Subgroups — C₇₁×#61951 #31882823

Order 18176.

2168 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3498	C₂	1	1	yes	maximal, characteristic
2	Q₈×#41×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3392	C₂	1	2	yes	maximal, complemented
4	C₇₁×#3362	C₂	1	2	yes	maximal, complemented
5	C₇₁×#3295	C₂	1	6	yes	maximal, complemented
6	D₄×SD₁₆×C₇₁	C₂	1	6	yes	maximal, complemented
7	C₇₁×#3246	C₂	1	3	yes	maximal, complemented
8	C₇₁×#3208	C₂	1	3	yes	maximal, complemented
9	C₇₁×#3103	C₂	1	3	yes	maximal, complemented
10	C₂×#460×C₇₁	C₂	1	3	yes	maximal, complemented
11	C₇₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₇₁	—	2	1	no	—
13	C₂²×SD₁₆×C₇₁	C₂²	1	6	yes	complemented
14	Q₈²×C₇₁	C₂²	1	1	yes	characteristic, complemented
15	#557×C₇₁	—	2	1	no	—
16	#557×C₇₁	C₂²	1	1	yes	characteristic, complemented
17	#554×C₇₁	—	2	3	no	—
18	#554×C₇₁	C₂²	1	3	yes	complemented
19	#550×C₇₁	—	2	1	no	—
20	#550×C₇₁	C₂²	1	2	yes	—
21	D₄×Q₈×C₇₁	C₂²	1	3	yes	complemented
22	D₄²×C₇₁	—	2	3	no	—
23	D₄²×C₇₁	C₂²	1	6	yes	—
24	#535×C₇₁	—	2	3	no	—
25	#533×C₇₁	—	2	3	no	—
26	#533×C₇₁	C₂²	1	3	yes	complemented
27	#532×C₇₁	C₂²	1	3	yes	—
28	C₂×#127×C₇₁	C₂²	1	3	yes	—
29	C₄×#41×C₇₁	—	2	1	no	—
30	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₇₁	C₂²	1	3	yes	complemented
32	#492×C₇₁	C₂²	1	6	yes	complemented
33	#486×C₇₁	C₂²	1	6	yes	complemented
34	#481×C₇₁	C₂²	1	6	yes	complemented
35	#476×C₇₁	C₂²	1	6	yes	complemented
36	#475×C₇₁	C₂²	1	1	yes	characteristic, complemented
37	#475×C₇₁	C₂²	1	3	yes	complemented
38	#465×C₇₁	C₂²	1	12	yes	complemented
39	#461×C₇₁	C₂²	1	6	yes	complemented
40	#460×C₇₁	C₂²	1	6	yes	complemented
41	#460×C₇₁	C₂²	1	3	yes	complemented
42	#460×C₇₁	C₂²	1	3	yes	complemented
43	#460×C₇₁	C₂²	1	6	yes	complemented
44	#460×C₇₁	C₂²	1	6	yes	complemented
45	#450×C₇₁	C₂²	1	12	yes	complemented
46	#448×C₇₁	C₂²	1	6	yes	complemented
47	C₈×Q₈×C₇₁	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₇₁	C₂²	1	6	yes	complemented
49	C₄×SD₁₆×C₇₁	C₂²	1	6	yes	complemented
50	C₈×D₄×C₇₁	C₂²	1	6	yes	complemented
51	#431×C₇₁	C₂²	1	3	yes	complemented
