Subgroups — C₇₁×#62001 #31882873

Order 18176.

2216 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#3483	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3391	C₂	1	2	yes	maximal, complemented
4	C₇₁×#3362	C₂	1	2	yes	maximal, complemented
5	C₇₁×#3325	C₂	1	6	yes	maximal, complemented
6	C₇₁×#3274	C₂	1	6	yes	maximal, complemented
7	C₇₁×#3253	C₂	1	3	yes	maximal, complemented
8	C₇₁×#3205	C₂	1	3	yes	maximal, complemented
9	C₇₁×#3106	C₂	1	3	yes	maximal, complemented
10	C₇₁×#3038	C₂	1	3	yes	maximal, complemented
11	C₇₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₇₁	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₇₁	C₂²	1	6	yes	complemented
14	Q₈²×C₇₁	C₂²	1	1	yes	characteristic, complemented
15	#555×C₇₁	—	2	3	no	—
16	#555×C₇₁	C₂²	1	3	yes	complemented
17	#550×C₇₁	—	2	1	no	—
18	#550×C₇₁	—	2	1	no	—
19	#550×C₇₁	C₂²	1	2	yes	—
20	#550×C₇₁	C₂²	1	3	yes	complemented
21	#550×C₇₁	C₂²	1	1	yes	characteristic, complemented
22	D₄²×C₇₁	—	2	3	no	—
23	D₄²×C₇₁	C₂²	1	6	yes	—
24	#535×C₇₁	—	2	3	no	—
25	#532×C₇₁	—	2	3	no	—
26	#532×C₇₁	C₂²	1	3	yes	—
27	#532×C₇₁	C₂²	1	3	yes	complemented
28	C₂×#127×C₇₁	C₂²	1	3	yes	—
29	C₄×#41×C₇₁	—	2	1	no	—
30	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
31	C₄×#41×C₇₁	C₂²	1	3	yes	complemented
32	#493×C₇₁	C₂²	1	6	yes	complemented
33	#486×C₇₁	C₂²	1	6	yes	complemented
34	#482×C₇₁	C₂²	1	6	yes	complemented
35	#477×C₇₁	C₂²	1	1	yes	characteristic, complemented
36	#476×C₇₁	C₂²	1	6	yes	complemented
37	#476×C₇₁	C₂²	1	3	yes	complemented
38	#469×C₇₁	C₂²	1	12	yes	complemented
39	#460×C₇₁	C₂²	1	3	yes	complemented
40	#460×C₇₁	C₂²	1	6	yes	complemented
41	#459×C₇₁	C₂²	1	6	yes	complemented
42	#459×C₇₁	C₂²	1	6	yes	complemented
43	#459×C₇₁	C₂²	1	3	yes	complemented
44	#459×C₇₁	C₂²	1	6	yes	complemented
45	#449×C₇₁	C₂²	1	6	yes	complemented
46	#447×C₇₁	C₂²	1	12	yes	complemented
47	C₈×Q₈×C₇₁	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₇₁	C₂²	1	6	yes	complemented
49	C₄×D₈×C₇₁	C₂²	1	6	yes	complemented
50	#436×C₇₁	C₂²	1	6	yes	complemented
51	#431×C₇₁	C₂²	1	3	yes	complemented
52	#423×C₇₁	C₂²	1	3	yes	complemented
53	#418×C₇₁	C₂²	1	6	yes	complemented
54	H(5,𝔽₂)×C₇₁	—	4	8	no	—
55	C₂×#41×C₇₁	—	2	1	no	—
56	C₂×#41×C₇₁	—	2	2	no	—
57	C₂×#41×C₇₁	—	2	3	no	—
58	C₂×#41×C₇₁	—	2	3	no	—
59	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
60	C₂×#41×C₇₁	C₂³	1	3	yes	—
61	C₂²×D₄×C₇₁	—	2	6	no	—
