Subgroups — C₇₁×#17630 #31838502

Order 18176.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3506	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2975	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1517	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#1513	C₂	1	4	yes	maximal, complemented
7	C₇₁×#1504	C₂	1	4	yes	maximal, complemented
8	C₇₁×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#1502	C₂	1	1	yes	maximal, characteristic, complemented
10	#559×C₇₁	—	2	2	no	—
11	#556×C₇₁	C₄	1	2	yes	complemented
12	#555×C₇₁	C₄	1	2	yes	complemented
13	#553×C₇₁	—	2	2	no	—
14	#550×C₇₁	C₄	1	2	yes	complemented
15	#547×C₇₁	C₄	1	2	yes	complemented
16	#538×C₇₁	—	2	2	no	—
17	#535×C₇₁	—	2	1	no	—
18	#533×C₇₁	C₂²	1	1	yes	characteristic
19	#532×C₇₁	C₂²	1	1	yes	characteristic
20	#525×C₇₁	C₂²	1	4	yes	—
21	#520×C₇₁	—	2	1	no	—
22	C₂×C₄×D₄×C₇₁	—	2	1	no	—
23	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
24	#436×C₇₁	—	2	2	no	—
25	#436×C₇₁	—	2	2	no	—
26	#431×C₇₁	C₂²	1	1	yes	characteristic
27	#431×C₇₁	C₂²	1	1	yes	characteristic
28	#407×C₇₁	C₂²	1	4	yes	—
29	#389×C₇₁	—	2	1	no	—
30	#385×C₇₁	—	2	2	no	—
31	#382×C₇₁	C₂²	1	1	yes	characteristic
32	#380×C₇₁	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
34	C₂×#95×C₇₁	C₂²	1	2	yes	—
35	#328×C₇₁	C₂²	1	2	yes	—
36	#328×C₇₁	C₂²	1	2	yes	—
37	#328×C₇₁	C₂²	1	4	yes	—
38	#327×C₇₁	C₂²	1	2	yes	—
39	#327×C₇₁	C₂²	1	2	yes	—
40	#327×C₇₁	C₂²	1	4	yes	—
41	C₂×#41×C₇₁	—	2	1	no	—
42	C₂×#41×C₇₁	C₂×C₄	1	4	yes	complemented
43	C₂²×D₄×C₇₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₇₁	—	2	2	no	—
45	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₇₁	—	2	4	no	—
47	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
48	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
49	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
50	#127×C₇₁	—	2	2	no	—
51	#127×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
52	#126×C₇₁	—	2	4	no	—
53	#125×C₇₁	—	2	1	no	—
54	#124×C₇₁	—	2	2	no	—
55	#124×C₇₁	—	2	4	no	—
56	#124×C₇₁	—	2	1	no	—
57	#123×C₇₁	—	2	4	no	—
58	#123×C₇₁	—	2	4	no	—
59	C₂²⋊Q₈×C₇₁	—	2	1	no	—
60	C₂²⋊Q₈×C₇₁	C₂×C₄	1	4	yes	complemented
61	C₂²⋊Q₈×C₇₁	C₂³	1	2	yes	—
62	#121×C₇₁	—	2	2	no	—
63	#121×C₇₁	—	2	2	no	—
64	#121×C₇₁	—	2	1	no	—
65	#121×C₇₁	—	2	4	no	—
66	#121×C₇₁	C₂×C₄	1	4	yes	complemented
67	#121×C₇₁	C₂³	1	2	yes	—
68	#120×C₇₁	—	2	4	no	—
69	C₄×Q₈×C₇₁	—	2	2	no	—
70	C₄×Q₈×C₇₁	—	2	1	no	—
71	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	complemented
72	C₄×D₄×C₇₁	—	2	4	no	—
73	C₄×D₄×C₇₁	—	2	4	no	—
74	C₄×D₄×C₇₁	—	2	2	no	—
75	C₄×D₄×C₇₁	—	2	1	no	—
76	C₄×D₄×C₇₁	—	4	4	no	—
77	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	complemented
80	C₄×D₄×C₇₁	D₄	1	2	yes	—
81	C₄×D₄×C₇₁	D₄	1	2	yes	—
82	#117×C₇₁	—	2	1	no	—
83	#117×C₇₁	—	2	2	no	—
84	#117×C₇₁	C₂×C₄	1	4	yes	complemented
85	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
89	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
