Subgroups — C₇₁×#17719 #31838591

Order 18176.

1050 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1552	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1551	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1550	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1547	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1542	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1540	C₂	1	2	yes	maximal, complemented
10	C₂³×C₈×C₇₁	—	2	1	no	—
11	#533×C₇₁	C₂²	1	1	yes	characteristic
12	#533×C₇₁	C₂²	1	1	yes	characteristic
13	#533×C₇₁	C₂²	1	1	yes	characteristic
14	#532×C₇₁	C₂²	1	1	yes	characteristic
15	#532×C₇₁	C₂²	1	1	yes	characteristic
16	#532×C₇₁	C₂²	1	1	yes	characteristic
17	#529×C₇₁	—	2	1	no	—
18	#529×C₇₁	—	2	1	no	—
19	#529×C₇₁	—	2	1	no	—
20	#529×C₇₁	—	2	1	no	—
21	#529×C₇₁	C₄	1	2	yes	—
22	#529×C₇₁	C₄	1	2	yes	—
23	#525×C₇₁	—	2	1	no	—
24	#525×C₇₁	—	2	1	no	—
25	#525×C₇₁	C₄	1	2	yes	—
26	C₄×#41×C₇₁	—	2	1	no	—
27	C₄×#41×C₇₁	—	2	1	no	—
28	C₄×#41×C₇₁	C₄	1	2	yes	—
29	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
30	#432×C₇₁	—	2	1	no	—
31	#432×C₇₁	C₂²	1	2	yes	—
32	#431×C₇₁	—	2	1	no	—
33	#431×C₇₁	C₂²	1	2	yes	—
34	#423×C₇₁	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
36	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
37	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
38	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
39	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
41	#407×C₇₁	—	2	1	no	—
42	C₄×M₁₆×C₇₁	—	2	1	no	—
43	#333×C₇₁	C₂²	1	4	yes	complemented
44	#332×C₇₁	C₂²	1	4	yes	complemented
45	#332×C₇₁	C₂²	1	4	yes	complemented
46	#331×C₇₁	C₂²	1	4	yes	complemented
47	C₂×#41×C₇₁	—	2	1	no	—
48	C₂×#41×C₇₁	—	2	1	no	—
49	C₂×#41×C₇₁	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₇₁	—	2	2	no	—
54	C₂²×C₈×C₇₁	—	2	2	no	—
55	C₂²×C₈×C₇₁	—	2	2	no	—
56	C₂²×C₈×C₇₁	—	2	2	no	—
57	C₂²×C₈×C₇₁	—	4	4	no	—
58	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
61	#127×C₇₁	C₂³	1	1	yes	characteristic
62	#126×C₇₁	—	2	2	no	—
63	#126×C₇₁	—	2	2	no	—
64	#125×C₇₁	—	2	1	no	—
65	#125×C₇₁	—	2	1	no	—
66	#125×C₇₁	C₂×C₄	1	2	yes	—
67	#124×C₇₁	—	2	1	no	—
68	#124×C₇₁	—	2	1	no	—
69	#124×C₇₁	C₂×C₄	1	2	yes	—
70	#123×C₇₁	—	2	2	no	—
71	#123×C₇₁	—	2	2	no	—
72	#123×C₇₁	—	2	2	no	—
73	C₂²⋊Q₈×C₇₁	—	2	1	no	—
74	C₂²⋊Q₈×C₇₁	—	2	1	no	—
75	C₂²⋊Q₈×C₇₁	—	2	1	no	—
76	C₂²⋊Q₈×C₇₁	—	2	1	no	—
77	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
78	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
79	#121×C₇₁	—	2	1	no	—
80	#121×C₇₁	—	2	1	no	—
81	#121×C₇₁	—	2	1	no	—
82	#121×C₇₁	—	2	1	no	—
83	#121×C₇₁	C₂×C₄	1	2	yes	—
84	#121×C₇₁	C₂×C₄	1	2	yes	—
85	#120×C₇₁	—	2	2	no	—
86	C₄×Q₈×C₇₁	—	2	1	no	—
87	C₄×Q₈×C₇₁	—	2	1	no	—
88	C₄×Q₈×C₇₁	—	2	1	no	—
89	C₄×Q₈×C₇₁	—	2	1	no	—
90	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
92	C₄×D₄×C₇₁	—	2	2	no	—
93	C₄×D₄×C₇₁	—	2	2	no	—
94	C₄×D₄×C₇₁	—	2	2	no	—
95	C₄×D₄×C₇₁	—	2	2	no	—
96	C₄×D₄×C₇₁	—	2	2	no	—
97	C₄×D₄×C₇₁	—	2	2	no	—
98	C₄×D₄×C₇₁	—	2	1	no	—
99	C₄×D₄×C₇₁	—	2	1	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	#117×C₇₁	—	2	1	no	—
105	#117×C₇₁	—	2	1	no	—
106	#117×C₇₁	—	2	2	no	—
