Subgroups — C₇₁×#22543 #31843415

Order 18176.

1138 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#559×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1983	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇₁×#1958	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1957	C₂	1	1	yes	maximal, characteristic
8	C₇₁×#1804	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#103×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#102×C₇₁	C₂	1	2	yes	maximal, complemented
12	#559×C₇₁	—	2	4	no	—
13	#559×C₇₁	C₄	1	8	yes	complemented
14	C₂×#127×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#124×C₇₁	—	2	1	no	—
16	C₂×#124×C₇₁	C₂²	1	2	yes	—
17	C₂×#123×C₇₁	—	2	2	no	—
18	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₇₁	—	2	1	no	—
21	C₂×C₄×D₄×C₇₁	C₂²	1	2	yes	—
22	C₈×Q₈×C₇₁	—	2	4	no	—
23	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
25	C₂×#103×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×#103×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₇₁	C₂²	1	2	yes	—
28	C₂×#102×C₇₁	C₂²	1	2	yes	—
29	C₂×#102×C₇₁	C₂²	1	2	yes	—
30	C₂×#102×C₇₁	C₂²	1	2	yes	—
31	C₂×#102×C₇₁	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
34	#391×C₇₁	—	2	1	no	—
35	#387×C₇₁	—	2	2	no	—
36	#384×C₇₁	C₂²	1	1	yes	characteristic
37	#382×C₇₁	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
41	#340×C₇₁	C₂²	1	1	yes	characteristic
42	#340×C₇₁	C₂²	1	2	yes	—
43	#340×C₇₁	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₇₁	—	2	1	no	—
47	C₂²×D₄×C₇₁	—	2	1	no	—
48	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₇₁	—	2	2	no	—
50	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
53	#127×C₇₁	—	2	2	no	—
54	#127×C₇₁	C₂×C₄	1	4	yes	—
55	#124×C₇₁	—	2	8	no	—
56	#124×C₇₁	—	2	4	no	—
57	#124×C₇₁	C₂×C₄	1	8	yes	—
58	#123×C₇₁	—	2	16	no	—
59	C₄×Q₈×C₇₁	—	2	2	no	—
60	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
61	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
62	C₄×Q₈×C₇₁	D₄	1	4	yes	—
63	C₄×D₄×C₇₁	—	2	8	no	—
64	C₄×D₄×C₇₁	—	2	4	no	—
65	C₄×D₄×C₇₁	C₂×C₄	1	8	yes	—
66	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₇₁	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₂×C₄²×C₇₁	—	2	1	no	—
79	C₂×C₄²×C₇₁	—	2	1	no	—
80	C₂×C₄²×C₇₁	—	2	2	no	—
81	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₇₁	—	2	4	no	—
84	C₄⋊C₈×C₇₁	—	2	2	no	—
85	#103×C₇₁	—	2	2	no	—
86	#103×C₇₁	—	2	2	no	—
87	#103×C₇₁	—	4	2	no	—
88	#102×C₇₁	—	2	4	no	—
89	#102×C₇₁	—	2	4	no	—
90	#102×C₇₁	—	4	2	no	—
91	#102×C₇₁	—	4	2	no	—
92	C₄×C₈×C₇₁	—	2	4	no	—
93	C₄×C₈×C₇₁	—	2	2	no	—
94	#95×C₇₁	—	2	2	no	—
95	#95×C₇₁	—	2	1	no	—
96	#95×C₇₁	—	2	2	no	—
97	#95×C₇₁	—	2	2	no	—
98	C₂⁴×C₇₁	—	2	2	no	—
99	C₂×Q₈×C₇₁	—	2	2	no	—
100	C₂×Q₈×C₇₁	—	2	2	no	—
101	C₂×Q₈×C₇₁	—	2	1	no	—
102	C₂×Q₈×C₇₁	—	2	1	no	—
