Subgroups — C₇₁×#19496 #31840368

Order 18176.

708 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#3106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1697	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1687	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1678	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1668	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₇₁	—	2	1	no	—
13	#532×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#127×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
16	#493×C₇₁	C₂²	1	2	yes	—
17	#486×C₇₁	C₂²	1	2	yes	—
18	#482×C₇₁	—	2	1	no	—
19	#477×C₇₁	C₂²	1	1	yes	characteristic, complemented
20	#476×C₇₁	C₂²	1	1	yes	characteristic, complemented
21	#460×C₇₁	—	2	1	no	—
22	#460×C₇₁	C₂²	1	2	yes	—
23	#460×C₇₁	C₂²	1	1	yes	characteristic, complemented
24	#459×C₇₁	—	2	1	no	—
25	#459×C₇₁	—	2	1	no	—
26	#459×C₇₁	—	2	1	no	—
27	#459×C₇₁	C₂²	1	2	yes	—
28	#459×C₇₁	C₂²	1	1	yes	characteristic, complemented
29	#449×C₇₁	—	2	1	no	—
30	#447×C₇₁	—	2	2	no	—
31	C₈×Q₈×C₇₁	C₂²	1	2	yes	complemented
32	C₄×SD₁₆×C₇₁	—	2	1	no	—
33	C₄×D₈×C₇₁	—	2	1	no	—
34	#436×C₇₁	—	2	2	no	—
35	#436×C₇₁	C₂²	1	2	yes	complemented
36	#431×C₇₁	—	2	1	no	—
37	#431×C₇₁	C₂²	1	1	yes	characteristic
38	#431×C₇₁	C₂²	1	1	yes	characteristic
39	#423×C₇₁	—	2	1	no	—
40	#423×C₇₁	C₂²	1	1	yes	characteristic
41	#423×C₇₁	C₂²	1	1	yes	characteristic
42	#418×C₇₁	—	2	1	no	—
43	#335×C₇₁	C₂²	1	2	yes	complemented
44	#334×C₇₁	C₂²	1	2	yes	complemented
45	#333×C₇₁	C₂²	1	2	yes	complemented
46	#331×C₇₁	C₂²	1	2	yes	—
47	#331×C₇₁	C₂²	1	2	yes	—
48	#331×C₇₁	C₂²	1	2	yes	complemented
49	C₂×#41×C₇₁	—	2	1	no	—
50	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₇₁	—	2	1	no	—
52	C₂²×D₄×C₇₁	—	2	1	no	—
53	C₂²×D₄×C₇₁	—	2	1	no	—
54	AGL(1,ℤ/8)×C₇₁	—	4	4	no	—
55	C₂×SD₁₆×C₇₁	—	2	1	no	—
56	C₂×SD₁₆×C₇₁	—	2	2	no	—
57	C₂×D₈×C₇₁	—	2	1	no	—
58	C₂×D₈×C₇₁	—	2	2	no	—
59	C₂×D₈×C₇₁	—	2	2	no	—
60	C₂×D₈×C₇₁	—	2	2	no	—
61	C₂×M₁₆×C₇₁	—	2	1	no	—
62	C₂×M₁₆×C₇₁	—	2	1	no	—
63	C₂×M₁₆×C₇₁	—	2	2	no	—
64	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
65	#127×C₇₁	—	2	1	no	—
66	#127×C₇₁	—	2	1	no	—
67	#127×C₇₁	—	2	1	no	—
68	#127×C₇₁	C₂³	1	2	yes	—
69	#127×C₇₁	C₂³	1	1	yes	characteristic
70	#124×C₇₁	—	2	1	no	—
71	#121×C₇₁	—	2	1	no	—
72	#121×C₇₁	—	2	1	no	—
73	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₇₁	—	2	1	no	—
76	C₄×D₄×C₇₁	—	2	1	no	—
77	C₄×D₄×C₇₁	—	2	1	no	—
78	C₄×D₄×C₇₁	—	2	1	no	—
79	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
