Subgroups — C₇₁×#60712 #31881584

Order 18176.

1950 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#3508	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#3182	C₂	1	4	yes	maximal, complemented
4	C₇₁×#3174	C₂	1	8	yes	maximal, complemented
5	C₇₁×#3167	C₂	1	4	yes	maximal, complemented
6	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#492×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₁×#3057	C₂	1	4	yes	maximal, complemented
11	C₇₁×#2949	C₂	1	4	yes	maximal, complemented
12	C₄×#131×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×SD₁₆×C₇₁	C₂²	1	4	yes	—
14	C₂×#131×C₇₁	C₂²	1	2	yes	—
15	#559×C₇₁	—	2	2	no	—
16	#557×C₇₁	—	2	2	no	—
17	#556×C₇₁	—	2	2	no	—
18	#552×C₇₁	—	2	4	no	—
19	#550×C₇₁	—	2	2	no	—
20	#546×C₇₁	—	2	4	no	—
21	#541×C₇₁	C₂²	1	4	yes	complemented
22	#540×C₇₁	C₂²	1	4	yes	—
23	#536×C₇₁	C₂²	1	2	yes	—
24	#535×C₇₁	C₂²	1	2	yes	—
25	#533×C₇₁	C₂²	1	1	yes	characteristic
26	#533×C₇₁	C₂²	1	1	yes	characteristic
27	#533×C₇₁	C₂²	1	4	yes	complemented
28	#532×C₇₁	C₂²	1	4	yes	—
29	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
30	C₂×#127×C₇₁	C₂²	1	1	yes	characteristic
31	#519×C₇₁	C₂²	1	2	yes	—
32	#518×C₇₁	C₂²	1	2	yes	—
33	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
34	#500×C₇₁	C₂²	1	2	yes	complemented
35	#499×C₇₁	C₂²	1	4	yes	complemented
36	#498×C₇₁	C₂²	1	2	yes	complemented
37	#492×C₇₁	C₂²	1	2	yes	—
38	#492×C₇₁	C₂²	1	4	yes	—
39	#492×C₇₁	C₂²	1	2	yes	—
40	#491×C₇₁	C₂²	1	4	yes	complemented
41	#490×C₇₁	C₂²	1	4	yes	—
42	#487×C₇₁	C₂²	1	4	yes	complemented
43	#486×C₇₁	C₂²	1	4	yes	—
44	#468×C₇₁	C₂²	1	16	yes	complemented
45	#465×C₇₁	C₂²	1	16	yes	complemented
46	#440×C₇₁	C₂²	1	16	yes	complemented
47	C₄×SD₁₆×C₇₁	C₂²	1	16	yes	complemented
48	#427×C₇₁	C₂²	1	4	yes	—
49	#419×C₇₁	C₂²	1	4	yes	—
50	#418×C₇₁	C₂²	1	4	yes	—
51	#405×C₇₁	C₂²	1	2	yes	—
52	#405×C₇₁	C₂²	1	4	yes	complemented
53	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×M₁₆×C₇₁	C₂²	1	2	yes	complemented
55	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	complemented
57	C₂×#41×C₇₁	—	2	4	no	—
58	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₇₁	—	2	2	no	—
61	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₇₁	—	2	16	no	—
63	C₂×SD₁₆×C₇₁	C₂³	1	8	yes	—
64	C₂×SD₁₆×C₇₁	C₂³	1	8	yes	—
65	#131×C₇₁	—	2	8	no	—
66	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
68	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
69	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₇₁	—	2	2	no	—
71	C₄⋊Q₈×C₇₁	—	2	2	no	—
72	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
76	#127×C₇₁	—	2	2	no	—
77	#127×C₇₁	—	2	2	no	—
78	#127×C₇₁	C₂³	1	2	yes	—
79	#127×C₇₁	C₂³	1	2	yes	—
80	#126×C₇₁	—	2	8	no	—
81	#125×C₇₁	—	2	8	no	—