62	C₂²×D₄×C₇₁	—	2	3	no	—
63	C₂²×D₄×C₇₁	—	2	3	no	—
64	C₂²×D₄×C₇₁	C₂³	1	6	yes	—
65	AGL(1,ℤ/8)×C₇₁	—	4	12	no	—
66	C₂×SD₁₆×C₇₁	C₂³	1	6	yes	—
67	C₂×D₈×C₇₁	C₂³	1	6	yes	—
68	C₂×D₈×C₇₁	C₂³	1	12	yes	—
69	C₂×M₁₆×C₇₁	C₂³	1	6	yes	—
70	C₄⋊Q₈×C₇₁	—	2	3	no	—
71	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₇₁	C₂³	1	3	yes	—
73	#127×C₇₁	—	2	3	no	—
74	#127×C₇₁	—	2	3	no	—
75	#127×C₇₁	—	2	3	no	—
76	#127×C₇₁	C₂³	1	3	yes	—
77	#127×C₇₁	C₂³	1	3	yes	—
78	#127×C₇₁	C₂³	1	6	yes	—
79	#124×C₇₁	—	2	6	no	—
80	#124×C₇₁	—	2	3	no	—
81	#124×C₇₁	—	2	3	no	—
82	#121×C₇₁	—	2	3	no	—
83	#121×C₇₁	—	2	3	no	—
84	#121×C₇₁	—	2	3	no	—
85	#121×C₇₁	—	2	3	no	—
86	#121×C₇₁	—	2	6	no	—
87	#121×C₇₁	—	2	6	no	—
88	#121×C₇₁	—	2	6	no	—
89	#121×C₇₁	C₂³	1	6	yes	—
90	#121×C₇₁	C₂³	1	6	yes	—
91	#120×C₇₁	—	2	12	no	—
92	C₄×Q₈×C₇₁	—	2	1	no	—
93	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
95	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₇₁	—	2	3	no	—
98	C₄×D₄×C₇₁	—	2	3	no	—
99	C₄×D₄×C₇₁	—	2	3	no	—
100	C₄×D₄×C₇₁	—	2	3	no	—
101	C₄×D₄×C₇₁	—	2	3	no	—
102	C₄×D₄×C₇₁	D₄	1	3	yes	—
103	C₄×D₄×C₇₁	C₂³	1	3	yes	—
104	C₄×D₄×C₇₁	C₂³	1	6	yes	—
105	C₄×D₄×C₇₁	C₂³	1	3	yes	—
106	C₄×D₄×C₇₁	C₂³	1	3	yes	—
107	#117×C₇₁	—	2	3	no	—
108	#117×C₇₁	—	2	3	no	—
109	#117×C₇₁	D₄	1	3	yes	—
110	#117×C₇₁	C₂³	1	3	yes	—
111	C₂×C₄²×C₇₁	—	2	3	no	—
112	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
113	#107×C₇₁	C₂³	1	2	yes	—
114	#106×C₇₁	C₂³	1	6	yes	—
115	C₄⋊C₈×C₇₁	C₂³	1	3	yes	—
116	C₄⋊C₈×C₇₁	C₂³	1	6	yes	—
117	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
118	#103×C₇₁	C₂³	1	2	yes	—
119	#102×C₇₁	C₂³	1	6	yes	—
120	#102×C₇₁	C₂³	1	12	yes	—
121	#102×C₇₁	C₂³	1	6	yes	—
122	#102×C₇₁	C₂³	1	6	yes	—
123	C₂²⋊C₈×C₇₁	C₂³	1	6	yes	—
124	C₄×C₈×C₇₁	C₂³	1	6	yes	—
125	C₂⁴×C₇₁	—	2	6	no	—
126	#41×C₇₁	—	2	4	no	—
127	#41×C₇₁	—	2	6	no	—
128	#41×C₇₁	—	4	3	no	—
129	#41×C₇₁	—	4	12	no	—
130	#41×C₇₁	—	4	4	no	—
131	#41×C₇₁	—	4	8	no	—
132	#41×C₇₁	—	8	6	no	—
133	C₂×Q₈×C₇₁	—	2	1	no	—
134	C₂×Q₈×C₇₁	—	2	3	no	—
135	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₇₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₇₁	—	2	3	no	—
138	C₂×D₄×C₇₁	—	2	6	no	—
139	C₂×D₄×C₇₁	—	2	6	no	—
140	C₂×D₄×C₇₁	—	2	3	no	—
141	C₂×D₄×C₇₁	—	2	3	no	—
142	C₂×D₄×C₇₁	—	2	6	no	—
143	C₂×D₄×C₇₁	—	2	3	no	—
144	C₂×D₄×C₇₁	—	2	6	no	—
145	C₂×D₄×C₇₁	—	2	6	no	—
146	C₂×D₄×C₇₁	—	2	3	no	—
147	C₂×D₄×C₇₁	—	4	6	no	—
