944	C₂×C₇₁	1	2	yes	—
174	#1921	C₂×C₇₁	1	2	yes	—
175	#1918	C₂×C₇₁	1	2	yes	—
176	#1776	C₂×C₇₁	1	2	yes	—
177	C₄×#107	C₂×C₇₁	1	1	yes	characteristic
178	#1770	C₂×C₇₁	1	1	yes	characteristic
179	C₇₁	#23165	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×#125	C₂²×C₇₁	1	2	yes	—
181	#499	C₄×C₇₁	1	8	yes	complemented
182	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
183	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
184	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
185	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
186	C₂×#106	C₂²×C₇₁	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
188	C₂×#97	C₂²×C₇₁	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
190	#389	—	2	2	no	—
191	#389	C₂²×C₇₁	1	2	yes	—
192	#389	C₂²×C₇₁	1	2	yes	—
193	#387	—	2	2	no	—
194	#387	—	2	2	no	—
195	C₄×C₄⋊C₄	—	2	2	no	—
196	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
197	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
198	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
199	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
200	#340	C₂²×C₇₁	1	2	yes	—
201	#340	C₂²×C₇₁	1	2	yes	—
202	#340	C₂²×C₇₁	1	2	yes	—
203	#340	C₂²×C₇₁	1	2	yes	—
204	#340	C₂²×C₇₁	1	2	yes	—
205	#336	C₂²×C₇₁	1	2	yes	—
206	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
207	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
209	#125	—	2	4	no	—
210	#125	C₂×C₄×C₇₁	1	8	yes	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
216	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
225	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	C₂×C₄×C₇₁	1	4	yes	—
228	#107	C₂×C₄×C₇₁	1	4	yes	—
229	#107	Q₈×C₇₁	1	4	yes	—
230	#106	C₂×C₄×C₇₁	1	8	yes	—
231	C₄⋊C₈	—	2	2	no	—
232	#97	—	2	4	no	—
233	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	D₄×C₇₁	1	2	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
255	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
256	C₂×C₈	C₂×Q₈×C₇₁	1	2	yes	—
257	C₂×C₈	#41×C₇₁	1	2	yes	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	8	2	no	—
269	C₄⋊C₄	C₂²×C₄×C₇₁	1	4	yes	—
270	C₄⋊C₄	C₂²×C₄×C₇₁	1	4	yes	—
271	C₄⋊C₄	C₂×Q₈×C₇₁	1	4	yes	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	4	no	—
275	C₄²	—	8	2	no	—
276	C₄²	—	8	2	no	—
277	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
278	C₄²	#41×C₇₁	1	2	yes	—
279	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
303	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
304	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
308	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
309	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₄×Q₈×C₇₁	1	4	yes	—
313	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
314	C₂²	#547×C₇₁	1	1	yes	characteristic
315	C₂²	D₄×Q₈×C₇₁	1	1	yes	characteristic
316	C₂²	#575×C₇₁	1	1	yes	characteristic
317	C₂²	#575×C₇₁	1	1	yes	characteristic
318	C₂²	#576×C₇₁	1	1	yes	characteristic
319	C₂²	#578×C₇₁	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	2	no	—
328	C₄	C₂×C₄×Q₈×C₇₁	1	1	yes	characteristic
329	C₄	#519×C₇₁	1	1	yes	characteristic
330	C₄	#556×C₇₁	1	1	yes	characteristic
331	C₄	#557×C₇₁	1	1	yes	characteristic
332	C₂	C₇₁×#2365	1	1	yes	characteristic
333	C₂	C₇₁×#2943	1	1	yes	characteristic
334	C₂	C₇₁×#2951	1	1	yes	characteristic
335	C₂	C₇₁×#3352	1	1	yes	characteristic
336	C₂	C₇₁×#3355	1	1	yes	characteristic
337	C₂	C₇₁×#3389	1	1	yes	characteristic
338	C₂	C₇₁×#3390	1	1	yes	characteristic

	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
11. #499×C₇₁	C₄	semidirect
162. #23165	C₇₁	direct (N × H)
179. C₇₁	#23165	direct (N × H)
181. #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×#1944	2
C₇₁×#1921	2
C₇₁×#1918	2
C₇₁×#1776	2
C₄×#107×C₇₁	2
C₇₁×#1770	2
#23165	71

Subgroups — C71×#23165 #31844037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23165 #31844037