Subgroups — C₇₁×#23174 #31844046

Order 18176.

1144 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#435×C₇₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#127×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1955	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#1870	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1869	C₂	1	2	yes	maximal, complemented
10	C₇₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×#127×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₇₁	C₂²	1	2	yes	—
19	#436×C₇₁	—	2	4	no	—
20	#435×C₇₁	—	2	8	no	—
21	C₈×D₄×C₇₁	—	2	4	no	—
22	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
32	C₂×#97×C₇₁	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
35	#390×C₇₁	C₄	1	2	yes	—
36	#390×C₇₁	C₄	1	1	yes	characteristic
37	#390×C₇₁	C₄	1	1	yes	characteristic
38	#390×C₇₁	C₄	1	2	yes	—
39	#384×C₇₁	C₄	1	1	yes	characteristic
40	C₄³×C₇₁	C₂²	1	1	yes	characteristic
41	#336×C₇₁	C₂²	1	1	yes	characteristic
42	#336×C₇₁	C₂²	1	1	yes	characteristic
43	#336×C₇₁	C₂²	1	2	yes	—
44	C₂²×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₇₁	C₂×C₄	1	2	yes	—
46	C₂²×D₄×C₇₁	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₇₁	—	2	4	no	—
51	C₂×M₁₆×C₇₁	—	2	4	no	—
52	C₂×M₁₆×C₇₁	—	4	2	no	—
53	C₂²×C₈×C₇₁	—	2	4	no	—
54	C₂²×C₈×C₇₁	—	2	4	no	—
55	C₂²×C₈×C₇₁	—	4	2	no	—
56	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
62	#127×C₇₁	—	2	4	no	—
63	C₄×D₄×C₇₁	—	2	2	no	—
64	C₄×D₄×C₇₁	—	2	4	no	—
65	C₄×D₄×C₇₁	—	2	2	no	—
66	C₄×D₄×C₇₁	—	2	2	no	—
67	C₄×D₄×C₇₁	—	2	4	no	—
68	C₄×D₄×C₇₁	—	2	2	no	—
69	C₄×D₄×C₇₁	—	4	2	no	—
70	C₄×D₄×C₇₁	—	4	2	no	—
71	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
76	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
77	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₇₁	—	2	2	no	—
89	C₄⋊C₈×C₇₁	—	2	2	no	—
90	C₄⋊C₈×C₇₁	D₄	1	4	yes	complemented
91	C₄⋊C₈×C₇₁	D₄	1	4	yes	complemented
92	C₂²⋊C₈×C₇₁	—	2	4	no	—
93	C₂²⋊C₈×C₇₁	—	2	4	no	—
94	C₂²⋊C₈×C₇₁	—	2	4	no	—
95	C₂²⋊C₈×C₇₁	—	2	4	no	—
96	C₂²⋊C₈×C₇₁	—	4	2	no	—
97	C₂²⋊C₈×C₇₁	—	4	2	no	—
98	#97×C₇₁	—	2	4	no	—
99	C₄×C₈×C₇₁	—	2	2	no	—
100	C₄×C₈×C₇₁	—	2	2	no	—
101	C₂⁴×C₇₁	C₂²×C₄	1	2	yes	—
102	C₂⁴×C₇₁	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₇₁	—	2	4	no	—
104	C₂×D₄×C₇₁	—	2	4	no	—
105	C₂×D₄×C₇₁	—	2	2	no	—
106	C₂×D₄×C₇₁	—	2	4	no	—
107	C₂×D₄×C₇₁	—	2	2	no	—
108	C₂×D₄×C₇₁	—	2	2	no	—
109	C₂×D₄×C₇₁	—	2	2	no	—
110	C₂×D₄×C₇₁	—	2	4	no	—
111	C₂×D₄×C₇₁	—	2	2	no	—
112	C₂×D₄×C₇₁	—	4	2	no	—
113	C₂×D₄×C₇₁	—	4	2	no	—
114	C₂×D₄×C₇₁	M₁₆	1	4	yes	complemented
115	C₂²×C₄×C₇₁	—	2	2	no	—
116	C₂²×C₄×C₇₁	—	2	2	no	—
117	C₂²×C₄×C₇₁	—	2	2	no	—
118	C₂²×C₄×C₇₁	—	2	2	no	—
119	C₂²×C₄×C₇₁	—	2	2	no	—
120	C₂²×C₄×C₇₁	—	2	2	no	—
121	C₂²×C₄×C₇₁	—	2	2	no	—
122	C₂²×C₄×C₇₁	—	2	2	no	—
123	C₂²×C₄×C₇₁	—	2	2	no	—
124	C₂²×C₄×C₇₁	—	2	2	no	—
125	C₂²×C₄×C₇₁	—	2	2	no	—
126	C₂²×C₄×C₇₁	—	2	2	no	—
127	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
136	M₁₆×C₇₁	—	4	8	no	—
137	C₂×C₈×C₇₁	—	2	1	no	—
138	C₂×C₈×C₇₁	—	2	1	no	—
139	C₂×C₈×C₇₁	—	2	2	no	—
