Subgroups — C₇₁×#17687 #31838559

Order 18176.

896 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3512	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×#1536	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×#1528	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#1528	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₁×#1525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1524	C₂	1	2	yes	maximal, complemented
12	#553×C₇₁	—	2	2	no	—
13	#553×C₇₁	—	2	2	no	—
14	#553×C₇₁	—	2	1	no	—
15	#553×C₇₁	—	2	1	no	—
16	#552×C₇₁	—	2	1	no	—
17	#552×C₇₁	—	2	1	no	—
18	#541×C₇₁	C₄	1	2	yes	—
19	#540×C₇₁	C₄	1	2	yes	—
20	#537×C₇₁	C₄	1	1	yes	characteristic
21	#537×C₇₁	C₄	1	1	yes	characteristic
22	#532×C₇₁	C₂²	1	1	yes	characteristic
23	#529×C₇₁	C₂²	1	2	yes	—
24	#529×C₇₁	C₂²	1	2	yes	—
25	C₂×#125×C₇₁	C₂²	1	1	yes	characteristic
26	#520×C₇₁	C₄	1	1	yes	characteristic
27	#520×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	2	no	—
32	#436×C₇₁	—	2	1	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	—
38	#432×C₇₁	C₂²	1	2	yes	—
39	C₄×M₁₆×C₇₁	C₂²	1	2	yes	—
40	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
42	C₂×#97×C₇₁	C₂²	1	2	yes	—
43	#330×C₇₁	C₂²	1	2	yes	—
44	#330×C₇₁	C₂²	1	2	yes	—
45	#330×C₇₁	C₂²	1	2	yes	—
46	#330×C₇₁	C₂²	1	2	yes	—
47	#329×C₇₁	C₂²	1	1	yes	characteristic
48	#329×C₇₁	C₂²	1	2	yes	—
49	#329×C₇₁	C₂²	1	1	yes	characteristic
50	#329×C₇₁	C₂²	1	1	yes	characteristic
51	#329×C₇₁	C₂²	1	1	yes	characteristic
52	#329×C₇₁	C₂²	1	2	yes	—
53	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×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	C₂×C₄	1	1	yes	characteristic
61	#127×C₇₁	C₂×C₄	1	1	yes	characteristic
62	#126×C₇₁	—	2	2	no	—
63	#126×C₇₁	—	2	2	no	—
64	#126×C₇₁	—	2	2	no	—
65	#126×C₇₁	—	2	2	no	—
66	#125×C₇₁	—	2	2	no	—
67	#125×C₇₁	—	2	2	no	—
68	#125×C₇₁	—	2	1	no	—
69	#125×C₇₁	—	2	1	no	—
70	#125×C₇₁	C₂×C₄	1	2	yes	—
71	#125×C₇₁	C₂³	1	2	yes	—
72	#124×C₇₁	C₂³	1	1	yes	characteristic
73	#124×C₇₁	C₂³	1	1	yes	characteristic
74	#123×C₇₁	—	2	2	no	—
75	#123×C₇₁	—	2	2	no	—
76	#123×C₇₁	—	2	2	no	—
77	#123×C₇₁	—	2	2	no	—
78	C₂²⋊Q₈×C₇₁	—	2	1	no	—
79	C₂²⋊Q₈×C₇₁	—	2	1	no	—
80	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
81	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
82	#121×C₇₁	—	2	1	no	—
83	#121×C₇₁	—	2	1	no	—
84	#121×C₇₁	—	2	2	no	—
85	#121×C₇₁	—	2	2	no	—
86	#121×C₇₁	—	2	1	no	—
87	#121×C₇₁	—	2	1	no	—
88	#121×C₇₁	—	2	2	no	—
89	#121×C₇₁	—	2	2	no	—
90	#121×C₇₁	C₂×C₄	1	2	yes	—
91	#121×C₇₁	C₂×C₄	1	2	yes	—
92	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
96	C₄×D₄×C₇₁	—	2	2	no	—
97	C₄×D₄×C₇₁	—	2	2	no	—
98	C₄×D₄×C₇₁	—	2	2	no	—
99	C₄×D₄×C₇₁	—	2	2	no	—
100	C₄×D₄×C₇₁	—	2	1	no	—
101	C₄×D₄×C₇₁	—	2	1	no	—
102	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
103	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
104	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
105	C₄×D₄×C₇₁	D₄	1	2	yes	—
106	C₄×D₄×C₇₁	D₄	1	2	yes	—
107	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
108	#117×C₇₁	—	2	1	no	—
109	#117×C₇₁	—	2	1	no	—