80	C₄×D₄×C₇₁	C₂³	1	1	yes	characteristic
81	#117×C₇₁	—	2	1	no	—
82	#117×C₇₁	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₇₁	—	2	1	no	—
84	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
85	#107×C₇₁	—	2	1	no	—
86	#106×C₇₁	—	2	1	no	—
87	C₄⋊C₈×C₇₁	—	2	2	no	—
88	C₄⋊C₈×C₇₁	—	2	1	no	—
89	C₄⋊C₈×C₇₁	—	2	1	no	—
90	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
91	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
92	C₄⋊C₈×C₇₁	C₂³	1	2	yes	—
93	C₄⋊C₈×C₇₁	C₂³	1	2	yes	—
94	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
96	#103×C₇₁	—	2	1	no	—
97	#102×C₇₁	—	2	1	no	—
98	#102×C₇₁	—	2	2	no	—
99	#102×C₇₁	—	2	2	no	—
100	#102×C₇₁	—	2	2	no	—
101	#102×C₇₁	—	2	2	no	—
102	#102×C₇₁	—	2	1	no	—
103	#102×C₇₁	—	2	1	no	—
104	C₂²⋊C₈×C₇₁	—	2	1	no	—
105	C₂²⋊C₈×C₇₁	—	2	2	no	—
106	C₂²⋊C₈×C₇₁	D₄	1	2	yes	—
107	C₄×C₈×C₇₁	—	2	2	no	—
108	C₄×C₈×C₇₁	—	2	1	no	—
109	C₄×C₈×C₇₁	C₂³	1	2	yes	—
110	C₂⁴×C₇₁	—	4	1	no	—
111	#41×C₇₁	—	2	2	no	—
112	#41×C₇₁	—	4	1	no	—
113	#41×C₇₁	—	8	2	no	—
114	C₂×Q₈×C₇₁	—	2	1	no	—
115	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×D₄×C₇₁	—	2	1	no	—
117	C₂×D₄×C₇₁	—	2	1	no	—
118	C₂×D₄×C₇₁	—	2	2	no	—
119	C₂×D₄×C₇₁	—	2	2	no	—
120	C₂×D₄×C₇₁	—	2	1	no	—
121	C₂×D₄×C₇₁	—	2	1	no	—
122	C₂×D₄×C₇₁	—	2	2	no	—
123	C₂×D₄×C₇₁	—	4	2	no	—
124	C₂×D₄×C₇₁	—	4	2	no	—
125	C₂×D₄×C₇₁	—	4	1	no	—
126	C₂×D₄×C₇₁	—	4	1	no	—
127	C₂×D₄×C₇₁	—	4	1	no	—
128	C₂×D₄×C₇₁	—	4	1	no	—
129	C₂×D₄×C₇₁	—	8	1	no	—
130	C₂×D₄×C₇₁	—	8	1	no	—
131	C₂×D₄×C₇₁	—	8	2	no	—
132	C₂×D₄×C₇₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₇₁	—	2	1	no	—
134	C₂²×C₄×C₇₁	—	2	1	no	—
135	C₂²×C₄×C₇₁	—	2	1	no	—
136	C₂²×C₄×C₇₁	—	4	1	no	—
137	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
140	SD₁₆×C₇₁	—	4	2	no	—
141	SD₁₆×C₇₁	—	4	4	no	—
142	D₈×C₇₁	—	4	2	no	—
143	D₈×C₇₁	—	4	4	no	—
144	D₈×C₇₁	—	8	2	no	—
145	D₈×C₇₁	—	8	2	no	—
146	M₁₆×C₇₁	—	4	2	no	—
147	M₁₆×C₇₁	—	8	2	no	—
148	M₁₆×C₇₁	—	8	1	no	—
149	C₂×C₈×C₇₁	—	2	1	no	—
150	C₂×C₈×C₇₁	—	2	1	no	—
151	C₂×C₈×C₇₁	—	2	2	no	—
152	C₂×C₈×C₇₁	—	2	2	no	—
153	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₇₁	—	2	1	no	—
156	C₄⋊C₄×C₇₁	—	2	1	no	—
157	C₄⋊C₄×C₇₁	—	2	1	no	—
158	C₄⋊C₄×C₇₁	—	2	1	no	—
159	C₄⋊C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
160	C₄⋊C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
161	C₂²⋊C₄×C₇₁	—	2	1	no	—
162	C₂²⋊C₄×C₇₁	—	2	1	no	—
163	C₂²⋊C₄×C₇₁	—	2	1	no	—
164	C₂²⋊C₄×C₇₁	—	4	1	no	—
165	C₂²⋊C₄×C₇₁	—	4	1	no	—
166	C₄²×C₇₁	—	2	1	no	—
167	C₄²×C₇₁	—	2	1	no	—
168	C₄²×C₇₁	—	2	1	no	—