107	#117×C₇₁	—	2	2	no	—
108	#117×C₇₁	—	2	2	no	—
109	#117×C₇₁	C₂×C₄	1	2	yes	—
110	C₂×C₄²×C₇₁	—	2	2	no	—
111	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
123	C₄⋊C₈×C₇₁	—	2	2	no	—
124	C₄⋊C₈×C₇₁	—	2	2	no	—
125	C₄⋊C₈×C₇₁	—	2	2	no	—
126	C₄⋊C₈×C₇₁	—	2	2	no	—
127	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
128	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
129	C₂²⋊C₈×C₇₁	—	2	2	no	—
130	C₂²⋊C₈×C₇₁	—	2	2	no	—
131	#97×C₇₁	—	2	2	no	—
132	C₄×C₈×C₇₁	—	2	2	no	—
133	C₂⁴×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
134	#41×C₇₁	—	4	4	no	—
135	#41×C₇₁	—	4	4	no	—
136	#41×C₇₁	—	4	2	no	—
137	#41×C₇₁	—	4	2	no	—
138	C₂×Q₈×C₇₁	—	2	1	no	—
139	C₂×Q₈×C₇₁	—	2	1	no	—
140	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
141	C₂×D₄×C₇₁	—	2	2	no	—
142	C₂×D₄×C₇₁	—	2	2	no	—
143	C₂×D₄×C₇₁	—	2	2	no	—
144	C₂×D₄×C₇₁	—	2	1	no	—
145	C₂×D₄×C₇₁	—	2	1	no	—
146	C₂×D₄×C₇₁	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₇₁	—	2	4	no	—
148	C₂²×C₄×C₇₁	—	2	2	no	—
149	C₂²×C₄×C₇₁	—	2	2	no	—
150	C₂²×C₄×C₇₁	—	2	2	no	—
151	C₂²×C₄×C₇₁	—	2	2	no	—
152	C₂²×C₄×C₇₁	—	2	2	no	—
153	C₂²×C₄×C₇₁	—	4	2	no	—
154	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
172	M₁₆×C₇₁	—	4	4	no	—
173	C₂×C₈×C₇₁	—	2	4	no	—
174	C₂×C₈×C₇₁	—	2	4	no	—
175	C₂×C₈×C₇₁	—	4	4	no	—
176	C₂×C₈×C₇₁	—	4	4	no	—
177	C₂×C₈×C₇₁	—	4	4	no	—
178	C₄⋊C₄×C₇₁	—	2	2	no	—
179	C₄⋊C₄×C₇₁	—	2	2	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	1	no	—
185	C₄⋊C₄×C₇₁	—	2	1	no	—
186	C₄⋊C₄×C₇₁	—	2	1	no	—
187	C₄⋊C₄×C₇₁	—	2	1	no	—
188	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
189	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
190	C₂²⋊C₄×C₇₁	—	2	2	no	—
191	C₂²⋊C₄×C₇₁	—	2	2	no	—
192	C₂²⋊C₄×C₇₁	—	2	2	no	—
193	C₂²⋊C₄×C₇₁	—	2	2	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	C₄²×C₇₁	—	2	2	no	—
200	C₄²×C₇₁	—	2	2	no	—
201	C₄²×C₇₁	—	2	2	no	—
202	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
218	C₂³×C₇₁	—	2	2	no	—
219	C₂³×C₇₁	—	2	2	no	—
220	C₂³×C₇₁	—	2	2	no	—
221	C₂³×C₇₁	—	2	2	no	—
222	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	Q₈×C₇₁	—	4	2	no	—
226	Q₈×C₇₁	—	4	2	no	—
227	Q₈×C₇₁	—	4	1	no	—
228	Q₈×C₇₁	—	4	1	no	—
229	D₄×C₇₁	—	4	2	no	—
230	D₄×C₇₁	—	4	4	no	—
231	D₄×C₇₁	—	4	4	no	—
232	D₄×C₇₁	—	4	4	no	—
233	D₄×C₇₁	—	4	2	no	—
234	D₄×C₇₁	—	4	1	no	—
235	D₄×C₇₁	—	4	1	no	—
236	C₂×C₄×C₇₁	—	2	2	no	—
237	C₂×C₄×C₇₁	—	2	2	no	—
238	C₂×C₄×C₇₁	—	2	2	no	—
239	C₂×C₄×C₇₁	—	2	2	no	—
240	C₂×C₄×C₇₁	—	2	2	no	—
241	C₂×C₄×C₇₁	—	2	2	no	—
242	C₂×C₄×C₇₁	—	2	1	no	—
243	C₂×C₄×C₇₁	—	2	1	no	—
244	C₂×C₄×C₇₁	—	2	1	no	—
245	C₂×C₄×C₇₁	—	2	1	no	—
246	C₂×C₄×C₇₁	—	2	2	no	—
247	C₂×C₄×C₇₁	—	2	2	no	—
248	C₂×C₄×C₇₁	—	2	2	no	—
249	C₂×C₄×C₇₁	—	2	2	no	—
250	C₂×C₄×C₇₁	—	2	2	no	—
251	C₂×C₄×C₇₁	—	2	2	no	—
252	C₂×C₄×C₇₁	—	2	2	no	—
253	C₂×C₄×C₇₁	—	4	1	no	—
254	C₂×C₄×C₇₁	—	4	1	no	—
255	C₂×C₄×C₇₁	—	4	1	no	—
256	C₂×C₄×C₇₁	—	4	1	no	—
257	C₂×C₄×C₇₁	—	4	4	no	—
258	C₂×C₄×C₇₁	—	4	4	no	—
259	C₂×C₄×C₇₁	—	4	4	no	—
260	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₇₁	#135	1	4	yes	—
270	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
271	C₂×C₄×C₇₁	#137	1	2	yes	—
