Subgroups — C₇₁×#23254 #31844126

Order 18176.

664 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1939	C₂	1	2	yes	maximal
3	C₇₁×#1939	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1939	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1917	C₂	1	2	yes	maximal
6	C₇₁×#1917	C₂	1	2	yes	maximal
7	C₇₁×#1771	C₂	1	2	yes	maximal
8	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
10	C₇₁×#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#97×C₇₁	C₂²	1	2	yes	—
20	C₂×#97×C₇₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
24	#389×C₇₁	C₄	1	1	yes	characteristic
25	#389×C₇₁	C₄	1	1	yes	characteristic
26	#387×C₇₁	C₄	1	2	yes	—
27	#387×C₇₁	C₄	1	2	yes	—
28	#382×C₇₁	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
33	C₄³×C₇₁	C₂²	1	1	yes	characteristic
34	#336×C₇₁	C₂²	1	1	yes	characteristic
35	#336×C₇₁	C₂²	1	1	yes	characteristic
36	#336×C₇₁	C₂²	1	2	yes	—
37	#336×C₇₁	C₂²	1	2	yes	—
38	#336×C₇₁	C₂²	1	2	yes	—
39	#336×C₇₁	C₂²	1	2	yes	—
40	#336×C₇₁	C₂²	1	2	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	#125×C₇₁	—	2	4	no	—
48	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
62	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₇₁	—	2	2	no	—
66	C₄⋊C₈×C₇₁	—	2	2	no	—
67	C₄⋊C₈×C₇₁	—	2	4	no	—
68	C₄⋊C₈×C₇₁	—	2	2	no	—
69	C₄⋊C₈×C₇₁	—	4	2	no	—
70	C₄⋊C₈×C₇₁	Q₈	1	4	yes	—
71	#97×C₇₁	—	2	4	no	—
72	#97×C₇₁	—	4	2	no	—
73	C₄×C₈×C₇₁	—	2	2	no	—
74	C₄×C₈×C₇₁	—	2	2	no	—
75	C₄×C₈×C₇₁	—	4	2	no	—
76	#95×C₇₁	C₂×C₄	1	2	yes	—
77	#95×C₇₁	C₂×C₄	1	2	yes	—
78	#95×C₇₁	C₂×C₄	1	2	yes	—
79	#95×C₇₁	C₂×C₄	1	2	yes	—
80	#95×C₇₁	C₂×C₄	1	2	yes	—
81	#95×C₇₁	C₂×C₄	1	2	yes	—
82	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₇₁	—	2	1	no	—
94	C₂×C₈×C₇₁	—	2	1	no	—
95	C₂×C₈×C₇₁	—	2	2	no	—
96	C₂×C₈×C₇₁	—	2	1	no	—
97	C₂×C₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	—	2	1	no	—
99	C₂×C₈×C₇₁	—	2	1	no	—
100	C₂×C₈×C₇₁	—	2	1	no	—
101	C₂×C₈×C₇₁	—	2	1	no	—
102	C₂×C₈×C₇₁	—	2	2	no	—
103	C₂×C₈×C₇₁	—	2	1	no	—
104	C₂×C₈×C₇₁	—	2	2	no	—
105	C₂×C₈×C₇₁	—	2	2	no	—
106	C₂×C₈×C₇₁	—	2	1	no	—
107	C₂×C₈×C₇₁	—	2	1	no	—
108	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
109	C₂×C₈×C₇₁	#41	1	2	yes	—
110	C₂×C₈×C₇₁	#41	1	2	yes	—
111	C₄⋊C₄×C₇₁	—	2	4	no	—
112	C₄⋊C₄×C₇₁	—	2	4	no	—
113	C₄⋊C₄×C₇₁	—	2	2	no	—
114	C₄⋊C₄×C₇₁	—	2	4	no	—
115	C₄⋊C₄×C₇₁	—	2	2	no	—
116	C₄⋊C₄×C₇₁	—	2	2	no	—
117	C₄⋊C₄×C₇₁	—	4	2	no	—
118	C₄⋊C₄×C₇₁	M₁₆	1	4	yes	—
119	C₄²×C₇₁	—	2	2	no	—
120	C₄²×C₇₁	—	2	2	no	—
121	C₄²×C₇₁	—	2	2	no	—
122	C₄²×C₇₁	—	2	2	no	—
123	C₄²×C₇₁	—	2	1	no	—
124	C₄²×C₇₁	—	2	1	no	—
125	C₄²×C₇₁	—	2	2	no	—
126	C₄²×C₇₁	—	2	2	no	—
127	C₄²×C₇₁	—	2	1	no	—
128	C₄²×C₇₁	—	2	2	no	—
129	C₄²×C₇₁	—	2	1	no	—
130	C₄²×C₇₁	—	4	1	no	—
131	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
132	C₄²×C₇₁	#41	1	2	yes	—
133	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₁	—	2	1	no	—
135	C₂×C₄×C₇₁	—	2	2	no	—
136	C₂×C₄×C₇₁	—	2	1	no	—
137	C₂×C₄×C₇₁	—	2	1	no	—
138	C₂×C₄×C₇₁	—	2	2	no	—
139	C₂×C₄×C₇₁	—	2	1	no	—
140	C₂×C₄×C₇₁	—	2	2	no	—
141	C₂×C₄×C₇₁	—	2	2	no	—
142	C₂×C₄×C₇₁	—	2	1	no	—
143	C₂×C₄×C₇₁	—	2	1	no	—
144	C₂×C₄×C₇₁	—	2	2	no	—
145	C₂×C₄×C₇₁	—	2	1	no	—
146	C₂×C₄×C₇₁	—	2	1	no	—
147	C₂×C₄×C₇₁	—	2	1	no	—
148	C₂×C₄×C₇₁	—	2	2	no	—
149	C₂×C₄×C₇₁	—	2	1	no	—
150	C₂×C₄×C₇₁	—	2	1	no	—
151	C₂×C₄×C₇₁	—	2	2	no	—
152	C₂×C₄×C₇₁	—	2	2	no	—
153	C₂×C₄×C₇₁	—	2	2	no	—
154	C₂×C₄×C₇₁	—	2	1	no	—
155	C₂×C₄×C₇₁	—	2	1	no	—
156	C₂×C₄×C₇₁	—	2	2	no	—