52	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	complemented
53	C₂×#102×C₇₁	C₂²	1	6	yes	complemented
54	H(5,𝔽₂)×C₇₁	—	4	8	no	—
55	C₂×#41×C₇₁	—	2	2	no	—
56	C₂×#41×C₇₁	—	2	3	no	—
57	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₇₁	C₂³	1	3	yes	—
59	C₂²×D₄×C₇₁	—	2	6	no	—
60	C₂²×D₄×C₇₁	—	2	3	no	—
61	C₂²×D₄×C₇₁	—	2	3	no	—
62	C₂²×D₄×C₇₁	C₂³	1	6	yes	—
63	C₂×SD₁₆×C₇₁	—	4	12	no	—
64	C₂×SD₁₆×C₇₁	C₂³	1	6	yes	—
65	C₂×SD₁₆×C₇₁	C₂³	1	6	yes	—
66	C₂×SD₁₆×C₇₁	C₂³	1	12	yes	—
67	C₂²×C₈×C₇₁	C₂³	1	6	yes	—
68	C₄⋊Q₈×C₇₁	—	2	3	no	—
69	C₄⋊Q₈×C₇₁	—	2	3	no	—
70	C₄⋊Q₈×C₇₁	C₂³	1	3	yes	—
71	C₄⋊Q₈×C₇₁	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₇₁	C₂³	1	3	yes	—
73	#127×C₇₁	—	2	3	no	—
74	#127×C₇₁	—	2	3	no	—
75	#127×C₇₁	C₂³	1	3	yes	—
76	#127×C₇₁	C₂³	1	6	yes	—
77	#125×C₇₁	—	2	3	no	—
78	#125×C₇₁	—	2	3	no	—
79	#125×C₇₁	—	2	3	no	—
80	#124×C₇₁	—	2	3	no	—
81	C₂²⋊Q₈×C₇₁	—	2	3	no	—
82	C₂²⋊Q₈×C₇₁	—	2	6	no	—
83	C₂²⋊Q₈×C₇₁	—	2	6	no	—
84	C₂²⋊Q₈×C₇₁	C₂³	1	6	yes	—
85	#121×C₇₁	—	2	3	no	—
86	#121×C₇₁	—	2	6	no	—
87	#121×C₇₁	—	2	6	no	—
88	#121×C₇₁	C₂³	1	6	yes	—
89	#120×C₇₁	—	2	12	no	—
90	C₄×Q₈×C₇₁	—	2	1	no	—
91	C₄×Q₈×C₇₁	—	2	1	no	—
92	C₄×Q₈×C₇₁	—	2	3	no	—
93	C₄×Q₈×C₇₁	—	2	3	no	—
94	C₄×Q₈×C₇₁	—	2	3	no	—
95	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
97	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
98	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
100	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₇₁	—	2	3	no	—
102	C₄×D₄×C₇₁	—	2	3	no	—
103	C₄×D₄×C₇₁	—	2	3	no	—
104	C₄×D₄×C₇₁	D₄	1	3	yes	—
105	C₄×D₄×C₇₁	C₂³	1	6	yes	—
106	C₄×D₄×C₇₁	C₂³	1	3	yes	—
107	#117×C₇₁	—	2	3	no	—
108	#117×C₇₁	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
110	C₂×C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
111	C₂×C₄²×C₇₁	—	2	3	no	—
112	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
113	#106×C₇₁	C₂³	1	2	yes	—
114	#106×C₇₁	C₂³	1	6	yes	—
115	C₄⋊C₈×C₇₁	C₂³	1	3	yes	—
116	C₄⋊C₈×C₇₁	C₂³	1	6	yes	—
117	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
118	#103×C₇₁	C₂³	1	2	yes	—
119	#103×C₇₁	C₂³	1	6	yes	—
120	#102×C₇₁	C₂³	1	6	yes	—
121	#102×C₇₁	C₂³	1	12	yes	—