272	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
273	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
275	C₈×C₇₁	—	4	4	no	—
276	C₈×C₇₁	—	4	4	no	—
277	C₂²×C₇₁	—	2	8	no	—
278	C₂²×C₇₁	—	2	1	no	—
279	C₂²×C₇₁	—	2	1	no	—
280	C₂²×C₇₁	—	2	1	no	—
281	C₂²×C₇₁	—	2	1	no	—
282	C₂²×C₇₁	—	2	1	no	—
283	C₂²×C₇₁	—	2	1	no	—
284	C₂²×C₇₁	—	4	2	no	—
285	C₂²×C₇₁	—	4	2	no	—
286	C₂²×C₇₁	—	4	2	no	—
287	C₂²×C₇₁	#413	1	2	yes	—
288	C₂²×C₇₁	#416	1	2	yes	—
289	C₂²×C₇₁	#417	1	2	yes	—
290	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
291	C₄×C₇₁	—	2	1	no	—
292	C₄×C₇₁	—	2	1	no	—
293	C₄×C₇₁	—	2	1	no	—
294	C₄×C₇₁	—	2	1	no	—
295	C₄×C₇₁	—	2	1	no	—
296	C₄×C₇₁	—	2	1	no	—
297	C₄×C₇₁	—	2	1	no	—
298	C₄×C₇₁	—	2	1	no	—
299	C₄×C₇₁	—	4	2	no	—
300	C₄×C₇₁	—	4	2	no	—
301	C₄×C₇₁	—	4	2	no	—
302	C₄×C₇₁	#416	1	2	yes	—
303	C₄×C₇₁	#417	1	2	yes	—
304	C₄×C₇₁	C₂×#135	1	2	yes	—
305	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₄×C₇₁	C₂×#137	1	1	yes	characteristic
307	#17719	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₇₁	—	2	2	no	—
309	C₂×C₇₁	—	2	2	no	—
310	C₂×C₇₁	—	2	2	no	—
311	C₂×C₇₁	—	8	2	no	—
312	C₂×C₇₁	C₂×#413	1	1	yes	characteristic
313	C₂×C₇₁	C₂×#416	1	1	yes	characteristic
314	C₂×C₇₁	C₂×#417	1	1	yes	characteristic
315	#3444	C₂×C₇₁	1	1	yes	characteristic
316	#2920	C₂×C₇₁	1	1	yes	characteristic, complemented
317	C₂²×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic, complemented
318	#1552	C₂×C₇₁	1	2	yes	complemented
319	#1551	C₂×C₇₁	1	2	yes	complemented
320	#1550	C₂×C₇₁	1	2	yes	complemented
321	#1547	C₂×C₇₁	1	2	yes	complemented
322	#1542	C₂×C₇₁	1	2	yes	complemented
323	#1540	C₂×C₇₁	1	2	yes	complemented
324	C₇₁	#17719	1	1	yes	characteristic, direct product factor, Sylow
325	C₂³×C₈	—	2	1	no	—
326	#533	C₂²×C₇₁	1	1	yes	characteristic
327	#533	C₂²×C₇₁	1	1	yes	characteristic
328	#533	C₂²×C₇₁	1	1	yes	characteristic
329	#532	C₂²×C₇₁	1	1	yes	characteristic
330	#532	C₂²×C₇₁	1	1	yes	characteristic
331	#532	C₂²×C₇₁	1	1	yes	characteristic
332	#529	—	2	1	no	—
333	#529	—	2	1	no	—
334	#529	—	2	1	no	—
335	#529	—	2	1	no	—
336	#529	C₄×C₇₁	1	2	yes	—
337	#529	C₄×C₇₁	1	2	yes	—
338	#525	—	2	1	no	—
339	#525	—	2	1	no	—
340	#525	C₄×C₇₁	1	2	yes	—
341	C₄×#41	—	2	1	no	—
342	C₄×#41	—	2	1	no	—
343	C₄×#41	C₄×C₇₁	1	2	yes	—
344	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
345	#432	—	2	1	no	—
346	#432	C₂²×C₇₁	1	2	yes	—
347	#431	—	2	1	no	—
348	#431	C₂²×C₇₁	1	2	yes	—
349	#423	C₂²×C₇₁	1	2	yes	—
350	C₂×C₄⋊C₈	—	2	1	no	—
351	C₂×C₄⋊C₈	—	2	1	no	—
352	C₂×C₄⋊C₈	—	2	1	no	—
353	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
354	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
355	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
356	#407	—	2	1	no	—
357	C₄×M₁₆	—	2	1	no	—
358	#333	C₂²×C₇₁	1	4	yes	complemented
359	#332	C₂²×C₇₁	1	4	yes	complemented
360	#332	C₂²×C₇₁	1	4	yes	complemented
361	#331	C₂²×C₇₁	1	4	yes	complemented
362	C₂×#41	—	2	1	no	—
363	C₂×#41	—	2	1	no	—
364	C₂×#41	C₂×C₄×C₇₁	1	2	yes	—
365	C₂³×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
366	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
367	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