148	C₂×D₄×C₇₁	—	4	6	no	—
149	C₂×D₄×C₇₁	—	4	12	no	—
150	C₂×D₄×C₇₁	—	4	24	no	—
151	C₂×D₄×C₇₁	—	8	6	no	—
152	C₂×D₄×C₇₁	C₂×D₄	1	3	yes	—
153	C₂×D₄×C₇₁	C₂⁴	1	6	yes	—
154	C₂×D₄×C₇₁	C₂⁴	1	3	yes	—
155	C₂²×C₄×C₇₁	—	2	3	no	—
156	C₂²×C₄×C₇₁	—	2	3	no	—
157	C₂²×C₄×C₇₁	—	2	3	no	—
158	C₂²×C₄×C₇₁	—	2	6	no	—
159	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
160	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
161	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
162	SD₁₆×C₇₁	—	2	12	no	—
163	D₈×C₇₁	—	4	6	no	—
164	D₈×C₇₁	—	4	12	no	—
165	M₁₆×C₇₁	—	4	6	no	—
166	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
167	C₂×C₈×C₇₁	C₂⁴	1	6	yes	—
168	C₄⋊C₄×C₇₁	—	2	1	no	—
169	C₄⋊C₄×C₇₁	—	2	3	no	—
170	C₄⋊C₄×C₇₁	—	2	3	no	—
171	C₄⋊C₄×C₇₁	—	2	1	no	—
172	C₄⋊C₄×C₇₁	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₇₁	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₇₁	C₂⁴	1	3	yes	—
175	C₄⋊C₄×C₇₁	C₂⁴	1	1	yes	characteristic
176	C₄⋊C₄×C₇₁	C₂⁴	1	2	yes	—
177	C₂²⋊C₄×C₇₁	—	2	3	no	—
178	C₂²⋊C₄×C₇₁	—	2	6	no	—
179	C₂²⋊C₄×C₇₁	—	2	3	no	—
180	C₂²⋊C₄×C₇₁	—	2	3	no	—
181	C₂²⋊C₄×C₇₁	—	2	6	no	—
182	C₂²⋊C₄×C₇₁	—	2	6	no	—
183	C₂²⋊C₄×C₇₁	C₂×D₄	1	6	yes	—
184	C₄²×C₇₁	—	2	1	no	—
185	C₄²×C₇₁	—	2	3	no	—
186	C₄²×C₇₁	—	2	3	no	—
187	C₄²×C₇₁	C₂×D₄	1	3	yes	—
188	C₄²×C₇₁	C₂×D₄	1	3	yes	—
189	C₄²×C₇₁	C₂⁴	1	3	yes	—
190	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
191	C₂³×C₇₁	—	2	3	no	—
192	C₂³×C₇₁	—	2	6	no	—
193	C₂³×C₇₁	—	4	12	no	—
194	C₂³×C₇₁	—	8	6	no	—
195	C₂³×C₇₁	—	8	6	no	—
196	C₂³×C₇₁	C₂²×D₄	1	3	yes	—
197	Q₈×C₇₁	—	2	1	no	—
198	Q₈×C₇₁	—	4	2	no	—
199	Q₈×C₇₁	—	8	3	no	—
200	Q₈×C₇₁	AGL(1,ℤ/8)	1	4	yes	complemented
201	Q₈×C₇₁	H(5,𝔽₂)	1	2	yes	complemented
202	D₄×C₇₁	—	2	6	no	—
203	D₄×C₇₁	—	2	3	no	—
204	D₄×C₇₁	—	2	6	no	—
205	D₄×C₇₁	—	2	3	no	—
206	D₄×C₇₁	—	2	6	no	—
207	D₄×C₇₁	—	4	6	no	—
208	D₄×C₇₁	—	4	6	no	—
209	D₄×C₇₁	—	4	12	no	—
210	D₄×C₇₁	—	4	12	no	—
211	D₄×C₇₁	—	4	6	no	—
212	D₄×C₇₁	—	4	6	no	—
213	D₄×C₇₁	—	8	6	no	—
214	D₄×C₇₁	—	8	6	no	—
215	D₄×C₇₁	—	8	3	no	—
216	D₄×C₇₁	—	8	6	no	—
217	C₂×C₄×C₇₁	—	2	1	no	—
218	C₂×C₄×C₇₁	—	2	6	no	—
219	C₂×C₄×C₇₁	—	2	6	no	—
220	C₂×C₄×C₇₁	—	2	3	no	—
221	C₂×C₄×C₇₁	—	2	3	no	—
222	C₂×C₄×C₇₁	—	2	1	no	—
223	C₂×C₄×C₇₁	—	2	3	no	—
224	C₂×C₄×C₇₁	—	2	2	no	—
225	C₂×C₄×C₇₁	—	4	6	no	—
226	C₂×C₄×C₇₁	—	4	12	no	—
227	C₂×C₄×C₇₁	—	4	6	no	—
228	C₂×C₄×C₇₁	—	4	6	no	—
229	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
