Subgroups — C₇₁×#17691 #31838563

Order 18176.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3510	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1537	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1535	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1532	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1529	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1527	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#1527	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₁×#1526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1524	C₂	1	2	yes	maximal, complemented
12	#559×C₇₁	—	2	1	no	—
13	#559×C₇₁	—	2	1	no	—
14	#556×C₇₁	—	2	1	no	—
15	#556×C₇₁	—	2	1	no	—
16	#551×C₇₁	—	2	2	no	—
17	#551×C₇₁	—	2	2	no	—
18	#542×C₇₁	C₄	1	2	yes	—
19	#542×C₇₁	C₄	1	2	yes	—
20	#536×C₇₁	C₄	1	1	yes	characteristic
21	#535×C₇₁	C₄	1	1	yes	characteristic
22	#533×C₇₁	C₂²	1	1	yes	characteristic
23	#529×C₇₁	C₂²	1	2	yes	—
24	#529×C₇₁	C₂²	1	2	yes	—
25	C₂×#124×C₇₁	C₂²	1	1	yes	characteristic
26	#519×C₇₁	C₄	1	1	yes	characteristic
27	#518×C₇₁	C₄	1	1	yes	characteristic
28	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
29	#446×C₇₁	—	2	1	no	—
30	C₈×Q₈×C₇₁	—	2	1	no	—
31	#436×C₇₁	—	2	1	no	—
32	#436×C₇₁	—	2	2	no	—
33	#435×C₇₁	—	2	2	no	—
34	C₈×D₄×C₇₁	—	2	1	no	—
35	#432×C₇₁	C₂²	1	1	yes	characteristic
36	#432×C₇₁	C₂²	1	1	yes	characteristic
37	#432×C₇₁	C₂²	1	2	yes	complemented
38	#432×C₇₁	C₂²	1	2	yes	complemented
39	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₇₁	C₂²	1	2	yes	complemented
42	C₂×#97×C₇₁	C₂²	1	2	yes	complemented
43	#330×C₇₁	C₂²	1	1	yes	characteristic, complemented
44	#330×C₇₁	C₂²	1	2	yes	complemented
45	#330×C₇₁	C₂²	1	1	yes	characteristic, complemented
46	#330×C₇₁	C₂²	1	1	yes	characteristic, complemented
47	#330×C₇₁	C₂²	1	1	yes	characteristic, complemented
48	#330×C₇₁	C₂²	1	2	yes	complemented
49	#329×C₇₁	C₂²	1	2	yes	—
50	#329×C₇₁	C₂²	1	2	yes	—
51	#329×C₇₁	C₂²	1	2	yes	—
52	#329×C₇₁	C₂²	1	2	yes	—
53	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₇₁	—	2	1	no	—
57	C₂×M₁₆×C₇₁	—	2	1	no	—
58	C₂×M₁₆×C₇₁	—	2	2	no	—
59	C₂²×C₈×C₇₁	—	2	2	no	—
60	C₄⋊Q₈×C₇₁	—	2	1	no	—
61	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
63	#127×C₇₁	—	2	1	no	—
64	#126×C₇₁	—	2	2	no	—
65	#126×C₇₁	—	2	2	no	—
66	#126×C₇₁	—	2	2	no	—
67	#126×C₇₁	—	2	2	no	—
68	#125×C₇₁	C₂³	1	1	yes	characteristic
69	#125×C₇₁	C₂³	1	1	yes	characteristic
70	#124×C₇₁	—	2	1	no	—
71	#124×C₇₁	—	2	1	no	—
72	#124×C₇₁	—	2	2	no	—
73	#124×C₇₁	—	2	2	no	—
74	#124×C₇₁	—	2	1	no	—
75	#124×C₇₁	—	2	1	no	—
76	#124×C₇₁	C₂×C₄	1	2	yes	—
77	#124×C₇₁	C₂³	1	2	yes	—
78	#123×C₇₁	—	2	2	no	—
79	#123×C₇₁	—	2	2	no	—
80	#123×C₇₁	—	2	2	no	—
81	#123×C₇₁	—	2	2	no	—
82	#123×C₇₁	—	2	2	no	—
83	C₂²⋊Q₈×C₇₁	—	2	1	no	—
84	C₂²⋊Q₈×C₇₁	—	2	1	no	—
85	C₂²⋊Q₈×C₇₁	—	2	2	no	—
86	C₂²⋊Q₈×C₇₁	—	2	2	no	—
87	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