368	C₂×M₁₆	—	2	2	no	—
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	—	2	2	no	—
371	C₂²×C₈	—	2	2	no	—
372	C₂²×C₈	—	4	4	no	—
373	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
374	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
375	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
376	#127	C₂³×C₇₁	1	1	yes	characteristic
377	#126	—	2	2	no	—
378	#126	—	2	2	no	—
379	#125	—	2	1	no	—
380	#125	—	2	1	no	—
381	#125	C₂×C₄×C₇₁	1	2	yes	—
382	#124	—	2	1	no	—
383	#124	—	2	1	no	—
384	#124	C₂×C₄×C₇₁	1	2	yes	—
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	#123	—	2	2	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
393	C₂²⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
394	#121	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	C₂×C₄×C₇₁	1	2	yes	—
399	#121	C₂×C₄×C₇₁	1	2	yes	—
400	#120	—	2	2	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	C₂×C₄×C₇₁	1	2	yes	—
406	C₄×Q₈	C₂×C₄×C₇₁	1	2	yes	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
418	C₄×D₄	C₂×C₄×C₇₁	1	2	yes	—
419	#117	—	2	1	no	—
420	#117	—	2	1	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	—	2	2	no	—
424	#117	C₂×C₄×C₇₁	1	2	yes	—
425	C₂×C₄²	—	2	2	no	—
426	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
427	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
428	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
429	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
430	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
432	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
434	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
438	C₄⋊C₈	—	2	2	no	—
439	C₄⋊C₈	—	2	2	no	—
440	C₄⋊C₈	—	2	2	no	—
441	C₄⋊C₈	—	2	2	no	—
442	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
443	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
444	C₂²⋊C₈	—	2	2	no	—
445	C₂²⋊C₈	—	2	2	no	—
446	#97	—	2	2	no	—
447	C₄×C₈	—	2	2	no	—
448	C₂⁴	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
449	#41	—	4	4	no	—
450	#41	—	4	4	no	—
451	#41	—	4	2	no	—
452	#41	—	4	2	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	C₂²×C₄×C₇₁	1	2	yes	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	C₂²×C₄×C₇₁	1	2	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
471	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
472	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
473	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
487	M₁₆	—	4	4	no	—
488	C₂×C₈	—	2	4	no	—
489	C₂×C₈	—	2	4	no	—
490	C₂×C₈	—	4	4	no	—
491	C₂×C₈	—	4	4	no	—
492	C₂×C₈	—	4	4	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
504	C₄⋊C₄	C₂²×C₄×C₇₁	1	2	yes	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
518	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
519	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
520	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
521	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
522	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
523	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
524	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
525	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