82	#125×C₇₁	—	2	4	no	—
83	#125×C₇₁	C₂³	1	2	yes	—
84	#125×C₇₁	C₂³	1	2	yes	—
85	#124×C₇₁	—	2	2	no	—
86	#124×C₇₁	—	2	2	no	—
87	#124×C₇₁	—	2	4	no	—
88	#123×C₇₁	—	2	4	no	—
89	#123×C₇₁	—	2	8	no	—
90	C₂²⋊Q₈×C₇₁	—	2	2	no	—
91	C₂²⋊Q₈×C₇₁	—	2	2	no	—
92	C₂²⋊Q₈×C₇₁	—	2	4	no	—
93	C₂²⋊Q₈×C₇₁	C₂³	1	8	yes	—
94	#121×C₇₁	—	2	4	no	—
95	#121×C₇₁	—	2	8	no	—
96	#121×C₇₁	C₂³	1	8	yes	—
97	#120×C₇₁	—	2	4	no	—
98	C₄×Q₈×C₇₁	—	2	4	no	—
99	C₄×Q₈×C₇₁	—	2	2	no	—
100	C₄×Q₈×C₇₁	—	2	2	no	—
101	C₄×Q₈×C₇₁	—	2	4	no	—
102	C₄×Q₈×C₇₁	D₄	1	2	yes	—
103	C₄×Q₈×C₇₁	C₂³	1	8	yes	—
104	C₄×D₄×C₇₁	—	2	4	no	—
105	C₄×D₄×C₇₁	—	2	4	no	—
106	C₄×D₄×C₇₁	D₄	1	4	yes	—
107	C₄×D₄×C₇₁	D₄	1	2	yes	—
108	C₄×D₄×C₇₁	C₂³	1	8	yes	—
109	#117×C₇₁	—	2	4	no	—
110	#117×C₇₁	—	2	4	no	—
111	#117×C₇₁	C₂³	1	2	yes	—
112	#117×C₇₁	C₂³	1	2	yes	—
113	#117×C₇₁	C₂³	1	1	yes	characteristic
114	#117×C₇₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
117	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
119	#107×C₇₁	C₂³	1	8	yes	—
120	#106×C₇₁	C₂³	1	8	yes	—
121	#103×C₇₁	C₂³	1	16	yes	—
122	#102×C₇₁	C₂³	1	16	yes	—
123	#97×C₇₁	C₂³	1	4	yes	—
124	#97×C₇₁	C₂³	1	2	yes	—
125	C₄×C₈×C₇₁	C₂³	1	4	yes	—
126	C₄×C₈×C₇₁	C₂³	1	2	yes	—
127	C₄×C₈×C₇₁	C₂³	1	2	yes	—
128	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
130	C₂⁴×C₇₁	—	2	2	no	—
131	#41×C₇₁	—	2	4	no	—
132	#41×C₇₁	—	8	8	no	—
133	C₂×Q₈×C₇₁	—	2	2	no	—
134	C₂×Q₈×C₇₁	—	2	2	no	—
135	C₂×Q₈×C₇₁	—	2	2	no	—
136	C₂×Q₈×C₇₁	—	2	8	no	—
137	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₇₁	C₂⁴	1	4	yes	—
139	C₂×D₄×C₇₁	—	2	8	no	—
140	C₂×D₄×C₇₁	—	2	2	no	—
141	C₂×D₄×C₇₁	—	2	4	no	—
142	C₂×D₄×C₇₁	—	2	4	no	—
143	C₂×D₄×C₇₁	—	2	8	no	—
144	C₂×D₄×C₇₁	—	8	4	no	—
145	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₇₁	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₇₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₇₁	—	2	2	no	—
149	C₂²×C₄×C₇₁	—	2	8	no	—
150	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
154	SD₁₆×C₇₁	—	2	16	no	—
155	SD₁₆×C₇₁	—	2	16	no	—
156	M₁₆×C₇₁	—	2	4	no	—
157	M₁₆×C₇₁	—	2	8	no	—
158	C₂×C₈×C₇₁	—	2	8	no	—
159	C₂×C₈×C₇₁	#41	1	8	yes	complemented
160	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
161	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
162	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₇₁	—	2	4	no	—
164	C₄⋊C₄×C₇₁	—	2	4	no	—
165	C₄⋊C₄×C₇₁	—	2	2	no	—
166	C₄⋊C₄×C₇₁	—	2	8	no	—
167	C₄⋊C₄×C₇₁	—	2	2	no	—
168	C₄⋊C₄×C₇₁	—	2	4	no	—
169	C₄⋊C₄×C₇₁	—	2	4	no	—
170	C₄⋊C₄×C₇₁	—	2	8	no	—
171	C₄⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
172	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
173	C₄⋊C₄×C₇₁	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₇₁	C₂⁴	1	4	yes	—
