Subgroups — C₇₁×#8257 #31829129

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1540	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1502	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₇₁	—	2	1	no	—
9	#532×C₇₁	C₂²	1	1	yes	characteristic
10	#436×C₇₁	—	2	1	no	—
11	#436×C₇₁	C₄	1	2	yes	complemented
12	C₈×D₄×C₇₁	—	2	1	no	—
13	C₈×D₄×C₇₁	C₄	1	2	yes	complemented
14	#431×C₇₁	C₂²	1	1	yes	characteristic
15	#431×C₇₁	C₂²	1	1	yes	characteristic
16	#423×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#408×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	—	2	1	no	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	#336×C₇₁	—	2	1	no	—
23	#332×C₇₁	C₄	1	2	yes	complemented
24	#331×C₇₁	C₄	1	2	yes	complemented
25	#327×C₇₁	C₄	1	4	yes	—
26	#322×C₇₁	—	2	1	no	—
27	#322×C₇₁	—	2	1	no	—
28	C₂×#41×C₇₁	—	2	1	no	—
29	#131×C₇₁	—	4	2	no	—
30	#131×C₇₁	—	4	2	no	—
31	C₂×M₁₆×C₇₁	—	2	1	no	—
32	C₂×M₁₆×C₇₁	—	2	2	no	—
33	C₂²×C₈×C₇₁	—	2	1	no	—
34	C₂²×C₈×C₇₁	—	2	2	no	—
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	#127×C₇₁	C₂×C₄	1	1	yes	characteristic
39	#124×C₇₁	—	2	1	no	—
40	#121×C₇₁	—	2	1	no	—
41	#121×C₇₁	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₇₁	—	2	1	no	—
43	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
45	C₂×C₄²×C₇₁	—	2	1	no	—
46	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₁	—	2	1	no	—
48	C₄⋊C₈×C₇₁	—	2	1	no	—
49	C₄⋊C₈×C₇₁	—	2	1	no	—
50	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
51	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
53	C₂²⋊C₈×C₇₁	—	2	1	no	—
54	C₂²⋊C₈×C₇₁	—	2	2	no	—
55	C₂²⋊C₈×C₇₁	—	2	2	no	—
56	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₇₁	—	2	1	no	—
58	C₄×C₈×C₇₁	—	2	1	no	—
59	C₄×C₈×C₇₁	—	4	2	no	—
60	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
64	#95×C₇₁	C₂×C₄	1	2	yes	—
65	#41×C₇₁	—	4	1	no	—
66	#41×C₇₁	—	4	1	no	—
67	#41×C₇₁	—	4	1	no	—
68	#41×C₇₁	—	4	1	no	—
69	C₂×Q₈×C₇₁	—	2	1	no	—
70	C₂×D₄×C₇₁	—	2	2	no	—
71	C₂×D₄×C₇₁	—	2	1	no	—
72	C₂×D₄×C₇₁	—	2	1	no	—
73	C₂×D₄×C₇₁	C₄²	1	2	yes	—
74	C₂²×C₄×C₇₁	—	2	2	no	—
75	C₂²×C₄×C₇₁	—	2	2	no	—
76	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
79	M₁₆×C₇₁	—	4	4	no	—
80	M₁₆×C₇₁	—	4	1	no	—
81	M₁₆×C₇₁	—	4	1	no	—
82	C₂×C₈×C₇₁	—	2	1	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	—	2	1	no	—
86	C₂×C₈×C₇₁	—	2	2	no	—
87	C₂×C₈×C₇₁	—	2	2	no	—
88	C₂×C₈×C₇₁	—	2	2	no	—
89	C₂×C₈×C₇₁	—	2	2	no	—
90	C₂×C₈×C₇₁	—	4	4	no	—
91	C₂×C₈×C₇₁	—	4	1	no	—
92	C₂×C₈×C₇₁	—	4	1	no	—
93	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₇₁	—	2	4	no	—
98	C₄⋊C₄×C₇₁	—	2	1	no	—
99	C₄⋊C₄×C₇₁	C₄²	1	2	yes	—
100	C₂²⋊C₄×C₇₁	—	2	2	no	—
101	C₂²⋊C₄×C₇₁	—	2	2	no	—
102	C₄²×C₇₁	—	4	1	no	—
103	C₄²×C₇₁	—	4	1	no	—
104	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₁	#41	1	1	yes	characteristic
108	C₂³×C₇₁	—	2	2	no	—
109	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
110	Q₈×C₇₁	—	4	1	no	—
111	Q₈×C₇₁	—	4	1	no	—
112	D₄×C₇₁	—	4	2	no	—
113	D₄×C₇₁	—	4	2	no	—
114	D₄×C₇₁	—	4	2	no	—
115	D₄×C₇₁	—	4	1	no	—
116	D₄×C₇₁	—	4	1	no	—
117	D₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	4	no	—
120	C₂×C₄×C₇₁	—	2	2	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	—	4	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	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	—	4	2	no	—
141	C₈×C₇₁	—	4	2	no	—
142	C₈×C₇₁	—	4	2	no	—
143	C₈×C₇₁	—	4	2	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	2	1	no	—
146	C₂²×C₇₁	—	4	2	no	—
147	C₂²×C₇₁	—	4	2	no	—
148	C₂²×C₇₁	—	4	2	no	—
149	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₇₁	#440	1	1	yes	characteristic
151	C₂²×C₇₁	#442	1	1	yes	characteristic
152	C₄×C₇₁	—	2	1	no	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	4	4	no	—
156	C₄×C₇₁	—	4	2	no	—
157	C₄×C₇₁	#417	1	1	yes	characteristic
158	C₄×C₇₁	#418	1	1	yes	characteristic
159	C₄×C₇₁	#440	1	1	yes	characteristic
160	C₄×C₇₁	#442	1	1	yes	characteristic
161	C₄×C₇₁	#443	1	2	yes	—
162	#8257	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₇₁	—	2	1	no	—
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	—	8	2	no	—
166	C₂×C₇₁	#1757	1	1	yes	characteristic
167	C₂×C₇₁	#1762	1	1	yes	characteristic
168	C₂×C₇₁	#1764	1	1	yes	characteristic