230	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
232	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
233	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
235	C₈×C₇₁	—	2	6	no	—
236	C₈×C₇₁	—	2	2	no	—
237	C₂²×C₇₁	—	2	3	no	—
238	C₂²×C₇₁	—	2	3	no	—
239	C₂²×C₇₁	—	2	3	no	—
240	C₂²×C₇₁	—	4	3	no	—
241	C₂²×C₇₁	—	4	6	no	—
242	C₂²×C₇₁	—	4	6	no	—
243	C₂²×C₇₁	—	4	3	no	—
244	C₂²×C₇₁	—	4	3	no	—
245	C₂²×C₇₁	—	4	6	no	—
246	C₂²×C₇₁	—	16	12	no	—
247	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
248	C₄×C₇₁	—	2	3	no	—
249	C₄×C₇₁	—	2	2	no	—
250	C₄×C₇₁	—	2	1	no	—
251	C₄×C₇₁	—	2	3	no	—
252	C₄×C₇₁	—	4	1	no	—
253	C₄×C₇₁	—	4	2	no	—
254	C₄×C₇₁	—	4	3	no	—
255	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
256	C₄×C₇₁	#576	1	2	yes	—
257	C₄×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
258	C₄×C₇₁	#585	1	1	yes	characteristic
259	#62001	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₇₁	—	4	3	no	—
261	C₂×C₇₁	—	8	3	no	—
262	C₂×C₇₁	—	8	6	no	—
263	C₂×C₇₁	#3488	1	1	yes	characteristic
264	C₂×C₇₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₂×C₇₁	C₂×#576	1	1	yes	characteristic
266	#3498	C₂×C₇₁	1	1	yes	characteristic, complemented
267	#3483	C₂×C₇₁	1	1	yes	characteristic, complemented
268	#3391	C₂×C₇₁	1	2	yes	complemented
269	#3362	C₂×C₇₁	1	2	yes	complemented
270	#3325	C₂×C₇₁	1	6	yes	complemented
271	#3274	C₂×C₇₁	1	6	yes	complemented
272	#3253	C₂×C₇₁	1	3	yes	complemented
273	#3205	C₂×C₇₁	1	3	yes	complemented
274	#3106	C₂×C₇₁	1	3	yes	complemented
275	#3038	C₂×C₇₁	1	3	yes	complemented
276	#2991	C₂×C₇₁	1	1	yes	characteristic, complemented
277	C₇₁	#62001	1	1	yes	characteristic, direct product factor, Sylow
278	C₂×H(5,𝔽₂)	—	2	1	no	—
279	C₂×AGL(1,ℤ/8)	C₂²×C₇₁	1	6	yes	complemented
280	Q₈²	C₂²×C₇₁	1	1	yes	characteristic, complemented
281	#555	—	2	3	no	—
282	#555	C₂²×C₇₁	1	3	yes	complemented
283	#550	—	2	1	no	—
284	#550	—	2	1	no	—
285	#550	C₂²×C₇₁	1	2	yes	—
286	#550	C₂²×C₇₁	1	3	yes	complemented
287	#550	C₂²×C₇₁	1	1	yes	characteristic, complemented
288	D₄²	—	2	3	no	—
289	D₄²	C₂²×C₇₁	1	6	yes	—
290	#535	—	2	3	no	—
291	#532	—	2	3	no	—
292	#532	C₂²×C₇₁	1	3	yes	—
293	#532	C₂²×C₇₁	1	3	yes	complemented
294	C₂×#127	C₂²×C₇₁	1	3	yes	—
295	C₄×#41	—	2	1	no	—
296	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
297	C₄×#41	C₂²×C₇₁	1	3	yes	complemented
298	#493	C₂²×C₇₁	1	6	yes	complemented