175	C₂²⋊C₄×C₇₁	—	2	2	no	—
176	C₂²⋊C₄×C₇₁	—	2	2	no	—
177	C₂²⋊C₄×C₇₁	—	2	8	no	—
178	C₂²⋊C₄×C₇₁	—	2	8	no	—
179	C₂²⋊C₄×C₇₁	—	2	8	no	—
180	C₂²⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
182	C₄²×C₇₁	—	2	8	no	—
183	C₄²×C₇₁	—	2	2	no	—
184	C₄²×C₇₁	—	2	2	no	—
185	C₄²×C₇₁	C₂×D₄	1	4	yes	—
186	C₄²×C₇₁	C₂×D₄	1	2	yes	—
187	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
188	C₄²×C₇₁	C₂⁴	1	2	yes	—
189	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
190	C₂³×C₇₁	—	2	4	no	—
191	C₂³×C₇₁	—	4	4	no	—
192	C₂³×C₇₁	—	8	4	no	—
193	C₂³×C₇₁	C₂²×D₄	1	2	yes	—
194	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₇₁	—	2	2	no	—
196	Q₈×C₇₁	—	2	8	no	—
197	Q₈×C₇₁	—	4	4	no	—
198	Q₈×C₇₁	—	4	4	no	—
199	Q₈×C₇₁	—	4	4	no	—
200	D₄×C₇₁	—	2	4	no	—
201	D₄×C₇₁	—	2	2	no	—
202	D₄×C₇₁	—	2	8	no	—
203	D₄×C₇₁	—	4	4	no	—
204	D₄×C₇₁	—	4	8	no	—
205	D₄×C₇₁	—	8	4	no	—
206	D₄×C₇₁	—	8	8	no	—
207	C₂×C₄×C₇₁	—	2	4	no	—
208	C₂×C₄×C₇₁	—	2	8	no	—
209	C₂×C₄×C₇₁	—	2	4	no	—
210	C₂×C₄×C₇₁	—	2	4	no	—
211	C₂×C₄×C₇₁	—	2	4	no	—
212	C₂×C₄×C₇₁	—	2	4	no	—
213	C₂×C₄×C₇₁	—	4	4	no	—
214	C₂×C₄×C₇₁	—	8	8	no	—
215	C₂×C₄×C₇₁	C₂²×D₄	1	4	yes	—
216	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₇₁	C₂×#41	1	4	yes	—
221	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
222	C₈×C₇₁	—	2	4	no	—
223	C₈×C₇₁	C₂×#41	1	4	yes	—
224	C₈×C₇₁	C₂×#41	1	4	yes	—
225	C₂²×C₇₁	—	2	2	no	—
226	C₂²×C₇₁	—	2	2	no	—
227	C₂²×C₇₁	—	2	4	no	—
228	C₂²×C₇₁	—	4	4	no	—
229	C₂²×C₇₁	—	4	8	no	—
230	C₂²×C₇₁	—	16	4	no	—
231	C₂²×C₇₁	#532	1	2	yes	—
232	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
233	C₄×C₇₁	—	2	2	no	—
234	C₄×C₇₁	—	2	4	no	—
235	C₄×C₇₁	—	2	2	no	—
236	C₄×C₇₁	—	4	4	no	—
237	C₄×C₇₁	—	4	4	no	—
238	C₄×C₇₁	—	4	4	no	—
239	C₄×C₇₁	#532	1	2	yes	—
240	C₄×C₇₁	#577	1	4	yes	—
241	C₄×C₇₁	C₂²×#41	1	2	yes	—
242	#60712	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₇₁	—	2	2	no	—
244	C₂×C₇₁	—	4	2	no	—
245	C₂×C₇₁	—	8	4	no	—
246	C₂×C₇₁	C₂×#532	1	1	yes	characteristic
247	C₂×C₇₁	C₂×#577	1	2	yes	—
248	#3509	C₂×C₇₁	1	1	yes	characteristic, complemented
249	#3508	C₂×C₇₁	1	1	yes	characteristic
250	#3182	C₂×C₇₁	1	4	yes	complemented
251	#3174	C₂×C₇₁	1	8	yes	complemented
252	#3167	C₂×C₇₁	1	4	yes	complemented
253	#3160	C₂×C₇₁	1	1	yes	characteristic, complemented
254	C₂×#492	C₂×C₇₁	1	1	yes	characteristic, complemented
255	#3139	C₂×C₇₁	1	1	yes	characteristic, complemented
256	#3138	C₂×C₇₁	1	1	yes	characteristic, complemented
257	#3057	C₂×C₇₁	1	4	yes	complemented
258	#2949	C₂×C₇₁	1	4	yes	complemented
259	C₄×#131	C₂×C₇₁	1	1	yes	characteristic, complemented
260	C₇₁	#60712	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²×SD₁₆	C₂²×C₇₁	1	4	yes	—
262	C₂×#131	C₂²×C₇₁	1	2	yes	—
263	#559	—	2	2	no	—