103	C₂×Q₈×C₇₁	—	2	1	no	—
104	C₂×Q₈×C₇₁	—	2	1	no	—
105	C₂×Q₈×C₇₁	—	2	1	no	—
106	C₂×Q₈×C₇₁	C₂²⋊C₄	1	4	yes	—
107	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₇₁	—	2	4	no	—
109	C₂×D₄×C₇₁	—	2	4	no	—
110	C₂×D₄×C₇₁	—	2	2	no	—
111	C₂×D₄×C₇₁	—	2	2	no	—
112	C₂×D₄×C₇₁	—	2	2	no	—
113	C₂×D₄×C₇₁	—	2	2	no	—
114	C₂×D₄×C₇₁	—	2	2	no	—
115	C₂×D₄×C₇₁	—	4	2	no	—
116	C₂×D₄×C₇₁	—	4	2	no	—
117	C₂×D₄×C₇₁	—	8	1	no	—
118	C₂×D₄×C₇₁	—	8	1	no	—
119	C₂×D₄×C₇₁	C₂²×C₄	1	4	yes	—
120	C₂²×C₄×C₇₁	—	2	1	no	—
121	C₂²×C₄×C₇₁	—	2	1	no	—
122	C₂²×C₄×C₇₁	—	2	1	no	—
123	C₂²×C₄×C₇₁	—	2	8	no	—
124	C₂²×C₄×C₇₁	—	2	1	no	—
125	C₂²×C₄×C₇₁	—	2	2	no	—
126	C₂²×C₄×C₇₁	—	8	2	no	—
127	C₂²×C₄×C₇₁	—	8	2	no	—
128	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₇₁	—	2	1	no	—
134	C₂×C₈×C₇₁	—	2	1	no	—
135	C₂×C₈×C₇₁	—	2	2	no	—
136	C₂×C₈×C₇₁	—	2	1	no	—
137	C₂×C₈×C₇₁	—	2	2	no	—
138	C₂×C₈×C₇₁	—	2	1	no	—
139	C₂×C₈×C₇₁	—	2	1	no	—
140	C₂×C₈×C₇₁	—	2	2	no	—
141	C₂×C₈×C₇₁	#41	1	2	yes	—
142	C₄⋊C₄×C₇₁	—	2	4	no	—
143	C₄⋊C₄×C₇₁	—	2	1	no	—
144	C₄⋊C₄×C₇₁	—	2	1	no	—
145	C₄⋊C₄×C₇₁	—	2	2	no	—
146	C₄⋊C₄×C₇₁	—	2	1	no	—
147	C₄⋊C₄×C₇₁	—	2	1	no	—
148	C₄⋊C₄×C₇₁	—	2	2	no	—
149	C₄⋊C₄×C₇₁	—	4	2	no	—
150	C₄⋊C₄×C₇₁	—	8	1	no	—
151	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
152	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	8	yes	—
153	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
154	C₄⋊C₄×C₇₁	#41	1	2	yes	—
155	C₂²⋊C₄×C₇₁	—	2	8	no	—
156	C₂²⋊C₄×C₇₁	—	2	8	no	—
157	C₂²⋊C₄×C₇₁	—	2	8	no	—
158	C₄²×C₇₁	—	2	4	no	—
159	C₄²×C₇₁	—	4	2	no	—
160	C₄²×C₇₁	—	8	2	no	—
161	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
162	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
163	C₄²×C₇₁	C₂×D₄	1	4	yes	—
164	C₄²×C₇₁	C₂×D₄	1	2	yes	—
165	C₂³×C₇₁	—	2	4	no	—
166	C₂³×C₇₁	—	4	2	no	—
167	C₂³×C₇₁	—	4	2	no	—
168	C₂³×C₇₁	—	4	2	no	—
169	C₂³×C₇₁	—	4	2	no	—
170	C₂³×C₇₁	—	4	2	no	—
171	C₂³×C₇₁	—	4	2	no	—
172	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₇₁	—	2	8	no	—
174	Q₈×C₇₁	—	2	2	no	—
175	Q₈×C₇₁	—	2	2	no	—
176	Q₈×C₇₁	—	4	1	no	—
177	Q₈×C₇₁	—	4	1	no	—
178	D₄×C₇₁	—	2	4	no	—
179	D₄×C₇₁	—	2	4	no	—
180	D₄×C₇₁	—	4	2	no	—
181	D₄×C₇₁	—	4	2	no	—
182	D₄×C₇₁	—	4	8	no	—
183	D₄×C₇₁	—	8	4	no	—
184	C₂×C₄×C₇₁	—	2	2	no	—
185	C₂×C₄×C₇₁	—	2	2	no	—
186	C₂×C₄×C₇₁	—	2	2	no	—
187	C₂×C₄×C₇₁	—	2	1	no	—
188	C₂×C₄×C₇₁	—	2	2	no	—
189	C₂×C₄×C₇₁	—	2	1	no	—
190	C₂×C₄×C₇₁	—	2	2	no	—
191	C₂×C₄×C₇₁	—	2	2	no	—
192	C₂×C₄×C₇₁	—	2	2	no	—
193	C₂×C₄×C₇₁	—	4	1	no	—
194	C₂×C₄×C₇₁	—	4	1	no	—
195	C₂×C₄×C₇₁	—	4	1	no	—
196	C₂×C₄×C₇₁	—	4	1	no	—
197	C₂×C₄×C₇₁	—	4	1	no	—
198	C₂×C₄×C₇₁	—	4	1	no	—
199	C₂×C₄×C₇₁	—	4	1	no	—
200	C₂×C₄×C₇₁	—	4	2	no	—
201	C₂×C₄×C₇₁	—	4	1	no	—
202	C₂×C₄×C₇₁	—	4	2	no	—
203	C₂×C₄×C₇₁	—	4	2	no	—
204	C₂×C₄×C₇₁	—	4	1	no	—
205	C₂×C₄×C₇₁	—	8	8	no	—