92	C₂×C₄²×C₇₁	—	2	1	no	—
93	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₇₁	—	2	1	no	—
95	C₄⋊C₈×C₇₁	—	2	1	no	—
96	C₂²⋊C₈×C₇₁	—	2	4	no	—
97	C₂²⋊C₈×C₇₁	C₂³	1	2	yes	—
98	C₂²⋊C₈×C₇₁	C₂³	1	2	yes	—
99	C₄×C₈×C₇₁	—	2	1	no	—
100	C₄×C₈×C₇₁	—	2	1	no	—
101	#95×C₇₁	—	2	2	no	—
102	#95×C₇₁	—	2	1	no	—
103	#95×C₇₁	C₂³	1	1	yes	characteristic
104	#95×C₇₁	C₂³	1	2	yes	—
105	#95×C₇₁	C₂³	1	1	yes	characteristic
106	C₂⁴×C₇₁	C₂×D₄	1	2	yes	—
107	#41×C₇₁	—	4	2	no	—
108	#41×C₇₁	—	4	2	no	—
109	#41×C₇₁	—	8	4	no	—
110	C₂×Q₈×C₇₁	—	2	1	no	—
111	C₂×Q₈×C₇₁	—	2	1	no	—
112	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₇₁	—	2	2	no	—
114	C₂×D₄×C₇₁	—	2	4	no	—
115	C₂×D₄×C₇₁	—	2	4	no	—
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	C₂²⋊C₄	1	2	yes	—
121	C₂×D₄×C₇₁	C₂²⋊C₄	1	2	yes	—
122	C₂×D₄×C₇₁	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₇₁	—	2	1	no	—
124	C₂²×C₄×C₇₁	—	2	2	no	—
125	C₂²×C₄×C₇₁	—	2	4	no	—
126	C₂²×C₄×C₇₁	—	2	2	no	—
127	C₂²×C₄×C₇₁	—	2	1	no	—
128	C₂²×C₄×C₇₁	—	4	4	no	—
129	C₂²×C₄×C₇₁	—	4	4	no	—
130	C₂²×C₄×C₇₁	—	8	2	no	—
131	C₂²×C₄×C₇₁	—	8	2	no	—
132	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₇₁	—	8	4	no	—
139	C₂×C₈×C₇₁	—	2	2	no	—
140	C₂×C₈×C₇₁	—	2	2	no	—
141	C₄⋊C₄×C₇₁	—	2	1	no	—
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	1	no	—
146	C₄⋊C₄×C₇₁	—	2	2	no	—
147	C₄⋊C₄×C₇₁	—	2	2	no	—
148	C₄⋊C₄×C₇₁	—	4	2	no	—
149	C₄⋊C₄×C₇₁	—	4	2	no	—
150	C₄⋊C₄×C₇₁	—	8	2	no	—
151	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
152	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
153	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
154	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
155	C₂²⋊C₄×C₇₁	—	2	4	no	—
156	C₂²⋊C₄×C₇₁	—	2	4	no	—
157	C₂²⋊C₄×C₇₁	—	2	2	no	—
158	C₂²⋊C₄×C₇₁	—	2	4	no	—
159	C₂²⋊C₄×C₇₁	—	2	2	no	—
160	C₂²⋊C₄×C₇₁	—	2	2	no	—
161	C₂²⋊C₄×C₇₁	—	4	4	no	—
162	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
163	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
164	C₄²×C₇₁	—	2	4	no	—
165	C₄²×C₇₁	—	2	2	no	—
166	C₄²×C₇₁	—	4	2	no	—
167	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
171	C₂³×C₇₁	—	2	2	no	—
172	C₂³×C₇₁	—	2	2	no	—
173	C₂³×C₇₁	—	2	2	no	—
174	C₂³×C₇₁	—	4	2	no	—
175	C₂³×C₇₁	—	4	2	no	—
176	C₂³×C₇₁	C₂×C₂²⋊C₄	1	4	yes	—
177	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₇₁	—	4	2	no	—
179	Q₈×C₇₁	—	4	2	no	—
180	Q₈×C₇₁	—	8	1	no	—
181	D₄×C₇₁	—	2	4	no	—
182	D₄×C₇₁	—	2	4	no	—
183	D₄×C₇₁	—	4	2	no	—
184	D₄×C₇₁	—	4	2	no	—
185	D₄×C₇₁	—	4	2	no	—
186	D₄×C₇₁	—	4	2	no	—
187	D₄×C₇₁	—	8	1	no	—
188	D₄×C₇₁	—	8	4	no	—
189	D₄×C₇₁	—	8	4	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	—	2	2	no	—
194	C₂×C₄×C₇₁	—	4	1	no	—
195	C₂×C₄×C₇₁	—	4	2	no	—
196	C₂×C₄×C₇₁	—	4	4	no	—
197	C₂×C₄×C₇₁	—	4	1	no	—
198	C₂×C₄×C₇₁	—	4	4	no	—
199	C₂×C₄×C₇₁	—	4	1	no	—
200	C₂×C₄×C₇₁	—	4	1	no	—
201	C₂×C₄×C₇₁	—	4	4	no	—
202	C₂×C₄×C₇₁	—	4	1	no	—
203	C₂×C₄×C₇₁	—	4	1	no	—
204	C₂×C₄×C₇₁	—	4	2	no	—