169	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₇₁	#41	1	1	yes	characteristic
172	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
173	C₂³×C₇₁	—	2	1	no	—
174	C₂³×C₇₁	—	4	1	no	—
175	C₂³×C₇₁	—	4	2	no	—
176	C₂³×C₇₁	—	8	2	no	—
177	C₂³×C₇₁	—	8	2	no	—
178	C₂³×C₇₁	—	8	2	no	—
179	C₂³×C₇₁	#121	1	1	yes	characteristic
180	Q₈×C₇₁	—	2	1	no	—
181	Q₈×C₇₁	—	8	1	no	—
182	Q₈×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
183	D₄×C₇₁	—	2	1	no	—
184	D₄×C₇₁	—	2	1	no	—
185	D₄×C₇₁	—	4	2	no	—
186	D₄×C₇₁	—	4	2	no	—
187	D₄×C₇₁	—	4	1	no	—
188	D₄×C₇₁	—	4	1	no	—
189	D₄×C₇₁	—	4	2	no	—
190	D₄×C₇₁	—	4	1	no	—
191	D₄×C₇₁	—	4	2	no	—
192	D₄×C₇₁	—	4	1	no	—
193	D₄×C₇₁	—	4	2	no	—
194	D₄×C₇₁	—	8	2	no	—
195	D₄×C₇₁	—	8	1	no	—
196	D₄×C₇₁	—	8	2	no	—
197	D₄×C₇₁	—	8	2	no	—
198	D₄×C₇₁	—	8	2	no	—
199	D₄×C₇₁	—	8	2	no	—
200	C₂×C₄×C₇₁	—	2	1	no	—
201	C₂×C₄×C₇₁	—	2	1	no	—
202	C₂×C₄×C₇₁	—	2	1	no	—
203	C₂×C₄×C₇₁	—	2	1	no	—
204	C₂×C₄×C₇₁	—	2	1	no	—
205	C₂×C₄×C₇₁	—	2	2	no	—
206	C₂×C₄×C₇₁	—	2	1	no	—
207	C₂×C₄×C₇₁	—	2	1	no	—
208	C₂×C₄×C₇₁	—	4	2	no	—
209	C₂×C₄×C₇₁	—	4	1	no	—
210	C₂×C₄×C₇₁	—	4	1	no	—
211	C₂×C₄×C₇₁	—	4	2	no	—
212	C₂×C₄×C₇₁	—	4	1	no	—
213	C₂×C₄×C₇₁	—	8	2	no	—
214	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
215	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
217	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
220	C₈×C₇₁	—	2	2	no	—
221	C₈×C₇₁	—	2	2	no	—
222	C₈×C₇₁	—	4	2	no	—
223	C₈×C₇₁	—	4	1	no	—
224	C₈×C₇₁	—	4	1	no	—
225	C₈×C₇₁	—	4	2	no	—
226	C₂²×C₇₁	—	2	1	no	—
227	C₂²×C₇₁	—	2	1	no	—
228	C₂²×C₇₁	—	2	1	no	—
229	C₂²×C₇₁	—	4	1	no	—
230	C₂²×C₇₁	—	4	1	no	—
231	C₂²×C₇₁	—	4	1	no	—
232	C₂²×C₇₁	—	8	2	no	—
233	C₂²×C₇₁	—	8	1	no	—
234	C₂²×C₇₁	—	8	2	no	—
235	C₂²×C₇₁	—	8	1	no	—
236	C₂²×C₇₁	—	8	1	no	—
237	C₂²×C₇₁	—	8	2	no	—
238	C₂²×C₇₁	—	16	4	no	—
239	C₂²×C₇₁	C₂×#121	1	1	yes	characteristic
240	C₄×C₇₁	—	2	1	no	—
241	C₄×C₇₁	—	2	1	no	—
242	C₄×C₇₁	—	2	2	no	—
243	C₄×C₇₁	—	2	1	no	—
244	C₄×C₇₁	—	4	1	no	—
245	C₄×C₇₁	—	4	1	no	—
246	C₄×C₇₁	—	8	1	no	—
247	C₄×C₇₁	#472	1	2	yes	—
248	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₄×C₇₁	#575	1	1	yes	characteristic
250	C₄×C₇₁	#576	1	1	yes	characteristic
251	C₄×C₇₁	#577	1	1	yes	characteristic
252	#19496	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₇₁	—	4	1	no	—
254	C₂×C₇₁	—	8	1	no	—
255	C₂×C₇₁	—	16	1	no	—
256	C₂×C₇₁	—	16	2	no	—
257	C₂×C₇₁	#3045	1	1	yes	characteristic
258	C₂×C₇₁	#3055	1	1	yes	characteristic
259	C₂×C₇₁	C₂×#472	1	1	yes	characteristic