206	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	4	yes	—
207	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	4	yes	—
208	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₇₁	#135	1	2	yes	—
211	C₂×C₄×C₇₁	#135	1	2	yes	—
212	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
218	C₈×C₇₁	—	2	2	no	—
219	C₈×C₇₁	—	4	2	no	—
220	C₈×C₇₁	—	4	1	no	—
221	C₂²×C₇₁	—	4	4	no	—
222	C₂²×C₇₁	—	4	4	no	—
223	C₂²×C₇₁	—	4	4	no	—
224	C₂²×C₇₁	—	8	2	no	—
225	C₂²×C₇₁	—	8	2	no	—
226	C₂²×C₇₁	—	8	2	no	—
227	C₂²×C₇₁	—	8	2	no	—
228	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
229	C₂²×C₇₁	#548	1	1	yes	characteristic
230	C₂²×C₇₁	#550	1	1	yes	characteristic
231	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
232	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
233	C₂²×C₇₁	#576	1	1	yes	characteristic
234	C₂²×C₇₁	#577	1	1	yes	characteristic
235	C₄×C₇₁	—	2	4	no	—
236	C₄×C₇₁	—	2	4	no	—
237	C₄×C₇₁	—	2	2	no	—
238	C₄×C₇₁	—	4	2	no	—
239	C₄×C₇₁	—	4	2	no	—
240	C₄×C₇₁	—	4	2	no	—
241	C₄×C₇₁	—	8	1	no	—
242	C₄×C₇₁	—	8	2	no	—
243	C₄×C₇₁	#416	1	4	yes	—
244	C₄×C₇₁	C₄×#41	1	1	yes	characteristic
245	C₄×C₇₁	#520	1	1	yes	characteristic
246	C₄×C₇₁	#548	1	1	yes	characteristic
247	C₄×C₇₁	#550	1	1	yes	characteristic
248	#22543	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂×C₇₁	—	8	4	no	—
250	C₂×C₇₁	#2341	1	1	yes	characteristic
251	C₂×C₇₁	C₂×#416	1	1	yes	characteristic
252	C₂×C₇₁	#2897	1	1	yes	characteristic
253	C₂×C₇₁	#3361	1	1	yes	characteristic
254	C₂×C₇₁	#3365	1	1	yes	characteristic
255	C₂×C₇₁	#3394	1	1	yes	characteristic
256	C₂×C₇₁	#3396	1	1	yes	characteristic
257	C₂×#559	C₂×C₇₁	1	1	yes	characteristic
258	C₂×C₈×Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
259	#2355	C₂×C₇₁	1	1	yes	characteristic, complemented
260	#1983	C₂×C₇₁	1	2	yes	complemented
261	#1982	C₂×C₇₁	1	1	yes	characteristic
262	#1958	C₂×C₇₁	1	2	yes	complemented
263	#1957	C₂×C₇₁	1	1	yes	characteristic
264	#1804	C₂×C₇₁	1	2	yes	complemented
265	#1803	C₂×C₇₁	1	1	yes	characteristic, complemented
266	C₄×#103	C₂×C₇₁	1	1	yes	characteristic, complemented
267	C₄×#102	C₂×C₇₁	1	2	yes	complemented
268	C₇₁	#22543	1	1	yes	characteristic, direct product factor, Sylow
269	#559	—	2	4	no	—
270	#559	C₄×C₇₁	1	8	yes	complemented
271	C₂×#127	C₂²×C₇₁	1	1	yes	characteristic
272	C₂×#124	—	2	1	no	—
273	C₂×#124	C₂²×C₇₁	1	2	yes	—
274	C₂×#123	—	2	2	no	—
275	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
276	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
277	C₂×C₄×D₄	—	2	1	no	—
278	C₂×C₄×D₄	C₂²×C₇₁	1	2	yes	—
279	C₈×Q₈	—	2	4	no	—
280	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
281	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
282	C₂×#103	C₂²×C₇₁	1	1	yes	characteristic
283	C₂×#103	C₂²×C₇₁	1	1	yes	characteristic