299	#486	C₂²×C₇₁	1	6	yes	complemented
300	#482	C₂²×C₇₁	1	6	yes	complemented
301	#477	C₂²×C₇₁	1	1	yes	characteristic, complemented
302	#476	C₂²×C₇₁	1	3	yes	complemented
303	#476	C₂²×C₇₁	1	6	yes	complemented
304	#469	C₂²×C₇₁	1	12	yes	complemented
305	#460	C₂²×C₇₁	1	3	yes	complemented
306	#460	C₂²×C₇₁	1	6	yes	complemented
307	#459	C₂²×C₇₁	1	3	yes	complemented
308	#459	C₂²×C₇₁	1	6	yes	complemented
309	#459	C₂²×C₇₁	1	6	yes	complemented
310	#459	C₂²×C₇₁	1	6	yes	complemented
311	#449	C₂²×C₇₁	1	6	yes	complemented
312	#447	C₂²×C₇₁	1	12	yes	complemented
313	C₈×Q₈	C₂²×C₇₁	1	2	yes	complemented
314	C₄×SD₁₆	C₂²×C₇₁	1	6	yes	complemented
315	C₄×D₈	C₂²×C₇₁	1	6	yes	complemented
316	#436	C₂²×C₇₁	1	6	yes	complemented
317	#431	C₂²×C₇₁	1	3	yes	complemented
318	#423	C₂²×C₇₁	1	3	yes	complemented
319	#418	C₂²×C₇₁	1	6	yes	complemented
320	H(5,𝔽₂)	—	4	8	no	—
321	C₂×#41	—	2	1	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	3	no	—
324	C₂×#41	—	2	3	no	—
325	C₂×#41	D₄×C₇₁	1	1	yes	characteristic
326	C₂×#41	C₂³×C₇₁	1	3	yes	—
327	C₂²×D₄	—	2	6	no	—
328	C₂²×D₄	—	2	3	no	—
329	C₂²×D₄	—	2	3	no	—
330	C₂²×D₄	C₂³×C₇₁	1	6	yes	—
331	AGL(1,ℤ/8)	—	4	12	no	—
332	C₂×SD₁₆	C₂³×C₇₁	1	6	yes	—
333	C₂×D₈	C₂³×C₇₁	1	12	yes	—
334	C₂×D₈	C₂³×C₇₁	1	6	yes	—
335	C₂×M₁₆	C₂³×C₇₁	1	6	yes	—
336	C₄⋊Q₈	—	2	3	no	—
337	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₇₁	1	3	yes	—
339	#127	—	2	3	no	—
340	#127	—	2	3	no	—
341	#127	—	2	3	no	—
342	#127	C₂³×C₇₁	1	3	yes	—
343	#127	C₂³×C₇₁	1	6	yes	—
344	#127	C₂³×C₇₁	1	3	yes	—
345	#124	—	2	6	no	—
346	#124	—	2	3	no	—
347	#124	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	—	2	3	no	—
350	#121	—	2	3	no	—
351	#121	—	2	3	no	—
352	#121	—	2	6	no	—
353	#121	—	2	6	no	—
354	#121	—	2	6	no	—
355	#121	C₂³×C₇₁	1	6	yes	—
356	#121	C₂³×C₇₁	1	6	yes	—
357	#120	—	2	12	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
360	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
361	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
362	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
363	C₄×D₄	—	2	3	no	—
364	C₄×D₄	—	2	3	no	—
365	C₄×D₄	—	2	3	no	—
366	C₄×D₄	—	2	3	no	—
367	C₄×D₄	—	2	3	no	—
368	C₄×D₄	D₄×C₇₁	1	3	yes	—
369	C₄×D₄	C₂³×C₇₁	1	3	yes	—
370	C₄×D₄	C₂³×C₇₁	1	3	yes	—
371	C₄×D₄	C₂³×C₇₁	1	6	yes	—
372	C₄×D₄	C₂³×C₇₁	1	3	yes	—
373	#117	—	2	3	no	—
374	#117	—	2	3	no	—
375	#117	D₄×C₇₁	1	3	yes	—
376	#117	C₂³×C₇₁	1	3	yes	—