264	#557	—	2	2	no	—
265	#556	—	2	2	no	—
266	#552	—	2	4	no	—
267	#550	—	2	2	no	—
268	#546	—	2	4	no	—
269	#541	C₂²×C₇₁	1	4	yes	complemented
270	#540	C₂²×C₇₁	1	4	yes	—
271	#536	C₂²×C₇₁	1	2	yes	—
272	#535	C₂²×C₇₁	1	2	yes	—
273	#533	C₂²×C₇₁	1	1	yes	characteristic
274	#533	C₂²×C₇₁	1	1	yes	characteristic
275	#533	C₂²×C₇₁	1	4	yes	complemented
276	#532	C₂²×C₇₁	1	4	yes	—
277	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
278	C₂×#127	C₂²×C₇₁	1	1	yes	characteristic
279	#519	C₂²×C₇₁	1	2	yes	—
280	#518	C₂²×C₇₁	1	2	yes	—
281	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
282	#500	C₂²×C₇₁	1	2	yes	complemented
283	#499	C₂²×C₇₁	1	4	yes	complemented
284	#498	C₂²×C₇₁	1	2	yes	complemented
285	#492	C₂²×C₇₁	1	2	yes	—
286	#492	C₂²×C₇₁	1	4	yes	—
287	#492	C₂²×C₇₁	1	2	yes	—
288	#491	C₂²×C₇₁	1	4	yes	complemented
289	#490	C₂²×C₇₁	1	4	yes	—
290	#487	C₂²×C₇₁	1	4	yes	complemented
291	#486	C₂²×C₇₁	1	4	yes	—
292	#468	C₂²×C₇₁	1	16	yes	complemented
293	#465	C₂²×C₇₁	1	16	yes	complemented
294	#440	C₂²×C₇₁	1	16	yes	complemented
295	C₄×SD₁₆	C₂²×C₇₁	1	16	yes	complemented
296	#427	C₂²×C₇₁	1	4	yes	—
297	#419	C₂²×C₇₁	1	4	yes	—
298	#418	C₂²×C₇₁	1	4	yes	—
299	#405	C₂²×C₇₁	1	2	yes	—
300	#405	C₂²×C₇₁	1	4	yes	complemented
301	C₄×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
302	C₄×M₁₆	C₂²×C₇₁	1	2	yes	complemented
303	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
304	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	complemented
305	C₂×#41	—	2	4	no	—
306	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₇₁	1	2	yes	—
308	C₂²×D₄	—	2	2	no	—
309	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
310	C₂×SD₁₆	—	2	16	no	—
311	C₂×SD₁₆	C₂³×C₇₁	1	8	yes	—
312	C₂×SD₁₆	C₂³×C₇₁	1	8	yes	—
313	#131	—	2	8	no	—
314	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
315	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
316	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
317	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
322	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
323	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
324	#127	—	2	2	no	—
325	#127	—	2	2	no	—
326	#127	C₂³×C₇₁	1	2	yes	—
327	#127	C₂³×C₇₁	1	2	yes	—
328	#126	—	2	8	no	—
329	#125	—	2	8	no	—
330	#125	—	2	4	no	—
331	#125	C₂³×C₇₁	1	2	yes	—
332	#125	C₂³×C₇₁	1	2	yes	—
333	#124	—	2	2	no	—
334	#124	—	2	2	no	—
335	#124	—	2	4	no	—
336	#123	—	2	4	no	—
337	#123	—	2	8	no	—
338	C₂²⋊Q₈	—	2	2	no	—
339	C₂²⋊Q₈	—	2	2	no	—
340	C₂²⋊Q₈	—	2	4	no	—
341	C₂²⋊Q₈	C₂³×C₇₁	1	8	yes	—
342	#121	—	2	4	no	—
343	#121	—	2	8	no	—
344	#121	C₂³×C₇₁	1	8	yes	—
345	#120	—	2	4	no	—
346	C₄×Q₈	—	2	4	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	4	no	—