110	#117×C₇₁	—	2	2	no	—
111	#117×C₇₁	—	2	2	no	—
112	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
113	#117×C₇₁	C₂×C₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
117	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
118	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
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	2	no	—
124	C₄⋊C₈×C₇₁	—	2	2	no	—
125	C₄⋊C₈×C₇₁	—	2	1	no	—
126	C₂²⋊C₈×C₇₁	—	2	2	no	—
127	C₂²⋊C₈×C₇₁	—	2	1	no	—
128	C₂²⋊C₈×C₇₁	—	2	1	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	2	no	—
139	C₄×C₈×C₇₁	—	2	1	no	—
140	#41×C₇₁	—	4	2	no	—
141	#41×C₇₁	—	4	2	no	—
142	#41×C₇₁	—	4	2	no	—
143	C₂×Q₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
145	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₇₁	—	2	1	no	—
147	C₂×D₄×C₇₁	—	2	1	no	—
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	—	2	1	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	1	yes	characteristic
155	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₇₁	—	2	1	no	—
157	C₂²×C₄×C₇₁	—	2	1	no	—
158	C₂²×C₄×C₇₁	—	2	1	no	—
159	C₂²×C₄×C₇₁	—	2	1	no	—
160	C₂²×C₄×C₇₁	—	2	2	no	—
161	C₂²×C₄×C₇₁	—	2	2	no	—
162	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
168	M₁₆×C₇₁	—	4	2	no	—
169	M₁₆×C₇₁	—	4	4	no	—
170	M₁₆×C₇₁	—	4	2	no	—
171	C₂×C₈×C₇₁	—	2	1	no	—
172	C₂×C₈×C₇₁	—	2	1	no	—
173	C₂×C₈×C₇₁	—	2	1	no	—
174	C₂×C₈×C₇₁	—	2	2	no	—
175	C₂×C₈×C₇₁	—	2	1	no	—
176	C₂×C₈×C₇₁	—	2	2	no	—
177	C₂×C₈×C₇₁	—	4	4	no	—
178	C₄⋊C₄×C₇₁	—	2	1	no	—
179	C₄⋊C₄×C₇₁	—	2	1	no	—
180	C₄⋊C₄×C₇₁	—	2	2	no	—
181	C₄⋊C₄×C₇₁	—	2	2	no	—
182	C₄⋊C₄×C₇₁	—	2	2	no	—
183	C₄⋊C₄×C₇₁	—	2	2	no	—
184	C₄⋊C₄×C₇₁	—	2	2	no	—
185	C₄⋊C₄×C₇₁	—	2	2	no	—
186	C₄⋊C₄×C₇₁	—	2	1	no	—
187	C₄⋊C₄×C₇₁	—	2	1	no	—
188	C₄⋊C₄×C₇₁	—	2	2	no	—
189	C₄⋊C₄×C₇₁	—	2	2	no	—
190	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
193	C₄⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
194	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
195	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
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	1	no	—
203	C₂²⋊C₄×C₇₁	—	2	1	no	—
204	C₂²⋊C₄×C₇₁	—	2	1	no	—
205	C₂²⋊C₄×C₇₁	—	2	1	no	—
206	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
207	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
208	C₂²⋊C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
209	C₄²×C₇₁	—	2	2	no	—
210	C₄²×C₇₁	—	2	2	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	1	no	—
219	C₂³×C₇₁	—	2	1	no	—
220	C₂³×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	Q₈×C₇₁	—	2	2	no	—
223	Q₈×C₇₁	—	4	1	no	—
224	Q₈×C₇₁	—	4	1	no	—
225	D₄×C₇₁	—	2	2	no	—
226	D₄×C₇₁	—	4	2	no	—
227	D₄×C₇₁	—	4	1	no	—
228	D₄×C₇₁	—	4	1	no	—
229	D₄×C₇₁	—	4	2	no	—
230	D₄×C₇₁	—	4	2	no	—
231	D₄×C₇₁	—	8	1	no	—
232	D₄×C₇₁	—	8	1	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	2	no	—
240	C₂×C₄×C₇₁	—	4	2	no	—
241	C₂×C₄×C₇₁	—	4	2	no	—
242	C₂×C₄×C₇₁	—	4	1	no	—
243	C₂×C₄×C₇₁	—	4	1	no	—
244	C₂×C₄×C₇₁	—	4	2	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	—	8	1	no	—
250	C₂×C₄×C₇₁	—	8	1	no	—