284	C₂×#103	C₂²×C₇₁	1	2	yes	—
285	C₂×#102	C₂²×C₇₁	1	2	yes	—
286	C₂×#102	C₂²×C₇₁	1	2	yes	—
287	C₂×#102	C₂²×C₇₁	1	2	yes	—
288	C₂×#102	C₂²×C₇₁	1	2	yes	—
289	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
291	#391	—	2	1	no	—
292	#387	—	2	2	no	—
293	#384	C₂²×C₇₁	1	1	yes	characteristic
294	#382	C₂²×C₇₁	1	2	yes	—
295	C₄×C₄⋊C₄	—	2	1	no	—
296	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
297	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
298	#340	C₂²×C₇₁	1	2	yes	—
299	#340	C₂²×C₇₁	1	1	yes	characteristic
300	#340	C₂²×C₇₁	1	1	yes	characteristic
301	C₂²×Q₈	D₄×C₇₁	1	1	yes	characteristic
302	C₂²×Q₈	C₂³×C₇₁	1	1	yes	characteristic
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
306	C₂³×C₄	—	2	2	no	—
307	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
308	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
310	#127	—	2	2	no	—
311	#127	C₂×C₄×C₇₁	1	4	yes	—
312	#124	—	2	8	no	—
313	#124	—	2	4	no	—
314	#124	C₂×C₄×C₇₁	1	8	yes	—
315	#123	—	2	16	no	—
316	C₄×Q₈	—	2	2	no	—
317	C₄×Q₈	C₂×C₄×C₇₁	1	4	yes	—
318	C₄×Q₈	C₂×C₄×C₇₁	1	4	yes	—
319	C₄×Q₈	D₄×C₇₁	1	4	yes	—
320	C₄×D₄	—	2	8	no	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	C₂×C₄×C₇₁	1	8	yes	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	D₄×C₇₁	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	—	2	1	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
339	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
340	C₄⋊C₈	—	2	4	no	—
341	C₄⋊C₈	—	2	2	no	—
342	#103	—	2	2	no	—
343	#103	—	2	2	no	—
344	#103	—	4	2	no	—
345	#102	—	2	4	no	—
346	#102	—	2	4	no	—
347	#102	—	4	2	no	—
348	#102	—	4	2	no	—
349	C₄×C₈	—	2	4	no	—
350	C₄×C₈	—	2	2	no	—
351	#95	—	2	2	no	—
352	#95	—	2	1	no	—
353	#95	—	2	2	no	—
354	#95	—	2	2	no	—
355	C₂⁴	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	C₂²⋊C₄×C₇₁	1	4	yes	—
364	C₂×Q₈	C₂²×C₄×C₇₁	1	2	yes	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	4	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	8	1	no	—
375	C₂×D₄	—	8	1	no	—
376	C₂×D₄	C₂²×C₄×C₇₁	1	4	yes	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	8	2	no	—
384	C₂²×C₄	—	8	2	no	—
385	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	#41×C₇₁	1	2	yes	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₄⋊C₄	—	8	1	no	—
408	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	4	yes	—
409	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	8	yes	—
410	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
411	C₄⋊C₄	#41×C₇₁	1	2	yes	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₄²	—	2	4	no	—
416	C₄²	—	4	2	no	—
417	C₄²	—	8	2	no	—
418	C₄²	C₂²×C₄×C₇₁	1	4	yes	—
419	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
420	C₄²	C₂×D₄×C₇₁	1	4	yes	—
421	C₄²	C₂×D₄×C₇₁	1	2	yes	—