350	C₄×Q₈	D₄×C₇₁	1	2	yes	—
351	C₄×Q₈	C₂³×C₇₁	1	8	yes	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	—	2	4	no	—
354	C₄×D₄	D₄×C₇₁	1	4	yes	—
355	C₄×D₄	D₄×C₇₁	1	2	yes	—
356	C₄×D₄	C₂³×C₇₁	1	8	yes	—
357	#117	—	2	4	no	—
358	#117	—	2	4	no	—
359	#117	C₂³×C₇₁	1	2	yes	—
360	#117	C₂³×C₇₁	1	2	yes	—
361	#117	C₂³×C₇₁	1	1	yes	characteristic
362	#117	C₂³×C₇₁	1	2	yes	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	—	2	4	no	—
365	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
367	#107	C₂³×C₇₁	1	8	yes	—
368	#106	C₂³×C₇₁	1	8	yes	—
369	#103	C₂³×C₇₁	1	16	yes	—
370	#102	C₂³×C₇₁	1	16	yes	—
371	#97	C₂³×C₇₁	1	2	yes	—
372	#97	C₂³×C₇₁	1	4	yes	—
373	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
374	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
375	C₄×C₈	C₂³×C₇₁	1	2	yes	—
376	C₄×C₈	C₂³×C₇₁	1	2	yes	—
377	C₄×C₈	C₂³×C₇₁	1	4	yes	—
378	C₂⁴	—	2	2	no	—
379	#41	—	2	4	no	—
380	#41	—	8	8	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	8	no	—
385	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
386	C₂×Q₈	C₂⁴×C₇₁	1	4	yes	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	8	4	no	—
393	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
394	C₂×D₄	C₂×D₄×C₇₁	1	1	yes	characteristic
395	C₂×D₄	C₂⁴×C₇₁	1	4	yes	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	8	no	—
398	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
402	SD₁₆	—	2	16	no	—
403	SD₁₆	—	2	16	no	—
404	M₁₆	—	2	4	no	—
405	M₁₆	—	2	8	no	—
406	C₂×C₈	—	2	8	no	—
407	C₂×C₈	#41×C₇₁	1	8	yes	complemented
408	C₂×C₈	C₂⁴×C₇₁	1	4	yes	—
409	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
410	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
420	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
421	C₄⋊C₄	C₂⁴×C₇₁	1	4	yes	—
422	C₄⋊C₄	C₂⁴×C₇₁	1	4	yes	—
423	C₂²⋊C₄	—	2	2	no	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	8	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
429	C₂²⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
430	C₄²	—	2	8	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	2	no	—
433	C₄²	C₂×D₄×C₇₁	1	4	yes	—
434	C₄²	C₂×D₄×C₇₁	1	2	yes	—
435	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
436	C₄²	C₂⁴×C₇₁	1	2	yes	—
437	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
438	C₂³	—	2	4	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	8	4	no	—
441	C₂³	C₂²×D₄×C₇₁	1	2	yes	—
442	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
443	Q₈	—	2	2	no	—
444	Q₈	—	2	8	no	—
445	Q₈	—	4	4	no	—
446	Q₈	—	4	4	no	—
447	Q₈	—	4	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	2	2	no	—
450	D₄	—	2	8	no	—
451	D₄	—	4	4	no	—
452	D₄	—	4	8	no	—
453	D₄	—	8	4	no	—
