Subgroups — C₇₁×#23190 #31844062

Order 18176.

1086 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#440×C₇₁	C₂	1	2	yes	maximal, complemented
3	C₇₁×#2366	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#2365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1973	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1880	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1877	C₂	1	2	yes	maximal, complemented
10	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#127×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₇₁	—	2	1	no	—
18	C₂×C₄×D₄×C₇₁	C₂²	1	2	yes	—
19	#492×C₇₁	C₄	1	8	yes	complemented
20	#440×C₇₁	—	2	8	no	—
21	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×#103×C₇₁	C₂²	1	2	yes	—
25	C₂×#103×C₇₁	C₂²	1	2	yes	—
26	C₂×#102×C₇₁	C₂²	1	2	yes	—
27	C₂×#102×C₇₁	C₂²	1	2	yes	—
28	C₂×#97×C₇₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
30	#391×C₇₁	C₂²	1	2	yes	—
31	#390×C₇₁	—	2	1	no	—
32	#390×C₇₁	C₂²	1	2	yes	—
33	#389×C₇₁	—	2	1	no	—
34	#387×C₇₁	—	2	2	no	—
35	#386×C₇₁	—	2	2	no	—
36	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
40	#340×C₇₁	C₂²	1	1	yes	characteristic
41	#340×C₇₁	C₂²	1	1	yes	characteristic
42	#336×C₇₁	C₂²	1	2	yes	—
43	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₇₁	—	2	1	no	—
45	C₂²×D₄×C₇₁	—	2	1	no	—
46	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₇₁	—	2	2	no	—
48	C₂×SD₁₆×C₇₁	—	2	4	no	—
49	C₂×SD₁₆×C₇₁	—	2	4	no	—
50	C₂×SD₁₆×C₇₁	—	4	2	no	—
51	C₂×SD₁₆×C₇₁	—	4	2	no	—
52	C₂×SD₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
53	C₂×SD₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
54	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₇₁	—	2	2	no	—
58	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	complemented
59	#127×C₇₁	—	2	2	no	—
60	#127×C₇₁	C₂×C₄	1	4	yes	—
61	C₄×Q₈×C₇₁	—	2	4	no	—
62	C₄×Q₈×C₇₁	—	4	2	no	—
63	C₄×D₄×C₇₁	—	2	4	no	—
64	C₄×D₄×C₇₁	—	4	2	no	—
65	C₄×D₄×C₇₁	—	4	4	no	—
66	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₂×C₄²×C₇₁	—	2	1	no	—
79	C₂×C₄²×C₇₁	—	2	2	no	—
80	C₂×C₄²×C₇₁	—	2	1	no	—
81	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
83	#107×C₇₁	—	2	2	no	—
84	#107×C₇₁	D₄	1	4	yes	complemented
85	C₄⋊C₈×C₇₁	—	2	2	no	—
86	#103×C₇₁	—	2	4	no	—
87	#103×C₇₁	—	4	2	no	—
88	#102×C₇₁	—	2	4	no	—
89	#102×C₇₁	—	4	2	no	—
90	#97×C₇₁	—	2	4	no	—
91	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	complemented
92	#95×C₇₁	—	2	2	no	—
93	#95×C₇₁	—	2	1	no	—
94	#95×C₇₁	—	2	1	no	—
95	#95×C₇₁	—	2	2	no	—
96	#95×C₇₁	D₄	1	2	yes	—
97	C₂⁴×C₇₁	—	2	2	no	—
98	C₂×Q₈×C₇₁	—	2	2	no	—
99	C₂×Q₈×C₇₁	—	2	2	no	—
100	C₂×Q₈×C₇₁	—	2	2	no	—
101	C₂×Q₈×C₇₁	—	2	2	no	—
102	C₂×Q₈×C₇₁	—	2	2	no	—
103	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
104	C₂×D₄×C₇₁	—	2	4	no	—
105	C₂×D₄×C₇₁	—	2	4	no	—