422	C₂³	—	2	4	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
430	Q₈	—	2	8	no	—
431	Q₈	—	2	2	no	—
432	Q₈	—	2	2	no	—
433	Q₈	—	4	1	no	—
434	Q₈	—	4	1	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	4	8	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	8	4	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	4	yes	—
464	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	4	yes	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
467	C₂×C₄	#135×C₇₁	1	2	yes	—
468	C₂×C₄	#135×C₇₁	1	2	yes	—
469	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
473	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
474	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
475	C₈	—	2	2	no	—
476	C₈	—	4	2	no	—
477	C₈	—	4	1	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	2	no	—
483	C₂²	—	8	2	no	—
484	C₂²	—	8	2	no	—
485	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
486	C₂²	#548×C₇₁	1	1	yes	characteristic
487	C₂²	#550×C₇₁	1	1	yes	characteristic
488	C₂²	C₂×#135×C₇₁	1	1	yes	characteristic
489	C₂²	C₂×#135×C₇₁	1	1	yes	characteristic
490	C₂²	#576×C₇₁	1	1	yes	characteristic
491	C₂²	#577×C₇₁	1	1	yes	characteristic
492	C₄	—	2	4	no	—
493	C₄	—	2	4	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	1	no	—
499	C₄	—	8	2	no	—
500	C₄	#416×C₇₁	1	4	yes	—
501	C₄	C₄×#41×C₇₁	1	1	yes	characteristic
502	C₄	#520×C₇₁	1	1	yes	characteristic
503	C₄	#548×C₇₁	1	1	yes	characteristic
504	C₄	#550×C₇₁	1	1	yes	characteristic
505	C₂	—	8	4	no	—
506	C₂	C₇₁×#2341	1	1	yes	characteristic
507	C₂	C₂×#416×C₇₁	1	1	yes	characteristic
508	C₂	C₇₁×#2897	1	1	yes	characteristic
509	C₂	C₇₁×#3361	1	1	yes	characteristic
510	C₂	C₇₁×#3365	1	1	yes	characteristic
511	C₂	C₇₁×#3394	1	1	yes	characteristic
512	C₂	C₇₁×#3396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂×#127×C₇₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄×Q₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₇₁	C₂	semidirect
3. C₇₁×#2355	C₂	semidirect
4. C₇₁×#1983	C₂	semidirect
6. C₇₁×#1958	C₂	semidirect
8. C₇₁×#1804	C₂	semidirect
9. C₇₁×#1803	C₂	semidirect
10. C₄×#103×C₇₁	C₂	semidirect
11. C₄×#102×C₇₁	C₂	semidirect
13. #559×C₇₁	C₄	semidirect
248. #22543	C₇₁	direct (N × H)
258. C₂×C₈×Q₈	C₂×C₇₁	semidirect
259. #2355	C₂×C₇₁	semidirect
260. #1983	C₂×C₇₁	semidirect
262. #1958	C₂×C₇₁	semidirect
264. #1804	C₂×C₇₁	semidirect
265. #1803	C₂×C₇₁	semidirect
266. C₄×#103	C₂×C₇₁	semidirect
267. C₄×#102	C₂×C₇₁	semidirect
268. C₇₁	#22543	direct (N × H)
270. #559	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×#559×C₇₁	2
C₂×C₈×Q₈×C₇₁	2
C₇₁×#2355	2
C₇₁×#1983	2
C₇₁×#1982	2
C₇₁×#1958	2
C₇₁×#1957	2
C₇₁×#1804	2
C₇₁×#1803	2
C₄×#103×C₇₁	2
C₄×#102×C₇₁	2
#22543	71

Subgroups — C71×#22543 #31843415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#22543 #31843415