88	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
89	#121×C₇₁	C₂×C₄	1	2	yes	—
90	#121×C₇₁	C₂×C₄	1	2	yes	—
91	#120×C₇₁	—	2	2	no	—
92	C₄×Q₈×C₇₁	—	2	1	no	—
93	C₄×Q₈×C₇₁	—	2	1	no	—
94	C₄×Q₈×C₇₁	—	2	1	no	—
95	C₄×Q₈×C₇₁	—	2	1	no	—
96	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
100	C₄×D₄×C₇₁	—	2	2	no	—
101	C₄×D₄×C₇₁	—	2	2	no	—
102	C₄×D₄×C₇₁	—	2	1	no	—
103	C₄×D₄×C₇₁	—	2	1	no	—
104	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
107	C₄×D₄×C₇₁	D₄	1	2	yes	—
108	C₄×D₄×C₇₁	D₄	1	2	yes	—
109	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
110	#117×C₇₁	—	2	2	no	—
111	#117×C₇₁	—	2	2	no	—
112	#117×C₇₁	—	2	2	no	—
113	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
114	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
115	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
116	#117×C₇₁	C₂×C₄	1	2	yes	—
117	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
120	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
122	C₄⋊C₈×C₇₁	—	2	1	no	—
123	C₄⋊C₈×C₇₁	—	2	1	no	—
124	C₄⋊C₈×C₇₁	—	2	2	no	—
125	C₄⋊C₈×C₇₁	—	2	2	no	—
126	C₂²⋊C₈×C₇₁	—	2	1	no	—
127	C₂²⋊C₈×C₇₁	—	2	1	no	—
128	C₂²⋊C₈×C₇₁	—	2	2	no	—
129	C₂²⋊C₈×C₇₁	—	2	2	no	—
130	#97×C₇₁	—	2	2	no	—
131	#97×C₇₁	C₂³	1	2	yes	—
132	#97×C₇₁	C₂³	1	2	yes	—
133	#97×C₇₁	C₂³	1	1	yes	characteristic
134	#97×C₇₁	C₂³	1	1	yes	characteristic
135	#97×C₇₁	C₂³	1	1	yes	characteristic
136	#97×C₇₁	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₇₁	—	2	1	no	—
138	C₄×C₈×C₇₁	—	2	1	no	—
139	C₄×C₈×C₇₁	—	2	2	no	—
140	C₂⁴×C₇₁	—	2	1	no	—
141	#41×C₇₁	—	4	2	no	—
142	C₂×Q₈×C₇₁	—	2	1	no	—
143	C₂×Q₈×C₇₁	—	2	1	no	—
144	C₂×Q₈×C₇₁	—	2	1	no	—
145	C₂×Q₈×C₇₁	—	4	2	no	—
146	C₂×Q₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₇₁	—	2	2	no	—
149	C₂×D₄×C₇₁	—	2	2	no	—
150	C₂×D₄×C₇₁	—	2	1	no	—
151	C₂×D₄×C₇₁	—	4	2	no	—
152	C₂×D₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂×D₄×C₇₁	C₂²⋊C₄	1	2	yes	—
154	C₂×D₄×C₇₁	C₂²×C₄	1	2	yes	—
155	C₂²×C₄×C₇₁	—	2	1	no	—
156	C₂²×C₄×C₇₁	—	2	2	no	—
157	C₂²×C₄×C₇₁	—	2	2	no	—
158	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
160	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
164	M₁₆×C₇₁	—	4	2	no	—
165	M₁₆×C₇₁	—	4	2	no	—
166	M₁₆×C₇₁	—	4	4	no	—
167	C₂×C₈×C₇₁	—	2	1	no	—
168	C₂×C₈×C₇₁	—	2	1	no	—
169	C₂×C₈×C₇₁	—	2	1	no	—
170	C₂×C₈×C₇₁	—	2	1	no	—
171	C₂×C₈×C₇₁	—	2	2	no	—
172	C₂×C₈×C₇₁	—	2	2	no	—
173	C₂×C₈×C₇₁	—	4	4	no	—
174	C₄⋊C₄×C₇₁	—	2	2	no	—
175	C₄⋊C₄×C₇₁	—	2	2	no	—
176	C₄⋊C₄×C₇₁	—	2	2	no	—
177	C₄⋊C₄×C₇₁	—	2	2	no	—
178	C₄⋊C₄×C₇₁	—	2	1	no	—
179	C₄⋊C₄×C₇₁	—	2	1	no	—
180	C₄⋊C₄×C₇₁	—	2	1	no	—
181	C₄⋊C₄×C₇₁	—	2	1	no	—
182	C₄⋊C₄×C₇₁	—	2	1	no	—
183	C₄⋊C₄×C₇₁	—	2	1	no	—
184	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
185	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
186	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
187	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