251	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
252	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
258	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
260	C₈×C₇₁	—	2	2	no	—
261	C₈×C₇₁	—	2	2	no	—
262	C₈×C₇₁	—	4	2	no	—
263	C₈×C₇₁	—	4	1	no	—
264	C₈×C₇₁	—	4	2	no	—
265	C₈×C₇₁	—	4	1	no	—
266	C₂²×C₇₁	—	2	1	no	—
267	C₂²×C₇₁	—	2	2	no	—
268	C₂²×C₇₁	—	2	1	no	—
269	C₂²×C₇₁	—	2	1	no	—
270	C₂²×C₇₁	—	2	2	no	—
271	C₂²×C₇₁	—	2	2	no	—
272	C₂²×C₇₁	—	4	1	no	—
273	C₂²×C₇₁	—	4	1	no	—
274	C₂²×C₇₁	—	4	1	no	—
275	C₂²×C₇₁	—	4	1	no	—
276	C₂²×C₇₁	—	4	1	no	—
277	C₂²×C₇₁	—	4	1	no	—
278	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
279	C₄×C₇₁	—	2	1	no	—
280	C₄×C₇₁	—	2	2	no	—
281	C₄×C₇₁	—	2	1	no	—
282	C₄×C₇₁	—	2	1	no	—
283	C₄×C₇₁	—	2	2	no	—
284	C₄×C₇₁	—	2	2	no	—
285	C₄×C₇₁	—	2	1	no	—
286	C₄×C₇₁	—	2	1	no	—
287	C₄×C₇₁	—	4	1	no	—
288	C₄×C₇₁	—	4	1	no	—
289	C₄×C₇₁	—	4	2	no	—
290	C₄×C₇₁	—	4	2	no	—
291	C₄×C₇₁	#413	1	2	yes	—
292	#17687	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
293	C₂×C₇₁	—	4	1	no	—
294	C₂×C₇₁	—	4	2	no	—
295	C₂×C₇₁	—	8	1	no	—
296	C₂×C₇₁	—	8	1	no	—
297	C₂×C₇₁	C₂×#413	1	1	yes	characteristic
298	C₂×C₇₁	#2901	1	1	yes	characteristic
299	C₂×C₇₁	#2901	1	1	yes	characteristic
300	#3512	C₂×C₇₁	1	1	yes	characteristic
301	#2971	C₂×C₇₁	1	1	yes	characteristic, complemented
302	#2970	C₂×C₇₁	1	1	yes	characteristic, complemented
303	#1536	C₂×C₇₁	1	1	yes	characteristic, complemented
304	#1535	C₂×C₇₁	1	2	yes	complemented
305	#1532	C₂×C₇₁	1	2	yes	complemented
306	#1529	C₂×C₇₁	1	2	yes	complemented
307	#1528	C₂×C₇₁	1	1	yes	characteristic, complemented
308	#1528	C₂×C₇₁	1	1	yes	characteristic, complemented
309	#1525	C₂×C₇₁	1	1	yes	characteristic, complemented
310	#1524	C₂×C₇₁	1	2	yes	complemented
311	C₇₁	#17687	1	1	yes	characteristic, direct product factor, Sylow
312	#553	—	2	2	no	—
313	#553	—	2	2	no	—
314	#553	—	2	1	no	—
315	#553	—	2	1	no	—
316	#552	—	2	1	no	—
317	#552	—	2	1	no	—
318	#541	C₄×C₇₁	1	2	yes	—
319	#540	C₄×C₇₁	1	2	yes	—
320	#537	C₄×C₇₁	1	1	yes	characteristic
321	#537	C₄×C₇₁	1	1	yes	characteristic
322	#532	C₂²×C₇₁	1	1	yes	characteristic
323	#529	C₂²×C₇₁	1	2	yes	—
324	#529	C₂²×C₇₁	1	2	yes	—
325	C₂×#125	C₂²×C₇₁	1	1	yes	characteristic
326	#520	C₄×C₇₁	1	1	yes	characteristic
327	#520	C₄×C₇₁	1	1	yes	characteristic
328	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
329	#446	—	2	1	no	—
330	C₈×Q₈	—	2	1	no	—
331	#436	—	2	2	no	—
332	#436	—	2	1	no	—
333	#435	—	2	2	no	—
334	C₈×D₄	—	2	1	no	—
335	#432	C₂²×C₇₁	1	1	yes	characteristic
336	#432	C₂²×C₇₁	1	1	yes	characteristic
337	#432	C₂²×C₇₁	1	2	yes	—
338	#432	C₂²×C₇₁	1	2	yes	—
339	C₄×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
340	C₄×M₁₆	C₂²×C₇₁	1	2	yes	—
341	C₄×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
342	C₂×#97	C₂²×C₇₁	1	2	yes	—
343	#330	C₂²×C₇₁	1	2	yes	—
344	#330	C₂²×C₇₁	1	2	yes	—
345	#330	C₂²×C₇₁	1	2	yes	—
346	#330	C₂²×C₇₁	1	2	yes	—
347	#329	C₂²×C₇₁	1	2	yes	—
348	#329	C₂²×C₇₁	1	2	yes	—
349	#329	C₂²×C₇₁	1	1	yes	characteristic
350	#329	C₂²×C₇₁	1	1	yes	characteristic