377	C₂×C₄²	—	2	3	no	—
378	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
379	#107	C₂³×C₇₁	1	2	yes	—
380	#106	C₂³×C₇₁	1	6	yes	—
381	C₄⋊C₈	C₂³×C₇₁	1	3	yes	—
382	C₄⋊C₈	C₂³×C₇₁	1	6	yes	—
383	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
384	#103	C₂³×C₇₁	1	2	yes	—
385	#102	C₂³×C₇₁	1	6	yes	—
386	#102	C₂³×C₇₁	1	12	yes	—
387	#102	C₂³×C₇₁	1	6	yes	—
388	#102	C₂³×C₇₁	1	6	yes	—
389	C₂²⋊C₈	C₂³×C₇₁	1	6	yes	—
390	C₄×C₈	C₂³×C₇₁	1	6	yes	—
391	C₂⁴	—	2	6	no	—
392	#41	—	2	4	no	—
393	#41	—	2	6	no	—
394	#41	—	4	3	no	—
395	#41	—	4	12	no	—
396	#41	—	4	4	no	—
397	#41	—	4	8	no	—
398	#41	—	8	6	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	3	no	—
401	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
402	C₂×Q₈	C₂⁴×C₇₁	1	1	yes	characteristic
403	C₂×D₄	—	2	3	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	3	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	2	6	no	—
409	C₂×D₄	—	2	3	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	6	no	—
412	C₂×D₄	—	2	3	no	—
413	C₂×D₄	—	4	6	no	—
414	C₂×D₄	—	4	6	no	—
415	C₂×D₄	—	4	12	no	—
416	C₂×D₄	—	4	24	no	—
417	C₂×D₄	—	8	6	no	—
418	C₂×D₄	C₂×D₄×C₇₁	1	3	yes	—
419	C₂×D₄	C₂⁴×C₇₁	1	6	yes	—
420	C₂×D₄	C₂⁴×C₇₁	1	3	yes	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	—	2	3	no	—
423	C₂²×C₄	—	2	3	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
426	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
427	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
428	SD₁₆	—	2	12	no	—
429	D₈	—	4	6	no	—
430	D₈	—	4	12	no	—
431	M₁₆	—	4	6	no	—
432	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
433	C₂×C₈	C₂⁴×C₇₁	1	6	yes	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	3	no	—
436	C₄⋊C₄	—	2	3	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	C₂×D₄×C₇₁	1	3	yes	—
439	C₄⋊C₄	C₂×D₄×C₇₁	1	3	yes	—
440	C₄⋊C₄	C₂⁴×C₇₁	1	3	yes	—
441	C₄⋊C₄	C₂⁴×C₇₁	1	1	yes	characteristic
442	C₄⋊C₄	C₂⁴×C₇₁	1	2	yes	—
443	C₂²⋊C₄	—	2	3	no	—
444	C₂²⋊C₄	—	2	6	no	—
445	C₂²⋊C₄	—	2	3	no	—
446	C₂²⋊C₄	—	2	3	no	—
447	C₂²⋊C₄	—	2	6	no	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	C₂×D₄×C₇₁	1	6	yes	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	3	no	—
452	C₄²	—	2	3	no	—
453	C₄²	C₂×D₄×C₇₁	1	3	yes	—
454	C₄²	C₂×D₄×C₇₁	1	3	yes	—
455	C₄²	C₂⁴×C₇₁	1	3	yes	—
456	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
457	C₂³	—	2	3	no	—
458	C₂³	—	2	6	no	—
459	C₂³	—	4	12	no	—
460	C₂³	—	8	6	no	—