188	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
192	C₂²⋊C₄×C₇₁	—	2	1	no	—
193	C₂²⋊C₄×C₇₁	—	2	1	no	—
194	C₂²⋊C₄×C₇₁	—	2	2	no	—
195	C₂²⋊C₄×C₇₁	—	2	2	no	—
196	C₂²⋊C₄×C₇₁	—	2	2	no	—
197	C₂²⋊C₄×C₇₁	—	2	2	no	—
198	C₂²⋊C₄×C₇₁	—	2	2	no	—
199	C₂²⋊C₄×C₇₁	—	2	2	no	—
200	C₂²⋊C₄×C₇₁	—	2	2	no	—
201	C₂²⋊C₄×C₇₁	—	2	2	no	—
202	C₂²⋊C₄×C₇₁	—	2	2	no	—
203	C₂²⋊C₄×C₇₁	—	2	2	no	—
204	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
205	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
206	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
207	C₄²×C₇₁	—	2	2	no	—
208	C₄²×C₇₁	—	2	2	no	—
209	C₄²×C₇₁	—	2	1	no	—
210	C₄²×C₇₁	—	2	1	no	—
211	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
212	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₇₁	C₂×D₄	1	2	yes	—
214	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₇₁	C₂×D₄	1	2	yes	—
217	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
218	C₂³×C₇₁	—	2	2	no	—
219	C₂³×C₇₁	—	4	2	no	—
220	C₂³×C₇₁	—	8	1	no	—
221	C₂³×C₇₁	—	8	1	no	—
222	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
224	Q₈×C₇₁	—	2	2	no	—
225	Q₈×C₇₁	—	4	2	no	—
226	Q₈×C₇₁	—	4	2	no	—
227	Q₈×C₇₁	—	8	1	no	—
228	Q₈×C₇₁	—	8	1	no	—
229	D₄×C₇₁	—	2	2	no	—
230	D₄×C₇₁	—	4	2	no	—
231	D₄×C₇₁	—	4	2	no	—
232	D₄×C₇₁	—	4	2	no	—
233	D₄×C₇₁	—	8	2	no	—
234	D₄×C₇₁	—	8	2	no	—
235	C₂×C₄×C₇₁	—	2	2	no	—
236	C₂×C₄×C₇₁	—	2	1	no	—
237	C₂×C₄×C₇₁	—	2	1	no	—
238	C₂×C₄×C₇₁	—	2	2	no	—
239	C₂×C₄×C₇₁	—	2	1	no	—
240	C₂×C₄×C₇₁	—	2	1	no	—
241	C₂×C₄×C₇₁	—	4	2	no	—
242	C₂×C₄×C₇₁	—	4	2	no	—
243	C₂×C₄×C₇₁	—	4	1	no	—
244	C₂×C₄×C₇₁	—	4	1	no	—
245	C₂×C₄×C₇₁	—	4	2	no	—
246	C₂×C₄×C₇₁	—	4	2	no	—
247	C₂×C₄×C₇₁	—	8	2	no	—
248	C₂×C₄×C₇₁	—	8	2	no	—
249	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
250	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
254	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
258	C₈×C₇₁	—	2	2	no	—
259	C₈×C₇₁	—	2	2	no	—
260	C₈×C₇₁	—	4	2	no	—
261	C₈×C₇₁	—	4	1	no	—
262	C₈×C₇₁	—	4	1	no	—
263	C₈×C₇₁	—	4	2	no	—
264	C₂²×C₇₁	—	2	1	no	—
265	C₂²×C₇₁	—	2	2	no	—
266	C₂²×C₇₁	—	2	2	no	—
267	C₂²×C₇₁	—	2	2	no	—
268	C₂²×C₇₁	—	2	1	no	—
269	C₂²×C₇₁	—	2	1	no	—
270	C₂²×C₇₁	—	4	2	no	—
271	C₂²×C₇₁	—	4	2	no	—
272	C₂²×C₇₁	—	4	2	no	—
273	C₂²×C₇₁	—	16	2	no	—
274	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
275	C₄×C₇₁	—	2	1	no	—
276	C₄×C₇₁	—	2	2	no	—
277	C₄×C₇₁	—	2	1	no	—
278	C₄×C₇₁	—	2	1	no	—
279	C₄×C₇₁	—	2	2	no	—
280	C₄×C₇₁	—	2	2	no	—
281	C₄×C₇₁	—	2	1	no	—
282	C₄×C₇₁	—	2	1	no	—
283	C₄×C₇₁	—	4	2	no	—
284	C₄×C₇₁	—	4	1	no	—
285	C₄×C₇₁	—	4	1	no	—
286	C₄×C₇₁	—	4	1	no	—
287	C₄×C₇₁	—	4	1	no	—
288	C₄×C₇₁	#413	1	2	yes	—
289	#17691	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
290	C₂×C₇₁	—	4	2	no	—
291	C₂×C₇₁	—	4	1	no	—
292	C₂×C₇₁	—	8	2	no	—
293	C₂×C₇₁	C₂×#413	1	1	yes	characteristic
294	C₂×C₇₁	#2901	1	1	yes	characteristic
295	C₂×C₇₁	#2901	1	1	yes	characteristic
296	#3510	C₂×C₇₁	1	1	yes	characteristic