454	D₄	—	8	8	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₂²×D₄×C₇₁	1	4	yes	—
464	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
468	C₂×C₄	C₂×#41×C₇₁	1	4	yes	—
469	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
470	C₈	—	2	4	no	—
471	C₈	C₂×#41×C₇₁	1	4	yes	—
472	C₈	C₂×#41×C₇₁	1	4	yes	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	8	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	16	4	no	—
479	C₂²	#532×C₇₁	1	2	yes	—
480	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	#532×C₇₁	1	2	yes	—
488	C₄	#577×C₇₁	1	4	yes	—
489	C₄	C₂²×#41×C₇₁	1	2	yes	—
490	C₂	—	2	2	no	—
491	C₂	—	4	2	no	—
492	C₂	—	8	4	no	—
493	C₂	C₂×#532×C₇₁	1	1	yes	characteristic
494	C₂	C₂×#577×C₇₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
prime radical	C₇₁×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₇₁×#3509	C₂	semidirect
3. C₇₁×#3182	C₂	semidirect
4. C₇₁×#3174	C₂	semidirect
5. C₇₁×#3167	C₂	semidirect
6. C₇₁×#3160	C₂	semidirect
7. C₂×#492×C₇₁	C₂	semidirect
8. C₇₁×#3139	C₂	semidirect
9. C₇₁×#3138	C₂	semidirect
10. C₇₁×#3057	C₂	semidirect
11. C₇₁×#2949	C₂	semidirect
12. C₄×#131×C₇₁	C₂	semidirect
21. #541×C₇₁	C₂²	semidirect
27. #533×C₇₁	C₂²	semidirect
34. #500×C₇₁	C₂²	semidirect
35. #499×C₇₁	C₂²	semidirect
36. #498×C₇₁	C₂²	semidirect
40. #491×C₇₁	C₂²	semidirect
42. #487×C₇₁	C₂²	semidirect
44. #468×C₇₁	C₂²	semidirect
45. #465×C₇₁	C₂²	semidirect
46. #440×C₇₁	C₂²	semidirect
47. C₄×SD₁₆×C₇₁	C₂²	semidirect
52. #405×C₇₁	C₂²	semidirect
54. C₄×M₁₆×C₇₁	C₂²	semidirect
56. C₂×C₄×C₈×C₇₁	C₂²	semidirect
159. C₂×C₈×C₇₁	#41	semidirect
242. #60712	C₇₁	direct (N × H)
248. #3509	C₂×C₇₁	semidirect
250. #3182	C₂×C₇₁	semidirect
251. #3174	C₂×C₇₁	semidirect
252. #3167	C₂×C₇₁	semidirect
253. #3160	C₂×C₇₁	semidirect
254. C₂×#492	C₂×C₇₁	semidirect
255. #3139	C₂×C₇₁	semidirect
256. #3138	C₂×C₇₁	semidirect
257. #3057	C₂×C₇₁	semidirect
258. #2949	C₂×C₇₁	semidirect
259. C₄×#131	C₂×C₇₁	semidirect
260. C₇₁	#60712	direct (N × H)
269. #541	C₂²×C₇₁	semidirect
275. #533	C₂²×C₇₁	semidirect
282. #500	C₂²×C₇₁	semidirect
283. #499	C₂²×C₇₁	semidirect
284. #498	C₂²×C₇₁	semidirect
288. #491	C₂²×C₇₁	semidirect
290. #487	C₂²×C₇₁	semidirect
292. #468	C₂²×C₇₁	semidirect
293. #465	C₂²×C₇₁	semidirect
294. #440	C₂²×C₇₁	semidirect
295. C₄×SD₁₆	C₂²×C₇₁	semidirect
300. #405	C₂²×C₇₁	semidirect
302. C₄×M₁₆	C₂²×C₇₁	semidirect
304. C₂×C₄×C₈	C₂²×C₇₁	semidirect
407. C₂×C₈	#41×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3509	2
C₇₁×#3508	2
C₇₁×#3182	2
C₇₁×#3174	2
C₇₁×#3167	2
C₇₁×#3160	2
C₂×#492×C₇₁	2
C₇₁×#3139	2
C₇₁×#3138	2
C₇₁×#3057	2
C₇₁×#2949	2
C₄×#131×C₇₁	2
#60712	71

Subgroups — C71×#60712 #31881584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#60712 #31881584