140	C₂×C₈×C₇₁	—	2	1	no	—
141	C₂×C₈×C₇₁	—	2	2	no	—
142	C₂×C₈×C₇₁	—	2	1	no	—
143	C₂×C₈×C₇₁	—	2	1	no	—
144	C₂×C₈×C₇₁	—	2	1	no	—
145	C₂×C₈×C₇₁	—	2	2	no	—
146	C₂×C₈×C₇₁	—	2	1	no	—
147	C₂×C₈×C₇₁	—	2	1	no	—
148	C₂×C₈×C₇₁	—	2	2	no	—
149	C₂×C₈×C₇₁	—	2	1	no	—
150	C₂×C₈×C₇₁	—	2	1	no	—
151	C₂×C₈×C₇₁	—	2	2	no	—
152	C₂×C₈×C₇₁	—	4	8	no	—
153	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
155	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₇₁	—	2	4	no	—
157	C₄⋊C₄×C₇₁	—	2	2	no	—
158	C₄⋊C₄×C₇₁	—	2	2	no	—
159	C₄⋊C₄×C₇₁	—	4	2	no	—
160	C₂²⋊C₄×C₇₁	—	2	4	no	—
161	C₂²⋊C₄×C₇₁	—	2	4	no	—
162	C₂²⋊C₄×C₇₁	—	2	4	no	—
163	C₂²⋊C₄×C₇₁	—	2	4	no	—
164	C₂²⋊C₄×C₇₁	—	4	2	no	—
165	C₂²⋊C₄×C₇₁	—	4	2	no	—
166	C₄²×C₇₁	—	2	2	no	—
167	C₄²×C₇₁	—	2	2	no	—
168	C₄²×C₇₁	—	2	2	no	—
169	C₄²×C₇₁	—	2	1	no	—
170	C₄²×C₇₁	—	2	1	no	—
171	C₄²×C₇₁	—	4	1	no	—
172	C₄²×C₇₁	C₂×D₄	1	2	yes	—
173	C₄²×C₇₁	C₂×D₄	1	2	yes	—
174	C₄²×C₇₁	#41	1	2	yes	—
175	C₄²×C₇₁	#41	1	2	yes	—
176	C₂³×C₇₁	—	2	2	no	—
177	C₂³×C₇₁	—	2	2	no	—
178	C₂³×C₇₁	—	2	2	no	—
179	C₂³×C₇₁	—	2	2	no	—
180	C₂³×C₇₁	—	2	2	no	—
181	C₂³×C₇₁	—	2	2	no	—
182	C₂³×C₇₁	—	2	2	no	—
183	C₂³×C₇₁	—	2	2	no	—
184	C₂³×C₇₁	—	2	2	no	—
185	C₂³×C₇₁	—	2	2	no	—
186	C₂³×C₇₁	—	2	2	no	—
187	C₂³×C₇₁	—	2	2	no	—
188	C₂³×C₇₁	—	2	2	no	—
189	C₂³×C₇₁	C₂×M₁₆	1	4	yes	—
190	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
191	D₄×C₇₁	—	2	8	no	—
192	D₄×C₇₁	—	4	8	no	—
193	D₄×C₇₁	—	4	8	no	—
194	D₄×C₇₁	—	4	4	no	—
195	C₂×C₄×C₇₁	—	2	1	no	—
196	C₂×C₄×C₇₁	—	2	1	no	—
197	C₂×C₄×C₇₁	—	2	1	no	—
198	C₂×C₄×C₇₁	—	2	1	no	—
199	C₂×C₄×C₇₁	—	2	1	no	—
200	C₂×C₄×C₇₁	—	2	1	no	—
201	C₂×C₄×C₇₁	—	2	1	no	—
202	C₂×C₄×C₇₁	—	2	1	no	—
203	C₂×C₄×C₇₁	—	2	2	no	—
204	C₂×C₄×C₇₁	—	2	2	no	—
205	C₂×C₄×C₇₁	—	2	2	no	—
206	C₂×C₄×C₇₁	—	2	2	no	—
207	C₂×C₄×C₇₁	—	2	1	no	—
208	C₂×C₄×C₇₁	—	2	2	no	—
209	C₂×C₄×C₇₁	—	2	2	no	—
210	C₂×C₄×C₇₁	—	2	1	no	—
211	C₂×C₄×C₇₁	—	2	1	no	—
212	C₂×C₄×C₇₁	—	4	2	no	—
213	C₂×C₄×C₇₁	—	4	2	no	—
214	C₂×C₄×C₇₁	—	4	2	no	—
215	C₂×C₄×C₇₁	—	4	2	no	—
216	C₂×C₄×C₇₁	—	4	2	no	—
217	C₂×C₄×C₇₁	—	4	2	no	—
218	C₂×C₄×C₇₁	—	4	2	no	—
219	C₂×C₄×C₇₁	—	4	2	no	—
220	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
221	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
222	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
223	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
224	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
225	C₂×C₄×C₇₁	#131	1	2	yes	—
226	C₂×C₄×C₇₁	#131	1	2	yes	—
227	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
233	C₈×C₇₁	—	2	2	no	—
234	C₈×C₇₁	—	2	4	no	—
235	C₈×C₇₁	—	2	2	no	—
236	C₈×C₇₁	—	4	1	no	—
237	C₈×C₇₁	—	4	1	no	—
238	C₈×C₇₁	—	4	2	no	—
239	C₂²×C₇₁	—	2	4	no	—
240	C₂²×C₇₁	—	2	4	no	—
241	C₂²×C₇₁	—	2	4	no	—
242	C₂²×C₇₁	—	4	2	no	—
243	C₂²×C₇₁	—	4	2	no	—
244	C₂²×C₇₁	—	4	2	no	—
245	C₂²×C₇₁	—	4	2	no	—
246	C₂²×C₇₁	—	4	2	no	—
247	C₂²×C₇₁	—	4	2	no	—
248	C₂²×C₇₁	—	4	2	no	—
249	C₂²×C₇₁	—	4	2	no	—
250	C₂²×C₇₁	—	4	2	no	—
251	C₂²×C₇₁	—	4	2	no	—
252	C₂²×C₇₁	—	4	2	no	—
253	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