297	#2971	C₂×C₇₁	1	1	yes	characteristic, complemented
298	#2970	C₂×C₇₁	1	1	yes	characteristic, complemented
299	#1537	C₂×C₇₁	1	1	yes	characteristic, complemented
300	#1535	C₂×C₇₁	1	2	yes	complemented
301	#1532	C₂×C₇₁	1	2	yes	complemented
302	#1529	C₂×C₇₁	1	2	yes	complemented
303	#1527	C₂×C₇₁	1	1	yes	characteristic, complemented
304	#1527	C₂×C₇₁	1	1	yes	characteristic, complemented
305	#1526	C₂×C₇₁	1	1	yes	characteristic, complemented
306	#1524	C₂×C₇₁	1	2	yes	complemented
307	C₇₁	#17691	1	1	yes	characteristic, direct product factor, Sylow
308	#559	—	2	1	no	—
309	#559	—	2	1	no	—
310	#556	—	2	1	no	—
311	#556	—	2	1	no	—
312	#551	—	2	2	no	—
313	#551	—	2	2	no	—
314	#542	C₄×C₇₁	1	2	yes	—
315	#542	C₄×C₇₁	1	2	yes	—
316	#536	C₄×C₇₁	1	1	yes	characteristic
317	#535	C₄×C₇₁	1	1	yes	characteristic
318	#533	C₂²×C₇₁	1	1	yes	characteristic
319	#529	C₂²×C₇₁	1	2	yes	—
320	#529	C₂²×C₇₁	1	2	yes	—
321	C₂×#124	C₂²×C₇₁	1	1	yes	characteristic
322	#519	C₄×C₇₁	1	1	yes	characteristic
323	#518	C₄×C₇₁	1	1	yes	characteristic
324	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
325	#446	—	2	1	no	—
326	C₈×Q₈	—	2	1	no	—
327	#436	—	2	1	no	—
328	#436	—	2	2	no	—
329	#435	—	2	2	no	—
330	C₈×D₄	—	2	1	no	—
331	#432	C₂²×C₇₁	1	1	yes	characteristic
332	#432	C₂²×C₇₁	1	1	yes	characteristic
333	#432	C₂²×C₇₁	1	2	yes	complemented
334	#432	C₂²×C₇₁	1	2	yes	complemented
335	C₄×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
336	C₄×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
337	C₄×M₁₆	C₂²×C₇₁	1	2	yes	complemented
338	C₂×#97	C₂²×C₇₁	1	2	yes	complemented
339	#330	C₂²×C₇₁	1	2	yes	complemented
340	#330	C₂²×C₇₁	1	2	yes	complemented
341	#330	C₂²×C₇₁	1	1	yes	characteristic, complemented
342	#330	C₂²×C₇₁	1	1	yes	characteristic, complemented
343	#330	C₂²×C₇₁	1	1	yes	characteristic, complemented
344	#330	C₂²×C₇₁	1	1	yes	characteristic, complemented
345	#329	C₂²×C₇₁	1	2	yes	—
346	#329	C₂²×C₇₁	1	2	yes	—
347	#329	C₂²×C₇₁	1	2	yes	—
348	#329	C₂²×C₇₁	1	2	yes	—
349	C₂×#41	C₂×C₄×C₇₁	1	1	yes	characteristic
350	C₂²×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
351	C₂²×D₄	C₂×C₄×C₇₁	1	1	yes	characteristic
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	2	no	—
355	C₂²×C₈	—	2	2	no	—
356	C₄⋊Q₈	—	2	1	no	—
357	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
358	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
359	#127	—	2	1	no	—
360	#126	—	2	2	no	—
361	#126	—	2	2	no	—
362	#126	—	2	2	no	—
363	#126	—	2	2	no	—
364	#125	C₂³×C₇₁	1	1	yes	characteristic
365	#125	C₂³×C₇₁	1	1	yes	characteristic
366	#124	—	2	1	no	—
367	#124	—	2	1	no	—
368	#124	—	2	2	no	—
369	#124	—	2	2	no	—
370	#124	—	2	1	no	—
371	#124	—	2	1	no	—
372	#124	C₂×C₄×C₇₁	1	2	yes	—
373	#124	C₂³×C₇₁	1	2	yes	—
374	#123	—	2	2	no	—
375	#123	—	2	2	no	—
376	#123	—	2	2	no	—
377	#123	—	2	2	no	—
378	#123	—	2	2	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