351	#329	C₂²×C₇₁	1	1	yes	characteristic
352	#329	C₂²×C₇₁	1	1	yes	characteristic
353	C₂×#41	C₂×C₄×C₇₁	1	1	yes	characteristic
354	C₂×#41	C₂×C₄×C₇₁	1	1	yes	characteristic
355	C₂×#41	C₂×C₄×C₇₁	1	1	yes	characteristic
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	2	2	no	—
359	C₂²×C₈	—	2	2	no	—
360	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
361	#127	C₂×C₄×C₇₁	1	1	yes	characteristic
362	#126	—	2	2	no	—
363	#126	—	2	2	no	—
364	#126	—	2	2	no	—
365	#126	—	2	2	no	—
366	#125	—	2	2	no	—
367	#125	—	2	2	no	—
368	#125	—	2	1	no	—
369	#125	—	2	1	no	—
370	#125	C₂×C₄×C₇₁	1	2	yes	—
371	#125	C₂³×C₇₁	1	2	yes	—
372	#124	C₂³×C₇₁	1	1	yes	characteristic
373	#124	C₂³×C₇₁	1	1	yes	characteristic
374	#123	—	2	2	no	—
375	#123	—	2	2	no	—
376	#123	—	2	2	no	—
377	#123	—	2	2	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
381	C₂²⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
382	#121	—	2	1	no	—
383	#121	—	2	1	no	—
384	#121	—	2	2	no	—
385	#121	—	2	2	no	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	#121	—	2	2	no	—
389	#121	—	2	2	no	—
390	#121	C₂×C₄×C₇₁	1	2	yes	—
391	#121	C₂×C₄×C₇₁	1	2	yes	—
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	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
403	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
404	C₄×D₄	D₄×C₇₁	1	1	yes	characteristic
405	C₄×D₄	D₄×C₇₁	1	2	yes	—
406	C₄×D₄	D₄×C₇₁	1	2	yes	—
407	C₄×D₄	D₄×C₇₁	1	1	yes	characteristic
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#117	—	2	2	no	—
411	#117	—	2	2	no	—
412	#117	C₂×C₄×C₇₁	1	1	yes	characteristic
413	#117	C₂×C₄×C₇₁	1	2	yes	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
422	C₄⋊C₈	—	2	1	no	—
423	C₄⋊C₈	—	2	2	no	—
424	C₄⋊C₈	—	2	2	no	—
425	C₄⋊C₈	—	2	1	no	—
426	C₂²⋊C₈	—	2	2	no	—
427	C₂²⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	1	no	—
429	C₂²⋊C₈	—	2	2	no	—
430	#97	—	2	2	no	—
431	#97	C₂³×C₇₁	1	1	yes	characteristic
432	#97	C₂³×C₇₁	1	2	yes	—
433	#97	C₂³×C₇₁	1	1	yes	characteristic
434	#97	C₂³×C₇₁	1	2	yes	—
435	#97	C₂³×C₇₁	1	1	yes	characteristic
436	#97	C₂³×C₇₁	1	1	yes	characteristic
437	C₄×C₈	—	2	1	no	—
438	C₄×C₈	—	2	2	no	—
439	C₄×C₈	—	2	1	no	—
440	#41	—	4	2	no	—
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	C₂×Q₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
444	C₂×Q₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
445	C₂×Q₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
453	C₂×D₄	C₂²⋊C₄×C₇₁	1	2	yes	—
454	C₂×D₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
455	C₂×D₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
463	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
464	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
468	M₁₆	—	4	2	no	—
469	M₁₆	—	4	4	no	—
470	M₁₆	—	4	2	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	4	4	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
491	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
492	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
493	C₄⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