205	C₂×C₄×C₇₁	—	4	2	no	—
206	C₂×C₄×C₇₁	—	4	1	no	—
207	C₂×C₄×C₇₁	—	4	2	no	—
208	C₂×C₄×C₇₁	—	8	4	no	—
209	C₂×C₄×C₇₁	—	8	8	no	—
210	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	4	yes	—
211	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
216	C₂×C₄×C₇₁	#137	1	2	yes	—
217	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
219	C₈×C₇₁	—	4	2	no	—
220	C₈×C₇₁	—	4	2	no	—
221	C₂²×C₇₁	—	2	1	no	—
222	C₂²×C₇₁	—	2	1	no	—
223	C₂²×C₇₁	—	2	4	no	—
224	C₂²×C₇₁	—	2	4	no	—
225	C₂²×C₇₁	—	2	4	no	—
226	C₂²×C₇₁	—	4	4	no	—
227	C₂²×C₇₁	—	4	4	no	—
228	C₂²×C₇₁	—	4	2	no	—
229	C₂²×C₇₁	—	4	2	no	—
230	C₂²×C₇₁	—	4	2	no	—
231	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂²×C₇₁	C₂×#137	1	1	yes	characteristic
234	C₄×C₇₁	—	2	1	no	—
235	C₄×C₇₁	—	2	1	no	—
236	C₄×C₇₁	—	2	4	no	—
237	C₄×C₇₁	—	2	1	no	—
238	C₄×C₇₁	—	2	1	no	—
239	C₄×C₇₁	—	4	2	no	—
240	C₄×C₇₁	—	4	2	no	—
241	C₄×C₇₁	—	4	2	no	—
242	C₄×C₇₁	—	4	2	no	—
243	C₄×C₇₁	—	8	1	no	—
244	C₄×C₇₁	—	8	2	no	—
245	C₄×C₇₁	#417	1	2	yes	—
246	C₄×C₇₁	#417	1	2	yes	—
247	#17630	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂×C₇₁	—	2	1	no	—
249	C₂×C₇₁	—	2	1	no	—
250	C₂×C₇₁	—	4	4	no	—
251	C₂×C₇₁	—	8	2	no	—
252	C₂×C₇₁	#2883	1	1	yes	characteristic
253	C₂×C₇₁	C₂×#417	1	1	yes	characteristic
254	C₂×C₇₁	#2901	1	1	yes	characteristic
255	#3506	C₂×C₇₁	1	1	yes	characteristic
256	#2975	C₂×C₇₁	1	1	yes	characteristic, complemented
257	#2349	C₂×C₇₁	1	1	yes	characteristic, complemented
258	#1518	C₂×C₇₁	1	1	yes	characteristic, complemented
259	#1517	C₂×C₇₁	1	1	yes	characteristic, complemented
260	#1513	C₂×C₇₁	1	4	yes	complemented
261	#1504	C₂×C₇₁	1	4	yes	complemented
262	#1503	C₂×C₇₁	1	1	yes	characteristic, complemented
263	#1502	C₂×C₇₁	1	1	yes	characteristic, complemented
264	C₇₁	#17630	1	1	yes	characteristic, direct product factor, Sylow
265	#559	—	2	2	no	—
266	#556	C₄×C₇₁	1	2	yes	complemented
267	#555	C₄×C₇₁	1	2	yes	complemented
268	#553	—	2	2	no	—
269	#550	C₄×C₇₁	1	2	yes	complemented
270	#547	C₄×C₇₁	1	2	yes	complemented
271	#538	—	2	2	no	—
272	#535	—	2	1	no	—
273	#533	C₂²×C₇₁	1	1	yes	characteristic
274	#532	C₂²×C₇₁	1	1	yes	characteristic
275	#525	C₂²×C₇₁	1	4	yes	—
276	#520	—	2	1	no	—
277	C₂×C₄×D₄	—	2	1	no	—
278	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
279	#436	—	2	2	no	—
280	#436	—	2	2	no	—
281	#431	C₂²×C₇₁	1	1	yes	characteristic
282	#431	C₂²×C₇₁	1	1	yes	characteristic
283	#407	C₂²×C₇₁	1	4	yes	—
284	#389	—	2	1	no	—
285	#385	—	2	2	no	—
286	#382	C₂²×C₇₁	1	1	yes	characteristic
287	#380	C₂²×C₇₁	1	2	yes	—
288	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
289	C₂×#95	C₂²×C₇₁	1	2	yes	—
290	#328	C₂²×C₇₁	1	2	yes	—
291	#328	C₂²×C₇₁	1	2	yes	—
292	#328	C₂²×C₇₁	1	4	yes	—
293	#327	C₂²×C₇₁	1	2	yes	—
294	#327	C₂²×C₇₁	1	2	yes	—
295	#327	C₂²×C₇₁	1	4	yes	—
296	C₂×#41	—	2	1	no	—
297	C₂×#41	C₂×C₄×C₇₁	1	4	yes	complemented
298	C₂²×D₄	D₄×C₇₁	1	1	yes	characteristic
299	C₂³×C₄	—	2	2	no	—