384	C₂²⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
385	#121	C₂×C₄×C₇₁	1	2	yes	—
386	#121	C₂×C₄×C₇₁	1	2	yes	—
387	#120	—	2	2	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	—	2	1	no	—
392	C₄×Q₈	C₂×C₄×C₇₁	1	2	yes	—
393	C₄×Q₈	C₂×C₄×C₇₁	1	2	yes	—
394	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
395	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	—	2	1	no	—
400	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
401	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
402	C₄×D₄	D₄×C₇₁	1	2	yes	—
403	C₄×D₄	D₄×C₇₁	1	2	yes	—
404	C₄×D₄	D₄×C₇₁	1	1	yes	characteristic
405	C₄×D₄	D₄×C₇₁	1	1	yes	characteristic
406	#117	—	2	2	no	—
407	#117	—	2	2	no	—
408	#117	—	2	2	no	—
409	#117	C₂×C₄×C₇₁	1	1	yes	characteristic
410	#117	C₂×C₄×C₇₁	1	1	yes	characteristic
411	#117	C₂×C₄×C₇₁	1	1	yes	characteristic
412	#117	C₂×C₄×C₇₁	1	2	yes	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	2	no	—
416	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
418	C₄⋊C₈	—	2	1	no	—
419	C₄⋊C₈	—	2	1	no	—
420	C₄⋊C₈	—	2	2	no	—
421	C₄⋊C₈	—	2	2	no	—
422	C₂²⋊C₈	—	2	1	no	—
423	C₂²⋊C₈	—	2	1	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	—	2	2	no	—
426	#97	—	2	2	no	—
427	#97	C₂³×C₇₁	1	1	yes	characteristic
428	#97	C₂³×C₇₁	1	2	yes	—
429	#97	C₂³×C₇₁	1	1	yes	characteristic
430	#97	C₂³×C₇₁	1	1	yes	characteristic
431	#97	C₂³×C₇₁	1	1	yes	characteristic
432	#97	C₂³×C₇₁	1	2	yes	—
433	C₄×C₈	—	2	1	no	—
434	C₄×C₈	—	2	1	no	—
435	C₄×C₈	—	2	2	no	—
436	C₂⁴	—	2	1	no	—
437	#41	—	4	2	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
443	C₂×Q₈	C₂²×C₄×C₇₁	1	2	yes	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	4	2	no	—
448	C₂×D₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
449	C₂×D₄	C₂²⋊C₄×C₇₁	1	2	yes	—
450	C₂×D₄	C₂²×C₄×C₇₁	1	2	yes	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
456	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
460	M₁₆	—	4	2	no	—
461	M₁₆	—	4	2	no	—
462	M₁₆	—	4	4	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	4	4	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
481	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
482	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
483	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
484	C₄⋊C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
485	C₄⋊C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
486	C₄⋊C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
487	C₄⋊C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
501	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
502	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
508	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
509	C₄²	C₂×D₄×C₇₁	1	2	yes	—
510	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₇₁	1	2	yes	—
513	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
514	C₂³	—	2	2	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	8	1	no	—