157	C₂×C₄×C₇₁	—	2	2	no	—
158	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
159	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
160	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
161	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
162	C₂×C₄×C₇₁	#131	1	2	yes	—
163	C₂×C₄×C₇₁	#131	1	2	yes	—
164	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
170	C₈×C₇₁	—	2	2	no	—
171	C₈×C₇₁	—	2	2	no	—
172	C₈×C₇₁	—	2	4	no	—
173	C₈×C₇₁	—	4	1	no	—
174	C₈×C₇₁	—	4	1	no	—
175	C₈×C₇₁	—	4	2	no	—
176	C₂²×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₇₁	C₄×#41	1	1	yes	characteristic
178	C₂²×C₇₁	#520	1	1	yes	characteristic
179	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
180	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
181	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
182	C₂²×C₇₁	#568	1	1	yes	characteristic
183	C₄×C₇₁	—	2	2	no	—
184	C₄×C₇₁	—	2	2	no	—
185	C₄×C₇₁	—	2	4	no	—
186	C₄×C₇₁	—	4	2	no	—
187	C₄×C₇₁	—	4	1	no	—
188	C₄×C₇₁	—	4	2	no	—
189	C₄×C₇₁	—	4	1	no	—
190	C₄×C₇₁	—	4	1	no	—
191	C₄×C₇₁	—	4	2	no	—
192	C₄×C₇₁	#446	1	4	yes	—
193	C₄×C₇₁	#552	1	1	yes	characteristic
194	C₄×C₇₁	#553	1	1	yes	characteristic
195	C₄×C₇₁	#554	1	1	yes	characteristic
196	C₄×C₇₁	#557	1	1	yes	characteristic
197	#23254	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₇₁	#2370	1	1	yes	characteristic
199	C₂×C₇₁	C₂×#446	1	1	yes	characteristic
200	C₂×C₇₁	#2962	1	1	yes	characteristic
201	C₂×C₇₁	#2966	1	1	yes	characteristic
202	C₂×C₇₁	#2971	1	1	yes	characteristic
203	C₂×C₇₁	#2990	1	1	yes	characteristic
204	C₂×C₇₁	#2993	1	1	yes	characteristic
205	C₄×#125	C₂×C₇₁	1	1	yes	characteristic
206	#1939	C₂×C₇₁	1	1	yes	characteristic
207	#1939	C₂×C₇₁	1	2	yes	—
208	#1939	C₂×C₇₁	1	1	yes	characteristic
209	#1917	C₂×C₇₁	1	2	yes	—
210	#1917	C₂×C₇₁	1	2	yes	—
211	#1771	C₂×C₇₁	1	2	yes	—
212	#1770	C₂×C₇₁	1	1	yes	characteristic
213	C₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic
214	#1767	C₂×C₇₁	1	1	yes	characteristic
215	C₄×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic
216	C₇₁	#23254	1	1	yes	characteristic, direct product factor, Sylow
217	C₂×#125	C₄×C₇₁	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
223	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
224	C₂×#97	C₂²×C₇₁	1	2	yes	—
225	C₂×#97	C₂²×C₇₁	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
227	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
229	#389	C₄×C₇₁	1	1	yes	characteristic
230	#389	C₄×C₇₁	1	1	yes	characteristic
231	#387	C₄×C₇₁	1	2	yes	—
232	#387	C₄×C₇₁	1	2	yes	—
233	#382	C₄×C₇₁	1	1	yes	characteristic
234	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
235	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
236	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
237	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
238	C₄³	C₂²×C₇₁	1	1	yes	characteristic
239	#336	C₂²×C₇₁	1	2	yes	—
240	#336	C₂²×C₇₁	1	1	yes	characteristic
241	#336	C₂²×C₇₁	1	2	yes	—
242	#336	C₂²×C₇₁	1	2	yes	—
243	#336	C₂²×C₇₁	1	2	yes	—
244	#336	C₂²×C₇₁	1	2	yes	—
245	#336	C₂²×C₇₁	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
249	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
250	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
252	#125	—	2	4	no	—
253	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
254	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