106	C₂×D₄×C₇₁	—	2	2	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	2	no	—
111	C₂×D₄×C₇₁	—	4	2	no	—
112	C₂×D₄×C₇₁	—	4	2	no	—
113	C₂×D₄×C₇₁	—	8	1	no	—
114	C₂×D₄×C₇₁	—	8	1	no	—
115	C₂×D₄×C₇₁	C₂²×C₄	1	4	yes	—
116	C₂²×C₄×C₇₁	—	2	2	no	—
117	C₂²×C₄×C₇₁	—	2	1	no	—
118	C₂²×C₄×C₇₁	—	2	1	no	—
119	C₂²×C₄×C₇₁	—	2	1	no	—
120	C₂²×C₄×C₇₁	—	2	1	no	—
121	C₂²×C₄×C₇₁	—	4	4	no	—
122	C₂²×C₄×C₇₁	—	4	4	no	—
123	C₂²×C₄×C₇₁	—	8	2	no	—
124	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
129	SD₁₆×C₇₁	—	2	16	no	—
130	SD₁₆×C₇₁	—	4	8	no	—
131	C₂×C₈×C₇₁	—	2	1	no	—
132	C₂×C₈×C₇₁	—	2	1	no	—
133	C₂×C₈×C₇₁	—	2	2	no	—
134	C₂×C₈×C₇₁	—	2	2	no	—
135	C₂×C₈×C₇₁	—	2	2	no	—
136	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₇₁	#41	1	2	yes	—
140	C₄⋊C₄×C₇₁	—	2	4	no	—
141	C₄⋊C₄×C₇₁	—	2	4	no	—
142	C₄⋊C₄×C₇₁	—	2	2	no	—
143	C₄⋊C₄×C₇₁	—	2	2	no	—
144	C₄⋊C₄×C₇₁	—	2	1	no	—
145	C₄⋊C₄×C₇₁	—	2	1	no	—
146	C₄⋊C₄×C₇₁	—	2	1	no	—
147	C₄⋊C₄×C₇₁	—	2	1	no	—
148	C₄⋊C₄×C₇₁	—	2	1	no	—
149	C₄⋊C₄×C₇₁	—	2	1	no	—
150	C₄⋊C₄×C₇₁	—	4	4	no	—
151	C₄⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
153	C₂²⋊C₄×C₇₁	—	4	4	no	—
154	C₂²⋊C₄×C₇₁	—	4	4	no	—
155	C₂²⋊C₄×C₇₁	—	8	2	no	—
156	C₄²×C₇₁	—	2	2	no	—
157	C₄²×C₇₁	—	2	2	no	—
158	C₄²×C₇₁	—	4	4	no	—
159	C₄²×C₇₁	—	4	2	no	—
160	C₄²×C₇₁	—	4	2	no	—
161	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
162	C₄²×C₇₁	#41	1	2	yes	—
163	C₂³×C₇₁	—	2	4	no	—
164	C₂³×C₇₁	—	4	2	no	—
165	C₂³×C₇₁	—	4	2	no	—
166	C₂³×C₇₁	—	4	2	no	—
167	C₂³×C₇₁	—	4	2	no	—
168	C₂³×C₇₁	—	4	2	no	—
169	C₂³×C₇₁	—	4	2	no	—
170	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₇₁	—	2	4	no	—
172	Q₈×C₇₁	—	2	4	no	—
173	Q₈×C₇₁	—	4	2	no	—
174	Q₈×C₇₁	—	4	2	no	—
175	D₄×C₇₁	—	2	4	no	—
176	D₄×C₇₁	—	2	4	no	—
177	D₄×C₇₁	—	4	2	no	—
178	D₄×C₇₁	—	4	2	no	—
179	D₄×C₇₁	—	4	8	no	—
180	D₄×C₇₁	—	8	4	no	—
181	C₂×C₄×C₇₁	—	2	1	no	—
182	C₂×C₄×C₇₁	—	2	1	no	—
183	C₂×C₄×C₇₁	—	2	4	no	—
184	C₂×C₄×C₇₁	—	2	2	no	—
185	C₂×C₄×C₇₁	—	2	2	no	—
186	C₂×C₄×C₇₁	—	2	2	no	—
187	C₂×C₄×C₇₁	—	2	2	no	—
188	C₂×C₄×C₇₁	—	2	2	no	—
189	C₂×C₄×C₇₁	—	2	2	no	—
190	C₂×C₄×C₇₁	—	2	2	no	—
191	C₂×C₄×C₇₁	—	2	2	no	—
192	C₂×C₄×C₇₁	—	4	2	no	—
193	C₂×C₄×C₇₁	—	4	2	no	—
194	C₂×C₄×C₇₁	—	4	1	no	—
195	C₂×C₄×C₇₁	—	4	1	no	—
196	C₂×C₄×C₇₁	—	4	1	no	—
197	C₂×C₄×C₇₁	—	4	1	no	—
198	C₂×C₄×C₇₁	—	4	1	no	—
199	C₂×C₄×C₇₁	—	4	1	no	—
200	C₂×C₄×C₇₁	—	4	1	no	—
201	C₂×C₄×C₇₁	—	4	1	no	—
202	C₂×C₄×C₇₁	—	4	1	no	—
203	C₂×C₄×C₇₁	—	4	1	no	—
204	C₂×C₄×C₇₁	—	8	8	no	—
205	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₇₁	#137	1	2	yes	—
209	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
214	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
215	C₈×C₇₁	—	2	2	no	—
216	C₈×C₇₁	—	4	2	no	—
217	C₈×C₇₁	C₄×D₄	1	4	yes	complemented
218	C₂²×C₇₁	—	4	4	no	—
219	C₂²×C₇₁	—	4	4	no	—
220	C₂²×C₇₁	—	4	4	no	—
221	C₂²×C₇₁	—	8	2	no	—
222	C₂²×C₇₁	—	8	2	no	—