260	#3205	C₂×C₇₁	1	1	yes	characteristic, complemented
261	#3106	C₂×C₇₁	1	1	yes	characteristic, complemented
262	#3038	C₂×C₇₁	1	1	yes	characteristic, complemented
263	#2991	C₂×C₇₁	1	1	yes	characteristic, complemented
264	#1697	C₂×C₇₁	1	2	yes	complemented
265	#1687	C₂×C₇₁	1	2	yes	complemented
266	#1678	C₂×C₇₁	1	2	yes	complemented
267	#1668	C₂×C₇₁	1	2	yes	complemented
268	#1574	C₂×C₇₁	1	1	yes	characteristic, complemented
269	#1560	C₂×C₇₁	1	1	yes	characteristic, complemented
270	#1545	C₂×C₇₁	1	1	yes	characteristic, complemented
271	C₇₁	#19496	1	1	yes	characteristic, direct product factor, Sylow
272	C₂×AGL(1,ℤ/8)	—	2	1	no	—
273	#532	C₂²×C₇₁	1	1	yes	characteristic
274	C₂×#127	C₂²×C₇₁	1	1	yes	characteristic
275	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
276	#493	C₂²×C₇₁	1	2	yes	—
277	#486	C₂²×C₇₁	1	2	yes	—
278	#482	—	2	1	no	—
279	#477	C₂²×C₇₁	1	1	yes	characteristic, complemented
280	#476	C₂²×C₇₁	1	1	yes	characteristic, complemented
281	#460	—	2	1	no	—
282	#460	C₂²×C₇₁	1	2	yes	—
283	#460	C₂²×C₇₁	1	1	yes	characteristic, complemented
284	#459	—	2	1	no	—
285	#459	—	2	1	no	—
286	#459	—	2	1	no	—
287	#459	C₂²×C₇₁	1	2	yes	—
288	#459	C₂²×C₇₁	1	1	yes	characteristic, complemented
289	#449	—	2	1	no	—
290	#447	—	2	2	no	—
291	C₈×Q₈	C₂²×C₇₁	1	2	yes	complemented
292	C₄×SD₁₆	—	2	1	no	—
293	C₄×D₈	—	2	1	no	—
294	#436	—	2	2	no	—
295	#436	C₂²×C₇₁	1	2	yes	complemented
296	#431	—	2	1	no	—
297	#431	C₂²×C₇₁	1	1	yes	characteristic
298	#431	C₂²×C₇₁	1	1	yes	characteristic
299	#423	—	2	1	no	—
300	#423	C₂²×C₇₁	1	1	yes	characteristic
301	#423	C₂²×C₇₁	1	1	yes	characteristic
302	#418	—	2	1	no	—
303	#335	C₂²×C₇₁	1	2	yes	complemented
304	#334	C₂²×C₇₁	1	2	yes	complemented
305	#333	C₂²×C₇₁	1	2	yes	complemented
306	#331	C₂²×C₇₁	1	2	yes	—
307	#331	C₂²×C₇₁	1	2	yes	—
308	#331	C₂²×C₇₁	1	2	yes	complemented
309	C₂×#41	—	2	1	no	—
310	C₂×#41	D₄×C₇₁	1	1	yes	characteristic
311	C₂²×D₄	—	2	1	no	—
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	—	2	1	no	—
314	AGL(1,ℤ/8)	—	4	4	no	—
315	C₂×SD₁₆	—	2	2	no	—
316	C₂×SD₁₆	—	2	1	no	—
317	C₂×D₈	—	2	2	no	—
318	C₂×D₈	—	2	1	no	—
319	C₂×D₈	—	2	2	no	—
320	C₂×D₈	—	2	2	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₂×M₁₆	—	2	2	no	—
324	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
325	#127	—	2	1	no	—
326	#127	—	2	1	no	—
327	#127	—	2	1	no	—
328	#127	C₂³×C₇₁	1	2	yes	—
329	#127	C₂³×C₇₁	1	1	yes	characteristic
330	#124	—	2	1	no	—
331	#121	—	2	1	no	—
332	#121	—	2	1	no	—
333	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