254	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₇₁	#518	1	1	yes	characteristic
256	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
257	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
258	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
259	C₂²×C₇₁	#568	1	1	yes	characteristic
260	C₄×C₇₁	—	2	2	no	—
261	C₄×C₇₁	—	2	2	no	—
262	C₄×C₇₁	—	2	2	no	—
263	C₄×C₇₁	—	2	2	no	—
264	C₄×C₇₁	—	4	2	no	—
265	C₄×C₇₁	—	4	2	no	—
266	C₄×C₇₁	—	4	1	no	—
267	C₄×C₇₁	#435	1	4	yes	complemented
268	C₄×C₇₁	D₄²	1	1	yes	characteristic
269	C₄×C₇₁	#550	1	1	yes	characteristic
270	C₄×C₇₁	#550	1	1	yes	characteristic
271	C₄×C₇₁	#559	1	1	yes	characteristic
272	#23174	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₇₁	—	4	4	no	—
274	C₂×C₇₁	—	8	2	no	—
275	C₂×C₇₁	#2366	1	1	yes	characteristic
276	C₂×C₇₁	C₂×#435	1	1	yes	characteristic
277	C₂×C₇₁	#2931	1	1	yes	characteristic
278	C₂×C₇₁	D₄×M₁₆	1	1	yes	characteristic
279	C₂×C₇₁	#2935	1	1	yes	characteristic
280	C₂×C₇₁	#2974	1	1	yes	characteristic
281	C₂×C₇₁	#2981	1	1	yes	characteristic
282	C₂×#436	C₂×C₇₁	1	1	yes	characteristic, complemented
283	C₂×#435	C₂×C₇₁	1	2	yes	complemented
284	C₂×C₈×D₄	C₂×C₇₁	1	1	yes	characteristic, complemented
285	C₄×#127	C₂×C₇₁	1	1	yes	characteristic
286	#1955	C₂×C₇₁	1	2	yes	complemented
287	#1954	C₂×C₇₁	1	1	yes	characteristic, complemented
288	#1954	C₂×C₇₁	1	1	yes	characteristic, complemented
289	#1870	C₂×C₇₁	1	2	yes	complemented
290	#1869	C₂×C₇₁	1	2	yes	complemented
291	#1767	C₂×C₇₁	1	1	yes	characteristic, complemented
292	C₄×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic, complemented
293	C₇₁	#23174	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
295	C₂³×C₈	C₂²×C₇₁	1	2	yes	—
296	C₂×#127	C₄×C₇₁	1	1	yes	characteristic
297	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂²×C₇₁	1	2	yes	—
299	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
300	C₂×C₄×D₄	C₂²×C₇₁	1	2	yes	—
301	#436	—	2	4	no	—
302	#435	—	2	8	no	—
303	C₈×D₄	—	2	4	no	—
304	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
305	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
308	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
309	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
310	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
311	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
312	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
313	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
314	C₂×#97	C₂²×C₇₁	1	2	yes	—
315	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
316	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
317	#390	C₄×C₇₁	1	2	yes	—
318	#390	C₄×C₇₁	1	1	yes	characteristic
319	#390	C₄×C₇₁	1	1	yes	characteristic
320	#390	C₄×C₇₁	1	2	yes	—
321	#384	C₄×C₇₁	1	1	yes	characteristic
322	C₄³	C₂²×C₇₁	1	1	yes	characteristic
323	#336	C₂²×C₇₁	1	1	yes	characteristic
324	#336	C₂²×C₇₁	1	2	yes	—
325	#336	C₂²×C₇₁	1	1	yes	characteristic
326	C₂²×D₄	C₂×C₄×C₇₁	1	1	yes	characteristic
327	C₂²×D₄	C₂×C₄×C₇₁	1	2	yes	—