300	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
301	C₂×M₁₆	—	2	4	no	—
302	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic, complemented
303	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic, complemented
304	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic, complemented
305	#127	—	2	2	no	—
306	#127	C₂×C₄×C₇₁	1	1	yes	characteristic, complemented
307	#126	—	2	4	no	—
308	#125	—	2	1	no	—
309	#124	—	2	2	no	—
310	#124	—	2	4	no	—
311	#124	—	2	1	no	—
312	#123	—	2	4	no	—
313	#123	—	2	4	no	—
314	C₂²⋊Q₈	—	2	1	no	—
315	C₂²⋊Q₈	C₂×C₄×C₇₁	1	4	yes	complemented
316	C₂²⋊Q₈	C₂³×C₇₁	1	2	yes	—
317	#121	—	2	2	no	—
318	#121	—	2	2	no	—
319	#121	—	2	1	no	—
320	#121	—	2	4	no	—
321	#121	C₂×C₄×C₇₁	1	4	yes	complemented
322	#121	C₂³×C₇₁	1	2	yes	—
323	#120	—	2	4	no	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×Q₈	—	2	1	no	—
326	C₄×Q₈	C₂×C₄×C₇₁	1	2	yes	complemented
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	4	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	1	no	—
331	C₄×D₄	—	4	4	no	—
332	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
333	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
334	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	complemented
335	C₄×D₄	D₄×C₇₁	1	2	yes	—
336	C₄×D₄	D₄×C₇₁	1	2	yes	—
337	#117	—	2	1	no	—
338	#117	—	2	2	no	—
339	#117	C₂×C₄×C₇₁	1	4	yes	complemented
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	4	no	—
352	C₂²⋊C₈	C₂³×C₇₁	1	2	yes	—
353	C₂²⋊C₈	C₂³×C₇₁	1	2	yes	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	2	1	no	—
356	#95	—	2	2	no	—
357	#95	—	2	1	no	—
358	#95	C₂³×C₇₁	1	1	yes	characteristic
359	#95	C₂³×C₇₁	1	2	yes	—
360	#95	C₂³×C₇₁	1	1	yes	characteristic
361	C₂⁴	C₂×D₄×C₇₁	1	2	yes	—
362	#41	—	4	2	no	—
363	#41	—	4	2	no	—
364	#41	—	8	4	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	C₂²×C₄×C₇₁	1	2	yes	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	C₂²⋊C₄×C₇₁	1	2	yes	—
376	C₂×D₄	C₂²⋊C₄×C₇₁	1	2	yes	—
377	C₂×D₄	C₂²×C₄×C₇₁	1	2	yes	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	C₂²×C₄×C₇₁	1	4	yes	—
388	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
391	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
393	M₁₆	—	8	4	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	8	2	no	—
406	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
407	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
408	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	4	yes	—
418	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	4	yes	—
419	C₄²	—	2	4	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	4	2	no	—
422	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
423	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	C₂×C₂²⋊C₄×C₇₁	1	4	yes	—
432	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	8	1	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	8	1	no	—
443	D₄	—	8	4	no	—