255	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
256	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
260	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
261	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
262	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
263	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
264	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
269	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	4	no	—
273	C₄⋊C₈	—	2	2	no	—
274	C₄⋊C₈	—	4	2	no	—
275	C₄⋊C₈	Q₈×C₇₁	1	4	yes	—
276	#97	—	2	4	no	—
277	#97	—	4	2	no	—
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	2	2	no	—
280	C₄×C₈	—	4	2	no	—
281	#95	C₂×C₄×C₇₁	1	2	yes	—
282	#95	C₂×C₄×C₇₁	1	2	yes	—
283	#95	C₂×C₄×C₇₁	1	2	yes	—
284	#95	C₂×C₄×C₇₁	1	2	yes	—
285	#95	C₂×C₄×C₇₁	1	2	yes	—
286	#95	C₂×C₄×C₇₁	1	2	yes	—
287	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
288	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
289	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
292	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
293	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
296	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
297	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	C₂×Q₈×C₇₁	1	4	yes	—
314	C₂×C₈	#41×C₇₁	1	2	yes	—
315	C₂×C₈	#41×C₇₁	1	2	yes	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	4	2	no	—
323	C₄⋊C₄	M₁₆×C₇₁	1	4	yes	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	4	1	no	—
336	C₄²	C₂×Q₈×C₇₁	1	2	yes	—
337	C₄²	#41×C₇₁	1	2	yes	—
338	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
364	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
365	C₂×C₄	C₂×M₁₆×C₇₁	1	4	yes	—
366	C₂×C₄	C₂×M₁₆×C₇₁	1	2	yes	—
367	C₂×C₄	#131×C₇₁	1	2	yes	—
368	C₂×C₄	#131×C₇₁	1	2	yes	—
369	C₂×C₄	C₂²×Q₈×C₇₁	1	1	yes	characteristic
370	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
373	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
374	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
375	C₈	—	2	2	no	—
376	C₈	—	2	2	no	—
377	C₈	—	2	4	no	—
378	C₈	—	4	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	2	no	—
381	C₂²	C₂×C₄×Q₈×C₇₁	1	1	yes	characteristic
382	C₂²	C₄×#41×C₇₁	1	1	yes	characteristic
383	C₂²	#520×C₇₁	1	1	yes	characteristic
384	C₂²	C₂²×M₁₆×C₇₁	1	1	yes	characteristic
385	C₂²	C₂×#131×C₇₁	1	1	yes	characteristic
386	C₂²	C₂×#131×C₇₁	1	1	yes	characteristic
387	C₂²	#568×C₇₁	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	4	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	4	2	no	—
397	C₄	#446×C₇₁	1	4	yes	—
398	C₄	#552×C₇₁	1	1	yes	characteristic
399	C₄	#553×C₇₁	1	1	yes	characteristic
400	C₄	#554×C₇₁	1	1	yes	characteristic
401	C₄	#557×C₇₁	1	1	yes	characteristic
402	C₂	C₇₁×#2370	1	1	yes	characteristic
403	C₂	C₂×#446×C₇₁	1	1	yes	characteristic
404	C₂	C₇₁×#2962	1	1	yes	characteristic
405	C₂	C₇₁×#2966	1	1	yes	characteristic
406	C₂	C₇₁×#2971	1	1	yes	characteristic
407	C₂	C₇₁×#2990	1	1	yes	characteristic
408	C₂	C₇₁×#2993	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
197. #23254	C₇₁	direct (N × H)
216. C₇₁	#23254	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#125×C₇₁	2
C₇₁×#1939	2
C₇₁×#1917	2
C₇₁×#1771	2
C₇₁×#1770	2
C₈×C₄⋊C₄×C₇₁	2
C₇₁×#1767	2
C₄×C₄⋊C₈×C₇₁	2
#23254	71

Subgroups — C71×#23254 #31844126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23254 #31844126