223	C₂²×C₇₁	—	8	2	no	—
224	C₂²×C₇₁	—	8	2	no	—
225	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂²×C₇₁	D₄²	1	1	yes	characteristic
227	C₂²×C₇₁	#550	1	1	yes	characteristic
228	C₂²×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂²×C₇₁	C₂×#137	1	1	yes	characteristic
230	C₂²×C₇₁	#577	1	1	yes	characteristic
231	C₂²×C₇₁	#577	1	1	yes	characteristic
232	C₄×C₇₁	—	2	2	no	—
233	C₄×C₇₁	—	4	4	no	—
234	C₄×C₇₁	—	4	2	no	—
235	C₄×C₇₁	—	4	2	no	—
236	C₄×C₇₁	—	4	2	no	—
237	C₄×C₇₁	—	4	2	no	—
238	C₄×C₇₁	—	8	1	no	—
239	C₄×C₇₁	—	8	1	no	—
240	C₄×C₇₁	#440	1	4	yes	complemented
241	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
242	C₄×C₇₁	#518	1	1	yes	characteristic
243	C₄×C₇₁	#550	1	1	yes	characteristic
244	C₄×C₇₁	#559	1	1	yes	characteristic
245	#23190	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂×C₇₁	—	8	4	no	—
247	C₂×C₇₁	#2366	1	1	yes	characteristic
248	C₂×C₇₁	C₂×#440	1	1	yes	characteristic
249	C₂×C₇₁	#2949	1	1	yes	characteristic
250	C₂×C₇₁	#3272	1	1	yes	characteristic
251	C₂×C₇₁	#3278	1	1	yes	characteristic
252	C₂×C₇₁	#3397	1	1	yes	characteristic
253	C₂×C₇₁	#3400	1	1	yes	characteristic
254	C₂×#492	C₂×C₇₁	1	1	yes	characteristic
255	C₂×#440	C₂×C₇₁	1	2	yes	complemented
256	#2366	C₂×C₇₁	1	1	yes	characteristic
257	#2365	C₂×C₇₁	1	1	yes	characteristic, complemented
258	#1982	C₂×C₇₁	1	1	yes	characteristic
259	#1981	C₂×C₇₁	1	1	yes	characteristic, complemented
260	#1973	C₂×C₇₁	1	2	yes	complemented
261	#1880	C₂×C₇₁	1	2	yes	complemented
262	#1877	C₂×C₇₁	1	2	yes	complemented
263	C₄×#107	C₂×C₇₁	1	1	yes	characteristic, complemented
264	#1770	C₂×C₇₁	1	1	yes	characteristic, complemented
265	C₇₁	#23190	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²×SD₁₆	C₂²×C₇₁	1	2	yes	—
267	C₂²×SD₁₆	C₂²×C₇₁	1	2	yes	—
268	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
269	C₂×#127	C₂²×C₇₁	1	1	yes	characteristic
270	C₂×C₄×Q₈	C₂²×C₇₁	1	2	yes	—
271	C₂×C₄×D₄	—	2	1	no	—
272	C₂×C₄×D₄	C₂²×C₇₁	1	2	yes	—
273	#492	C₄×C₇₁	1	8	yes	complemented
274	#440	—	2	8	no	—
275	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
276	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
278	C₂×#103	C₂²×C₇₁	1	2	yes	—
279	C₂×#103	C₂²×C₇₁	1	2	yes	—
280	C₂×#102	C₂²×C₇₁	1	2	yes	—
281	C₂×#102	C₂²×C₇₁	1	2	yes	—
282	C₂×#97	C₂²×C₇₁	1	2	yes	—
283	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
284	#391	C₂²×C₇₁	1	2	yes	—
285	#390	—	2	1	no	—
286	#390	C₂²×C₇₁	1	2	yes	—
287	#389	—	2	1	no	—
288	#387	—	2	2	no	—
289	#386	—	2	2	no	—
290	C₄×C₄⋊C₄	—	2	1	no	—
291	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
292	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
293	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
294	#340	C₂²×C₇₁	1	1	yes	characteristic
295	#340	C₂²×C₇₁	1	1	yes	characteristic
296	#336	C₂²×C₇₁	1	2	yes	—