334	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	D₄×C₇₁	1	1	yes	characteristic
340	C₄×D₄	C₂³×C₇₁	1	1	yes	characteristic
341	#117	—	2	1	no	—
342	#117	D₄×C₇₁	1	1	yes	characteristic
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
345	#107	—	2	1	no	—
346	#106	—	2	1	no	—
347	C₄⋊C₈	—	2	2	no	—
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	D₄×C₇₁	1	1	yes	characteristic
351	C₄⋊C₈	D₄×C₇₁	1	1	yes	characteristic
352	C₄⋊C₈	C₂³×C₇₁	1	2	yes	—
353	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
354	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
355	C₄⋊C₈	C₂³×C₇₁	1	2	yes	—
356	#103	—	2	1	no	—
357	#102	—	2	1	no	—
358	#102	—	2	2	no	—
359	#102	—	2	2	no	—
360	#102	—	2	2	no	—
361	#102	—	2	2	no	—
362	#102	—	2	1	no	—
363	#102	—	2	1	no	—
364	C₂²⋊C₈	—	2	1	no	—
365	C₂²⋊C₈	—	2	2	no	—
366	C₂²⋊C₈	D₄×C₇₁	1	2	yes	—
367	C₄×C₈	—	2	2	no	—
368	C₄×C₈	—	2	1	no	—
369	C₄×C₈	C₂³×C₇₁	1	2	yes	—
370	C₂⁴	—	4	1	no	—
371	#41	—	2	2	no	—
372	#41	—	4	1	no	—
373	#41	—	8	2	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	8	1	no	—
390	C₂×D₄	—	8	1	no	—
391	C₂×D₄	—	8	2	no	—
392	C₂×D₄	C₂×D₄×C₇₁	1	1	yes	characteristic
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
399	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
400	SD₁₆	—	4	2	no	—
401	SD₁₆	—	4	4	no	—
402	D₈	—	4	2	no	—
403	D₈	—	4	4	no	—
404	D₈	—	8	2	no	—
405	D₈	—	8	2	no	—
406	M₁₆	—	4	2	no	—
407	M₁₆	—	8	2	no	—
408	M₁₆	—	8	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
414	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
420	C₄⋊C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
431	C₄²	#41×C₇₁	1	1	yes	characteristic
432	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
433	C₂³	—	2	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	—	8	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	#121×C₇₁	1	1	yes	characteristic
440	Q₈	—	2	1	no	—
441	Q₈	—	8	1	no	—
442	Q₈	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
443	D₄	—	2	1	no	—
444	D₄	—	2	1	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	1	no	—
448	D₄	—	4	1	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	1	no	—
453	D₄	—	4	2	no	—
454	D₄	—	8	2	no	—
455	D₄	—	8	1	no	—
456	D₄	—	8	2	no	—
457	D₄	—	8	2	no	—
458	D₄	—	8	2	no	—
459	D₄	—	8	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	8	2	no	—
474	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
475	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
476	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	2	no	—