517	C₂³	—	8	1	no	—
518	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
519	C₂³	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
520	Q₈	—	2	2	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	4	2	no	—
523	Q₈	—	8	1	no	—
524	Q₈	—	8	1	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	2	no	—
529	D₄	—	8	2	no	—
530	D₄	—	8	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	8	2	no	—
544	C₂×C₄	—	8	2	no	—
545	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
546	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
547	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
548	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
549	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
550	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
551	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	4	2	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	2	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	16	2	no	—
570	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	1	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	#413×C₇₁	1	2	yes	—
585	C₂	—	4	1	no	—
586	C₂	—	4	2	no	—
587	C₂	—	8	2	no	—
588	C₂	C₂×#413×C₇₁	1	1	yes	characteristic
589	C₂	C₇₁×#2901	1	1	yes	characteristic
590	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	C₄²	C₂⁴×C₇₁
prime radical	#535×C₇₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
2. C₇₁×#2971	C₂	semidirect
3. C₇₁×#2970	C₂	semidirect
4. C₇₁×#1537	C₂	semidirect
5. C₇₁×#1535	C₂	semidirect
6. C₇₁×#1532	C₂	semidirect
7. C₇₁×#1529	C₂	semidirect
8. C₇₁×#1527	C₂	semidirect
9. C₇₁×#1527	C₂	semidirect
10. C₇₁×#1526	C₂	semidirect
11. C₇₁×#1524	C₂	semidirect
37. #432×C₇₁	C₂²	semidirect
38. #432×C₇₁	C₂²	semidirect
41. C₄×M₁₆×C₇₁	C₂²	semidirect
42. C₂×#97×C₇₁	C₂²	semidirect
43. #330×C₇₁	C₂²	semidirect
44. #330×C₇₁	C₂²	semidirect
45. #330×C₇₁	C₂²	semidirect
46. #330×C₇₁	C₂²	semidirect
47. #330×C₇₁	C₂²	semidirect
48. #330×C₇₁	C₂²	semidirect
289. #17691	C₇₁	direct (N × H)
297. #2971	C₂×C₇₁	semidirect
298. #2970	C₂×C₇₁	semidirect
299. #1537	C₂×C₇₁	semidirect
300. #1535	C₂×C₇₁	semidirect
301. #1532	C₂×C₇₁	semidirect
302. #1529	C₂×C₇₁	semidirect
303. #1527	C₂×C₇₁	semidirect
304. #1527	C₂×C₇₁	semidirect
305. #1526	C₂×C₇₁	semidirect
306. #1524	C₂×C₇₁	semidirect
307. C₇₁	#17691	direct (N × H)
333. #432	C₂²×C₇₁	semidirect
334. #432	C₂²×C₇₁	semidirect
337. C₄×M₁₆	C₂²×C₇₁	semidirect
338. C₂×#97	C₂²×C₇₁	semidirect
339. #330	C₂²×C₇₁	semidirect
340. #330	C₂²×C₇₁	semidirect
341. #330	C₂²×C₇₁	semidirect
342. #330	C₂²×C₇₁	semidirect
343. #330	C₂²×C₇₁	semidirect
344. #330	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×#3510	2
C₇₁×#2971	2
C₇₁×#2970	2
C₇₁×#1537	2
C₇₁×#1535	2
C₇₁×#1532	2
C₇₁×#1529	2
C₇₁×#1527	2
C₇₁×#1526	2
C₇₁×#1524	2
#17691	71

Subgroups — C71×#17691 #31838563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#17691 #31838563