444	D₄	—	8	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	1	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	—	4	2	no	—
463	C₂×C₄	—	8	4	no	—
464	C₂×C₄	—	8	8	no	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	4	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
467	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
468	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
469	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
470	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
471	C₂×C₄	#137×C₇₁	1	2	yes	—
472	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
474	C₈	—	4	2	no	—
475	C₈	—	4	2	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	4	4	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
487	C₂²	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
488	C₂²	C₂×#137×C₇₁	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	1	no	—
500	C₄	#417×C₇₁	1	2	yes	—
501	C₄	#417×C₇₁	1	2	yes	—
502	C₂	—	2	1	no	—
503	C₂	—	2	1	no	—
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₇₁×#2883	1	1	yes	characteristic
507	C₂	C₂×#417×C₇₁	1	1	yes	characteristic
508	C₂	C₇₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₇₁×#3506	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄×D₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2975	C₂	semidirect
3. C₇₁×#2349	C₂	semidirect
4. C₇₁×#1518	C₂	semidirect
5. C₇₁×#1517	C₂	semidirect
6. C₇₁×#1513	C₂	semidirect
7. C₇₁×#1504	C₂	semidirect
8. C₇₁×#1503	C₂	semidirect
9. C₇₁×#1502	C₂	semidirect
11. #556×C₇₁	C₄	semidirect
12. #555×C₇₁	C₄	semidirect
14. #550×C₇₁	C₄	semidirect
15. #547×C₇₁	C₄	semidirect
42. C₂×#41×C₇₁	C₂×C₄	semidirect
47. C₄⋊Q₈×C₇₁	C₂×C₄	semidirect
48. C₄⋊Q₈×C₇₁	C₂×C₄	semidirect
49. C₄⋊Q₈×C₇₁	C₂×C₄	semidirect
51. #127×C₇₁	C₂×C₄	semidirect
60. C₂²⋊Q₈×C₇₁	C₂×C₄	semidirect
66. #121×C₇₁	C₂×C₄	semidirect
71. C₄×Q₈×C₇₁	C₂×C₄	semidirect
79. C₄×D₄×C₇₁	C₂×C₄	semidirect
84. #117×C₇₁	C₂×C₄	semidirect
247. #17630	C₇₁	direct (N × H)
256. #2975	C₂×C₇₁	semidirect
257. #2349	C₂×C₇₁	semidirect
258. #1518	C₂×C₇₁	semidirect
259. #1517	C₂×C₇₁	semidirect
260. #1513	C₂×C₇₁	semidirect
261. #1504	C₂×C₇₁	semidirect
262. #1503	C₂×C₇₁	semidirect
263. #1502	C₂×C₇₁	semidirect
264. C₇₁	#17630	direct (N × H)
266. #556	C₄×C₇₁	semidirect
267. #555	C₄×C₇₁	semidirect
269. #550	C₄×C₇₁	semidirect
270. #547	C₄×C₇₁	semidirect
297. C₂×#41	C₂×C₄×C₇₁	semidirect
302. C₄⋊Q₈	C₂×C₄×C₇₁	semidirect
303. C₄⋊Q₈	C₂×C₄×C₇₁	semidirect
304. C₄⋊Q₈	C₂×C₄×C₇₁	semidirect
306. #127	C₂×C₄×C₇₁	semidirect
315. C₂²⋊Q₈	C₂×C₄×C₇₁	semidirect
321. #121	C₂×C₄×C₇₁	semidirect
326. C₄×Q₈	C₂×C₄×C₇₁	semidirect
334. C₄×D₄	C₂×C₄×C₇₁	semidirect
339. #117	C₂×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×#3506	2
C₇₁×#2975	2
C₇₁×#2349	2
C₇₁×#1518	2
C₇₁×#1517	2
C₇₁×#1513	2
C₇₁×#1504	2
C₇₁×#1503	2
C₇₁×#1502	2
#17630	71

Subgroups — C71×#17630 #31838502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#17630 #31838502