461	C₂³	—	8	6	no	—
462	C₂³	C₂²×D₄×C₇₁	1	3	yes	—
463	Q₈	—	2	1	no	—
464	Q₈	—	4	2	no	—
465	Q₈	—	8	3	no	—
466	Q₈	AGL(1,ℤ/8)×C₇₁	1	4	yes	complemented
467	Q₈	H(5,𝔽₂)×C₇₁	1	2	yes	complemented
468	D₄	—	2	6	no	—
469	D₄	—	2	3	no	—
470	D₄	—	2	6	no	—
471	D₄	—	2	3	no	—
472	D₄	—	2	6	no	—
473	D₄	—	4	6	no	—
474	D₄	—	4	6	no	—
475	D₄	—	4	12	no	—
476	D₄	—	4	12	no	—
477	D₄	—	4	6	no	—
478	D₄	—	4	6	no	—
479	D₄	—	8	6	no	—
480	D₄	—	8	6	no	—
481	D₄	—	8	3	no	—
482	D₄	—	8	6	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	6	no	—
485	C₂×C₄	—	2	6	no	—
486	C₂×C₄	—	2	3	no	—
487	C₂×C₄	—	2	3	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	3	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	4	6	no	—
492	C₂×C₄	—	4	12	no	—
493	C₂×C₄	—	4	6	no	—
494	C₂×C₄	—	4	6	no	—
495	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
496	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
498	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
499	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
500	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
501	C₈	—	2	6	no	—
502	C₈	—	2	2	no	—
503	C₂²	—	2	3	no	—
504	C₂²	—	2	3	no	—
505	C₂²	—	2	3	no	—
506	C₂²	—	4	6	no	—
507	C₂²	—	4	3	no	—
508	C₂²	—	4	3	no	—
509	C₂²	—	4	6	no	—
510	C₂²	—	4	3	no	—
511	C₂²	—	4	6	no	—
512	C₂²	—	16	12	no	—
513	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	3	no	—
516	C₄	—	2	3	no	—
517	C₄	—	2	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	3	no	—
521	C₄	C₂×AGL(1,ℤ/8)×C₇₁	1	6	yes	—
522	C₄	#576×C₇₁	1	2	yes	—
523	C₄	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
524	C₄	#585×C₇₁	1	1	yes	characteristic
525	C₂	—	4	3	no	—
526	C₂	—	8	6	no	—
527	C₂	—	8	3	no	—
528	C₂	C₇₁×#3488	1	1	yes	characteristic
529	C₂	C₂²×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
530	C₂	C₂×#576×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
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×#3498	C₂	semidirect
2. C₇₁×#3483	C₂	semidirect
3. C₇₁×#3391	C₂	semidirect
4. C₇₁×#3362	C₂	semidirect
5. C₇₁×#3325	C₂	semidirect
6. C₇₁×#3274	C₂	semidirect
7. C₇₁×#3253	C₂	semidirect
8. C₇₁×#3205	C₂	semidirect
9. C₇₁×#3106	C₂	semidirect
10. C₇₁×#3038	C₂	semidirect
11. C₇₁×#2991	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₇₁	C₂²	semidirect
14. Q₈²×C₇₁	C₂²	semidirect
16. #555×C₇₁	C₂²	semidirect
20. #550×C₇₁	C₂²	semidirect
21. #550×C₇₁	C₂²	semidirect