482	C₈	—	4	2	no	—
483	C₈	—	4	1	no	—
484	C₈	—	4	1	no	—
485	C₈	—	4	2	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	4	1	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	8	1	no	—
494	C₂²	—	8	2	no	—
495	C₂²	—	8	1	no	—
496	C₂²	—	8	1	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	16	4	no	—
499	C₂²	C₂×#121×C₇₁	1	1	yes	characteristic
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	1	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	1	no	—
506	C₄	—	8	1	no	—
507	C₄	#472×C₇₁	1	2	yes	—
508	C₄	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
509	C₄	#575×C₇₁	1	1	yes	characteristic
510	C₄	#576×C₇₁	1	1	yes	characteristic
511	C₄	#577×C₇₁	1	1	yes	characteristic
512	C₂	—	4	1	no	—
513	C₂	—	8	1	no	—
514	C₂	—	16	2	no	—
515	C₂	—	16	1	no	—
516	C₂	C₇₁×#3045	1	1	yes	characteristic
517	C₂	C₇₁×#3055	1	1	yes	characteristic
518	C₂	C₂×#472×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3205	C₂	semidirect
2. C₇₁×#3106	C₂	semidirect
3. C₇₁×#3038	C₂	semidirect
4. C₇₁×#2991	C₂	semidirect
5. C₇₁×#1697	C₂	semidirect
6. C₇₁×#1687	C₂	semidirect
7. C₇₁×#1678	C₂	semidirect
8. C₇₁×#1668	C₂	semidirect
9. C₇₁×#1574	C₂	semidirect
10. C₇₁×#1560	C₂	semidirect
11. C₇₁×#1545	C₂	semidirect
19. #477×C₇₁	C₂²	semidirect
20. #476×C₇₁	C₂²	semidirect
23. #460×C₇₁	C₂²	semidirect
28. #459×C₇₁	C₂²	semidirect
31. C₈×Q₈×C₇₁	C₂²	semidirect
35. #436×C₇₁	C₂²	semidirect
43. #335×C₇₁	C₂²	semidirect
44. #334×C₇₁	C₂²	semidirect
45. #333×C₇₁	C₂²	semidirect
48. #331×C₇₁	C₂²	semidirect
252. #19496	C₇₁	direct (N × H)
260. #3205	C₂×C₇₁	semidirect
261. #3106	C₂×C₇₁	semidirect
262. #3038	C₂×C₇₁	semidirect
263. #2991	C₂×C₇₁	semidirect
264. #1697	C₂×C₇₁	semidirect
265. #1687	C₂×C₇₁	semidirect
266. #1678	C₂×C₇₁	semidirect
267. #1668	C₂×C₇₁	semidirect
268. #1574	C₂×C₇₁	semidirect
269. #1560	C₂×C₇₁	semidirect
270. #1545	C₂×C₇₁	semidirect
271. C₇₁	#19496	direct (N × H)
279. #477	C₂²×C₇₁	semidirect
280. #476	C₂²×C₇₁	semidirect
283. #460	C₂²×C₇₁	semidirect
288. #459	C₂²×C₇₁	semidirect
291. C₈×Q₈	C₂²×C₇₁	semidirect
295. #436	C₂²×C₇₁	semidirect
303. #335	C₂²×C₇₁	semidirect
304. #334	C₂²×C₇₁	semidirect
305. #333	C₂²×C₇₁	semidirect
308. #331	C₂²×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3205	2
C₇₁×#3106	2
C₇₁×#3038	2
C₇₁×#2991	2
C₇₁×#1697	2
C₇₁×#1687	2
C₇₁×#1678	2
C₇₁×#1668	2
C₇₁×#1574	2
C₇₁×#1560	2
C₇₁×#1545	2
#19496	71

Subgroups — C71×#19496 #31840368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#19496 #31840368