527	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
528	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
530	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
532	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
538	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
539	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
540	Q₈	—	4	2	no	—
541	Q₈	—	4	2	no	—
542	Q₈	—	4	1	no	—
543	Q₈	—	4	1	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	4	no	—
546	D₄	—	4	4	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	2	no	—
549	D₄	—	4	1	no	—
550	D₄	—	4	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	4	no	—
575	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
578	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
579	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
580	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
581	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
582	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
584	C₂×C₄	#135×C₇₁	1	4	yes	—
585	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
586	C₂×C₄	#137×C₇₁	1	2	yes	—
587	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
589	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
590	C₈	—	4	4	no	—
591	C₈	—	4	4	no	—
592	C₂²	—	2	8	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	#413×C₇₁	1	2	yes	—
603	C₂²	#416×C₇₁	1	2	yes	—
604	C₂²	#417×C₇₁	1	2	yes	—
605	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	4	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	#416×C₇₁	1	2	yes	—
618	C₄	#417×C₇₁	1	2	yes	—
619	C₄	C₂×#135×C₇₁	1	2	yes	—
620	C₄	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
621	C₄	C₂×#137×C₇₁	1	1	yes	characteristic
622	C₂	—	2	2	no	—
623	C₂	—	2	2	no	—
624	C₂	—	2	2	no	—
625	C₂	—	8	2	no	—
626	C₂	C₂×#413×C₇₁	1	1	yes	characteristic
627	C₂	C₂×#416×C₇₁	1	1	yes	characteristic
628	C₂	C₂×#417×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₄²	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂²×C₂²⋊C₄
prime radical	C₇₁×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2920	C₂	semidirect
3. C₂²×C₄⋊C₈×C₇₁	C₂	semidirect
4. C₇₁×#1552	C₂	semidirect
5. C₇₁×#1551	C₂	semidirect
6. C₇₁×#1550	C₂	semidirect
7. C₇₁×#1547	C₂	semidirect
8. C₇₁×#1542	C₂	semidirect
9. C₇₁×#1540	C₂	semidirect
43. #333×C₇₁	C₂²	semidirect
44. #332×C₇₁	C₂²	semidirect
45. #332×C₇₁	C₂²	semidirect
46. #331×C₇₁	C₂²	semidirect
307. #17719	C₇₁	direct (N × H)
316. #2920	C₂×C₇₁	semidirect
317. C₂²×C₄⋊C₈	C₂×C₇₁	semidirect
318. #1552	C₂×C₇₁	semidirect
319. #1551	C₂×C₇₁	semidirect
320. #1550	C₂×C₇₁	semidirect
321. #1547	C₂×C₇₁	semidirect
322. #1542	C₂×C₇₁	semidirect
323. #1540	C₂×C₇₁	semidirect
324. C₇₁	#17719	direct (N × H)
358. #333	C₂²×C₇₁	semidirect
359. #332	C₂²×C₇₁	semidirect
360. #332	C₂²×C₇₁	semidirect
361. #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×#3444	2
C₇₁×#2920	2
C₂²×C₄⋊C₈×C₇₁	2
C₇₁×#1552	2
C₇₁×#1551	2
C₇₁×#1550	2
C₇₁×#1547	2
C₇₁×#1542	2
C₇₁×#1540	2
#17719	71

Subgroups — C71×#17719 #31838591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#17719 #31838591