297	C₂²×Q₈	C₂³×C₇₁	1	2	yes	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	—	2	1	no	—
300	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
301	C₂³×C₄	—	2	2	no	—
302	C₂×SD₁₆	—	2	4	no	—
303	C₂×SD₁₆	—	2	4	no	—
304	C₂×SD₁₆	—	4	2	no	—
305	C₂×SD₁₆	—	4	2	no	—
306	C₂×SD₁₆	C₂×C₄×C₇₁	1	8	yes	complemented
307	C₂×SD₁₆	C₂×C₄×C₇₁	1	8	yes	complemented
308	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
309	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	C₂×C₄×C₇₁	1	4	yes	complemented
313	#127	—	2	2	no	—
314	#127	C₂×C₄×C₇₁	1	4	yes	—
315	C₄×Q₈	—	2	4	no	—
316	C₄×Q₈	—	4	2	no	—
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	4	2	no	—
319	C₄×D₄	—	4	4	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
324	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	2	2	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
336	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
337	#107	—	2	2	no	—
338	#107	D₄×C₇₁	1	4	yes	complemented
339	C₄⋊C₈	—	2	2	no	—
340	#103	—	2	4	no	—
341	#103	—	4	2	no	—
342	#102	—	2	4	no	—
343	#102	—	4	2	no	—
344	#97	—	2	4	no	—
345	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	complemented
346	#95	—	2	2	no	—
347	#95	—	2	1	no	—
348	#95	—	2	1	no	—
349	#95	—	2	2	no	—
350	#95	D₄×C₇₁	1	2	yes	—
351	C₂⁴	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	C₂²×C₄×C₇₁	1	4	yes	—
358	C₂×D₄	—	2	4	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	8	1	no	—
368	C₂×D₄	—	8	1	no	—
369	C₂×D₄	C₂²×C₄×C₇₁	1	4	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	4	4	no	—
376	C₂²×C₄	—	4	4	no	—
377	C₂²×C₄	—	8	2	no	—
378	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
382	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
383	SD₁₆	—	2	16	no	—
384	SD₁₆	—	4	8	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
391	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
392	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
393	C₂×C₈	#41×C₇₁	1	2	yes	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₄⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
406	C₄⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
407	C₂²⋊C₄	—	4	4	no	—
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	8	2	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	2	no	—
412	C₄²	—	4	4	no	—
413	C₄²	—	4	2	no	—
414	C₄²	—	4	2	no	—
415	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
416	C₄²	#41×C₇₁	1	2	yes	—
417	C₂³	—	2	4	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
425	Q₈	—	2	4	no	—
426	Q₈	—	2	4	no	—
427	Q₈	—	4	2	no	—
428	Q₈	—	4	2	no	—
429	D₄	—	2	4	no	—
430	D₄	—	2	4	no	—
431	D₄	—	4	8	no	—
432	D₄	—	4	2	no	—
433	D₄	—	4	2	no	—
434	D₄	—	8	4	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	8	8	no	—
459	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