169	#2931	C₂×C₇₁	1	1	yes	characteristic
170	C₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic, complemented
171	#1540	C₂×C₇₁	1	1	yes	characteristic
172	#1502	C₂×C₇₁	1	1	yes	characteristic
173	#1452	C₂×C₇₁	1	1	yes	characteristic, complemented
174	#1303	C₂×C₇₁	1	1	yes	characteristic, complemented
175	#1301	C₂×C₇₁	1	1	yes	characteristic, complemented
176	C₇₁	#8257	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×#131	—	2	1	no	—
178	#532	C₂²×C₇₁	1	1	yes	characteristic
179	#436	—	2	1	no	—
180	#436	C₄×C₇₁	1	2	yes	complemented
181	C₈×D₄	—	2	1	no	—
182	C₈×D₄	C₄×C₇₁	1	2	yes	complemented
183	#431	C₂²×C₇₁	1	1	yes	characteristic
184	#431	C₂²×C₇₁	1	1	yes	characteristic
185	#423	C₂²×C₇₁	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
187	#408	—	2	1	no	—
188	C₂×C₄×C₈	—	2	1	no	—
189	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
191	#336	—	2	1	no	—
192	#332	C₄×C₇₁	1	2	yes	complemented
193	#331	C₄×C₇₁	1	2	yes	complemented
194	#327	C₄×C₇₁	1	4	yes	—
195	#322	—	2	1	no	—
196	#322	—	2	1	no	—
197	C₂×#41	—	2	1	no	—
198	#131	—	4	2	no	—
199	#131	—	4	2	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
206	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
207	#127	C₂×C₄×C₇₁	1	1	yes	characteristic
208	#124	—	2	1	no	—
209	#121	—	2	1	no	—
210	#121	C₂×C₄×C₇₁	1	2	yes	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	C₂×C₄×C₇₁	1	2	yes	—
220	C₄⋊C₈	C₂×C₄×C₇₁	1	2	yes	—
221	C₄⋊C₈	C₂×C₄×C₇₁	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	C₂×C₄×C₇₁	1	2	yes	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	4	2	no	—
229	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
231	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
232	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
233	#95	C₂×C₄×C₇₁	1	2	yes	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	#41	—	4	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄²×C₇₁	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
247	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
248	M₁₆	—	4	4	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	C₄²×C₇₁	1	2	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
274	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
276	C₄²	#41×C₇₁	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₄×D₄×C₇₁	1	1	yes	characteristic
279	Q₈	—	4	1	no	—
280	Q₈	—	4	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₂×C₄²×C₇₁	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
297	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
301	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
302	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
303	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
304	C₂×C₄	#137×C₇₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₄×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
319	C₂²	#440×C₇₁	1	1	yes	characteristic
320	C₂²	#442×C₇₁	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	2	no	—
326	C₄	#417×C₇₁	1	1	yes	characteristic
327	C₄	#418×C₇₁	1	1	yes	characteristic
328	C₄	#440×C₇₁	1	1	yes	characteristic
329	C₄	#442×C₇₁	1	1	yes	characteristic
330	C₄	#443×C₇₁	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₇₁×#1757	1	1	yes	characteristic
335	C₂	C₇₁×#1762	1	1	yes	characteristic
336	C₂	C₇₁×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	C₄×C₂²⋊C₄
prime radical	#532×C₇₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×C₄⋊C₄×C₇₁	C₂	semidirect
5. C₇₁×#1452	C₂	semidirect
6. C₇₁×#1303	C₂	semidirect
7. C₇₁×#1301	C₂	semidirect
11. #436×C₇₁	C₄	semidirect
13. C₈×D₄×C₇₁	C₄	semidirect
23. #332×C₇₁	C₄	semidirect
24. #331×C₇₁	C₄	semidirect
162. #8257	C₇₁	direct (N × H)
170. C₈×C₄⋊C₄	C₂×C₇₁	semidirect
173. #1452	C₂×C₇₁	semidirect
174. #1303	C₂×C₇₁	semidirect
175. #1301	C₂×C₇₁	semidirect
176. C₇₁	#8257	direct (N × H)
180. #436	C₄×C₇₁	semidirect
182. C₈×D₄	C₄×C₇₁	semidirect
192. #332	C₄×C₇₁	semidirect
193. #331	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#2931	2
C₈×C₄⋊C₄×C₇₁	2
C₇₁×#1540	2
C₇₁×#1502	2
C₇₁×#1452	2
C₇₁×#1303	2
C₇₁×#1301	2
#8257	71

Subgroups — C71×#8257 #31829129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#8257 #31829129