328	C₂²×D₄	C₂×C₄×C₇₁	1	2	yes	—
329	C₂²×D₄	C₂×C₄×C₇₁	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
331	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
332	C₂×M₁₆	—	2	4	no	—
333	C₂×M₁₆	—	2	4	no	—
334	C₂×M₁₆	—	4	2	no	—
335	C₂²×C₈	—	2	4	no	—
336	C₂²×C₈	—	2	4	no	—
337	C₂²×C₈	—	4	2	no	—
338	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
341	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
343	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
344	#127	—	2	4	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	—	2	4	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	2	4	no	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	—	4	2	no	—
352	C₄×D₄	—	4	2	no	—
353	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
354	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
359	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
360	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
361	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
362	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
363	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
364	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
365	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
366	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
370	C₄⋊C₈	—	2	2	no	—
371	C₄⋊C₈	—	2	2	no	—
372	C₄⋊C₈	D₄×C₇₁	1	4	yes	complemented
373	C₄⋊C₈	D₄×C₇₁	1	4	yes	complemented
374	C₂²⋊C₈	—	2	4	no	—
375	C₂²⋊C₈	—	2	4	no	—
376	C₂²⋊C₈	—	2	4	no	—
377	C₂²⋊C₈	—	2	4	no	—
378	C₂²⋊C₈	—	4	2	no	—
379	C₂²⋊C₈	—	4	2	no	—
380	#97	—	2	4	no	—
381	C₄×C₈	—	2	2	no	—
382	C₄×C₈	—	2	2	no	—
383	C₂⁴	C₂²×C₄×C₇₁	1	2	yes	—
384	C₂⁴	C₂²×C₄×C₇₁	1	2	yes	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	M₁₆×C₇₁	1	4	yes	complemented
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
410	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
411	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
412	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
413	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
414	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
415	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
416	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
418	M₁₆	—	4	8	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	4	8	no	—
435	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
436	C₂×C₈	C₂×D₄×C₇₁	1	4	yes	—
437	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
438	C₄⋊C₄	—	2	4	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	4	2	no	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	4	2	no	—
447	C₂²⋊C₄	—	4	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	2	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	4	1	no	—
454	C₄²	C₂×D₄×C₇₁	1	2	yes	—
455	C₄²	C₂×D₄×C₇₁	1	2	yes	—
456	C₄²	#41×C₇₁	1	2	yes	—
457	C₄²	#41×C₇₁	1	2	yes	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	2	no	—
471	C₂³	C₂×M₁₆×C₇₁	1	4	yes	—
472	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