460	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
461	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
462	C₂×C₄	#137×C₇₁	1	2	yes	—
463	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
467	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
468	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	4	2	no	—
471	C₈	C₄×D₄×C₇₁	1	4	yes	complemented
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
480	C₂²	D₄²×C₇₁	1	1	yes	characteristic
481	C₂²	#550×C₇₁	1	1	yes	characteristic
482	C₂²	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
483	C₂²	C₂×#137×C₇₁	1	1	yes	characteristic
484	C₂²	#577×C₇₁	1	1	yes	characteristic
485	C₂²	#577×C₇₁	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	4	4	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	1	no	—
493	C₄	—	8	1	no	—
494	C₄	#440×C₇₁	1	4	yes	complemented
495	C₄	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
496	C₄	#518×C₇₁	1	1	yes	characteristic
497	C₄	#550×C₇₁	1	1	yes	characteristic
498	C₄	#559×C₇₁	1	1	yes	characteristic
499	C₂	—	8	4	no	—
500	C₂	C₇₁×#2366	1	1	yes	characteristic
501	C₂	C₂×#440×C₇₁	1	1	yes	characteristic
502	C₂	C₇₁×#2949	1	1	yes	characteristic
503	C₂	C₇₁×#3272	1	1	yes	characteristic
504	C₂	C₇₁×#3278	1	1	yes	characteristic
505	C₂	C₇₁×#3397	1	1	yes	characteristic
506	C₂	C₇₁×#3400	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂×#127×C₇₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄×C₄⋊C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#440×C₇₁	C₂	semidirect
4. C₇₁×#2365	C₂	semidirect
6. C₇₁×#1981	C₂	semidirect
7. C₇₁×#1973	C₂	semidirect
8. C₇₁×#1880	C₂	semidirect
9. C₇₁×#1877	C₂	semidirect
10. C₄×#107×C₇₁	C₂	semidirect
11. C₇₁×#1770	C₂	semidirect
19. #492×C₇₁	C₄	semidirect
52. C₂×SD₁₆×C₇₁	C₂×C₄	semidirect
53. C₂×SD₁₆×C₇₁	C₂×C₄	semidirect
58. C₄⋊Q₈×C₇₁	C₂×C₄	semidirect
84. #107×C₇₁	D₄	semidirect
91. C₄×C₈×C₇₁	C₂×C₄	semidirect
217. C₈×C₇₁	C₄×D₄	semidirect
240. C₄×C₇₁	#440	semidirect
245. #23190	C₇₁	direct (N × H)
255. C₂×#440	C₂×C₇₁	semidirect
257. #2365	C₂×C₇₁	semidirect
259. #1981	C₂×C₇₁	semidirect
260. #1973	C₂×C₇₁	semidirect
261. #1880	C₂×C₇₁	semidirect
262. #1877	C₂×C₇₁	semidirect
263. C₄×#107	C₂×C₇₁	semidirect
264. #1770	C₂×C₇₁	semidirect
265. C₇₁	#23190	direct (N × H)
273. #492	C₄×C₇₁	semidirect
306. C₂×SD₁₆	C₂×C₄×C₇₁	semidirect
307. C₂×SD₁₆	C₂×C₄×C₇₁	semidirect
312. C₄⋊Q₈	C₂×C₄×C₇₁	semidirect
338. #107	D₄×C₇₁	semidirect
345. C₄×C₈	C₂×C₄×C₇₁	semidirect
471. C₈	C₄×D₄×C₇₁	semidirect
494. C₄	#440×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₇₁	2
C₂×#440×C₇₁	2
C₇₁×#2366	2
C₇₁×#2365	2
C₇₁×#1982	2
C₇₁×#1981	2
C₇₁×#1973	2
C₇₁×#1880	2
C₇₁×#1877	2
C₄×#107×C₇₁	2
C₇₁×#1770	2
#23190	71

Subgroups — C71×#23190 #31844062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23190 #31844062