494	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
495	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
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	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
507	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
508	C₂²⋊C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
509	C₄²	—	2	2	no	—
510	C₄²	—	2	2	no	—
511	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
512	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₇₁	1	2	yes	—
514	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
515	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
516	C₄²	C₂×D₄×C₇₁	1	2	yes	—
517	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
521	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
522	Q₈	—	2	2	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	4	1	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	1	no	—
528	D₄	—	4	1	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	2	no	—
531	D₄	—	8	1	no	—
532	D₄	—	8	1	no	—
533	D₄	—	8	2	no	—
534	D₄	—	8	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	8	2	no	—
548	C₂×C₄	—	8	2	no	—
549	C₂×C₄	—	8	1	no	—
550	C₂×C₄	—	8	1	no	—
551	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
552	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
553	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
554	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
555	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
556	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
557	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
560	C₈	—	2	2	no	—
561	C₈	—	2	2	no	—
562	C₈	—	4	2	no	—
563	C₈	—	4	1	no	—
564	C₈	—	4	2	no	—
565	C₈	—	4	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
579	C₄	—	2	1	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	2	no	—
584	C₄	—	2	2	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	2	no	—
591	C₄	#413×C₇₁	1	2	yes	—
592	C₂	—	4	1	no	—
593	C₂	—	4	2	no	—
594	C₂	—	8	1	no	—
595	C₂	—	8	1	no	—
596	C₂	C₂×#413×C₇₁	1	1	yes	characteristic
597	C₂	C₇₁×#2901	1	1	yes	characteristic
598	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	C₇₁×#3512	C₂
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×#1536	C₂	semidirect
5. C₇₁×#1535	C₂	semidirect
6. C₇₁×#1532	C₂	semidirect
7. C₇₁×#1529	C₂	semidirect
8. C₇₁×#1528	C₂	semidirect
9. C₇₁×#1528	C₂	semidirect
10. C₇₁×#1525	C₂	semidirect
11. C₇₁×#1524	C₂	semidirect
292. #17687	C₇₁	direct (N × H)
301. #2971	C₂×C₇₁	semidirect
302. #2970	C₂×C₇₁	semidirect
303. #1536	C₂×C₇₁	semidirect
304. #1535	C₂×C₇₁	semidirect
305. #1532	C₂×C₇₁	semidirect
306. #1529	C₂×C₇₁	semidirect
307. #1528	C₂×C₇₁	semidirect
308. #1528	C₂×C₇₁	semidirect
309. #1525	C₂×C₇₁	semidirect
310. #1524	C₂×C₇₁	semidirect
311. C₇₁	#17687	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₁×#3512	2
C₇₁×#2971	2
C₇₁×#2970	2
C₇₁×#1536	2
C₇₁×#1535	2
C₇₁×#1532	2
C₇₁×#1529	2
C₇₁×#1528	2
C₇₁×#1525	2
C₇₁×#1524	2
#17687	71

Subgroups — C71×#17687 #31838559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#17687 #31838559