473	D₄	—	2	8	no	—
474	D₄	—	4	8	no	—
475	D₄	—	4	8	no	—
476	D₄	—	4	4	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
503	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
504	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
505	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
506	C₂×C₄	C₂×M₁₆×C₇₁	1	2	yes	—
507	C₂×C₄	#131×C₇₁	1	2	yes	—
508	C₂×C₄	#131×C₇₁	1	2	yes	—
509	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
511	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
512	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
513	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
514	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	2	4	no	—
517	C₈	—	2	2	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	1	no	—
520	C₈	—	4	2	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
536	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
537	C₂²	#518×C₇₁	1	1	yes	characteristic
538	C₂²	C₂²×M₁₆×C₇₁	1	1	yes	characteristic
539	C₂²	C₂×#131×C₇₁	1	1	yes	characteristic
540	C₂²	C₂×#131×C₇₁	1	1	yes	characteristic
541	C₂²	#568×C₇₁	1	1	yes	characteristic
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	1	no	—
549	C₄	#435×C₇₁	1	4	yes	complemented
550	C₄	D₄²×C₇₁	1	1	yes	characteristic
551	C₄	#550×C₇₁	1	1	yes	characteristic
552	C₄	#550×C₇₁	1	1	yes	characteristic
553	C₄	#559×C₇₁	1	1	yes	characteristic
554	C₂	—	4	4	no	—
555	C₂	—	8	2	no	—
556	C₂	C₇₁×#2366	1	1	yes	characteristic
557	C₂	C₂×#435×C₇₁	1	1	yes	characteristic
558	C₂	C₇₁×#2931	1	1	yes	characteristic
559	C₂	D₄×M₁₆×C₇₁	1	1	yes	characteristic
560	C₂	C₇₁×#2935	1	1	yes	characteristic
561	C₂	C₇₁×#2974	1	1	yes	characteristic
562	C₂	C₇₁×#2981	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂³×C₇₁	C₂³×C₄
prime radical	C₂×#127×C₇₁	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
1. C₂×#436×C₇₁	C₂	semidirect
2. C₂×#435×C₇₁	C₂	semidirect
3. C₂×C₈×D₄×C₇₁	C₂	semidirect
5. C₇₁×#1955	C₂	semidirect
6. C₇₁×#1954	C₂	semidirect
7. C₇₁×#1954	C₂	semidirect
8. C₇₁×#1870	C₂	semidirect
9. C₇₁×#1869	C₂	semidirect
10. C₇₁×#1767	C₂	semidirect
11. C₄×C₄⋊C₈×C₇₁	C₂	semidirect
90. C₄⋊C₈×C₇₁	D₄	semidirect
91. C₄⋊C₈×C₇₁	D₄	semidirect
114. C₂×D₄×C₇₁	M₁₆	semidirect
267. C₄×C₇₁	#435	semidirect
272. #23174	C₇₁	direct (N × H)
282. C₂×#436	C₂×C₇₁	semidirect
283. C₂×#435	C₂×C₇₁	semidirect
284. C₂×C₈×D₄	C₂×C₇₁	semidirect
286. #1955	C₂×C₇₁	semidirect
287. #1954	C₂×C₇₁	semidirect
288. #1954	C₂×C₇₁	semidirect
289. #1870	C₂×C₇₁	semidirect
290. #1869	C₂×C₇₁	semidirect
291. #1767	C₂×C₇₁	semidirect
292. C₄×C₄⋊C₈	C₂×C₇₁	semidirect
293. C₇₁	#23174	direct (N × H)
372. C₄⋊C₈	D₄×C₇₁	semidirect
373. C₄⋊C₈	D₄×C₇₁	semidirect
396. C₂×D₄	M₁₆×C₇₁	semidirect
549. C₄	#435×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₇₁	2
C₂×#435×C₇₁	2
C₂×C₈×D₄×C₇₁	2
C₄×#127×C₇₁	2
C₇₁×#1955	2
C₇₁×#1954	2
C₇₁×#1870	2
C₇₁×#1869	2
C₇₁×#1767	2
C₄×C₄⋊C₈×C₇₁	2
#23174	71

Subgroups — C71×#23174 #31844046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23174 #31844046