27. #532×C₇₁	C₂²	semidirect
31. C₄×#41×C₇₁	C₂²	semidirect
32. #493×C₇₁	C₂²	semidirect
33. #486×C₇₁	C₂²	semidirect
34. #482×C₇₁	C₂²	semidirect
35. #477×C₇₁	C₂²	semidirect
36. #476×C₇₁	C₂²	semidirect
37. #476×C₇₁	C₂²	semidirect
38. #469×C₇₁	C₂²	semidirect
39. #460×C₇₁	C₂²	semidirect
40. #460×C₇₁	C₂²	semidirect
41. #459×C₇₁	C₂²	semidirect
42. #459×C₇₁	C₂²	semidirect
43. #459×C₇₁	C₂²	semidirect
44. #459×C₇₁	C₂²	semidirect
45. #449×C₇₁	C₂²	semidirect
46. #447×C₇₁	C₂²	semidirect
47. C₈×Q₈×C₇₁	C₂²	semidirect
48. C₄×SD₁₆×C₇₁	C₂²	semidirect
49. C₄×D₈×C₇₁	C₂²	semidirect
50. #436×C₇₁	C₂²	semidirect
51. #431×C₇₁	C₂²	semidirect
52. #423×C₇₁	C₂²	semidirect
53. #418×C₇₁	C₂²	semidirect
200. Q₈×C₇₁	AGL(1,ℤ/8)	semidirect
201. Q₈×C₇₁	H(5,𝔽₂)	semidirect
259. #62001	C₇₁	direct (N × H)
266. #3498	C₂×C₇₁	semidirect
267. #3483	C₂×C₇₁	semidirect
268. #3391	C₂×C₇₁	semidirect
269. #3362	C₂×C₇₁	semidirect
270. #3325	C₂×C₇₁	semidirect
271. #3274	C₂×C₇₁	semidirect
272. #3253	C₂×C₇₁	semidirect
273. #3205	C₂×C₇₁	semidirect
274. #3106	C₂×C₇₁	semidirect
275. #3038	C₂×C₇₁	semidirect
276. #2991	C₂×C₇₁	semidirect
277. C₇₁	#62001	direct (N × H)
279. C₂×AGL(1,ℤ/8)	C₂²×C₇₁	semidirect
280. Q₈²	C₂²×C₇₁	semidirect
282. #555	C₂²×C₇₁	semidirect
286. #550	C₂²×C₇₁	semidirect
287. #550	C₂²×C₇₁	semidirect
293. #532	C₂²×C₇₁	semidirect
297. C₄×#41	C₂²×C₇₁	semidirect
298. #493	C₂²×C₇₁	semidirect
299. #486	C₂²×C₇₁	semidirect
300. #482	C₂²×C₇₁	semidirect
301. #477	C₂²×C₇₁	semidirect
302. #476	C₂²×C₇₁	semidirect
303. #476	C₂²×C₇₁	semidirect
304. #469	C₂²×C₇₁	semidirect
305. #460	C₂²×C₇₁	semidirect
306. #460	C₂²×C₇₁	semidirect
307. #459	C₂²×C₇₁	semidirect
308. #459	C₂²×C₇₁	semidirect
309. #459	C₂²×C₇₁	semidirect
310. #459	C₂²×C₇₁	semidirect
311. #449	C₂²×C₇₁	semidirect
312. #447	C₂²×C₇₁	semidirect
313. C₈×Q₈	C₂²×C₇₁	semidirect
314. C₄×SD₁₆	C₂²×C₇₁	semidirect
315. C₄×D₈	C₂²×C₇₁	semidirect
316. #436	C₂²×C₇₁	semidirect
317. #431	C₂²×C₇₁	semidirect
318. #423	C₂²×C₇₁	semidirect
319. #418	C₂²×C₇₁	semidirect
466. Q₈	AGL(1,ℤ/8)×C₇₁	semidirect
467. Q₈	H(5,𝔽₂)×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3498	2
C₇₁×#3483	2
C₇₁×#3391	2
C₇₁×#3362	2
C₇₁×#3325	2
C₇₁×#3274	2
C₇₁×#3253	2
C₇₁×#3205	2
C₇₁×#3106	2
C₇₁×#3038	2
C₇₁×#2991	2
#62001	71

Subgroups — C71×#62001 #31882873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#62001 #31882873