122	#102×C₇₁	C₂³	1	6	yes	—
123	C₂²⋊C₈×C₇₁	C₂³	1	6	yes	—
124	C₄×C₈×C₇₁	C₂³	1	6	yes	—
125	C₂⁴×C₇₁	—	2	6	no	—
126	#41×C₇₁	—	4	8	no	—
127	#41×C₇₁	—	4	12	no	—
128	#41×C₇₁	SD₁₆	1	8	yes	complemented
129	C₂×Q₈×C₇₁	—	2	3	no	—
130	C₂×Q₈×C₇₁	—	2	1	no	—
131	C₂×Q₈×C₇₁	—	2	3	no	—
132	C₂×Q₈×C₇₁	—	2	6	no	—
133	C₂×Q₈×C₇₁	—	4	3	no	—
134	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₇₁	C₂⁴	1	3	yes	—
136	C₂×Q₈×C₇₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₇₁	—	2	6	no	—
138	C₂×D₄×C₇₁	—	2	3	no	—
139	C₂×D₄×C₇₁	—	2	6	no	—
140	C₂×D₄×C₇₁	—	2	3	no	—
141	C₂×D₄×C₇₁	—	2	6	no	—
142	C₂×D₄×C₇₁	—	2	6	no	—
143	C₂×D₄×C₇₁	—	4	6	no	—
144	C₂×D₄×C₇₁	—	4	6	no	—
145	C₂×D₄×C₇₁	—	4	24	no	—
146	C₂×D₄×C₇₁	—	4	12	no	—
147	C₂×D₄×C₇₁	—	8	6	no	—
148	C₂×D₄×C₇₁	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₇₁	C₂⁴	1	6	yes	—
150	C₂²×C₄×C₇₁	—	2	3	no	—
151	C₂²×C₄×C₇₁	—	2	3	no	—
152	C₂²×C₄×C₇₁	—	2	6	no	—
153	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
156	SD₁₆×C₇₁	—	2	12	no	—
157	SD₁₆×C₇₁	—	4	12	no	—
158	SD₁₆×C₇₁	—	4	6	no	—
159	C₂×C₈×C₇₁	—	4	6	no	—
160	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
161	C₂×C₈×C₇₁	C₂⁴	1	6	yes	—
162	C₄⋊C₄×C₇₁	—	2	1	no	—
163	C₄⋊C₄×C₇₁	—	2	3	no	—
164	C₄⋊C₄×C₇₁	—	2	3	no	—
165	C₄⋊C₄×C₇₁	—	2	3	no	—
166	C₄⋊C₄×C₇₁	—	2	3	no	—
167	C₄⋊C₄×C₇₁	—	2	3	no	—
168	C₄⋊C₄×C₇₁	—	2	3	no	—
169	C₄⋊C₄×C₇₁	—	2	1	no	—
170	C₄⋊C₄×C₇₁	—	2	3	no	—
171	C₄⋊C₄×C₇₁	—	2	6	no	—
172	C₄⋊C₄×C₇₁	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₇₁	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₇₁	C₂⁴	1	1	yes	characteristic
175	C₄⋊C₄×C₇₁	C₂⁴	1	3	yes	—
176	C₄⋊C₄×C₇₁	C₂⁴	1	2	yes	—
177	C₂²⋊C₄×C₇₁	—	2	6	no	—
178	C₂²⋊C₄×C₇₁	—	2	6	no	—
179	C₂²⋊C₄×C₇₁	—	2	6	no	—
180	C₂²⋊C₄×C₇₁	C₂×D₄	1	6	yes	—
181	C₄²×C₇₁	—	2	1	no	—
182	C₄²×C₇₁	—	2	3	no	—
183	C₄²×C₇₁	—	2	3	no	—
184	C₄²×C₇₁	—	2	3	no	—
185	C₄²×C₇₁	C₂×D₄	1	3	yes	—
186	C₄²×C₇₁	C₂×D₄	1	3	yes	—
187	C₄²×C₇₁	C₂⁴	1	3	yes	—
188	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
189	C₂³×C₇₁	—	2	6	no	—
190	C₂³×C₇₁	—	4	12	no	—
191	C₂³×C₇₁	—	8	6	no	—
192	C₂³×C₇₁	—	8	6	no	—
193	C₂³×C₇₁	C₂²×D₄	1	3	yes	—
194	Q₈×C₇₁	—	2	1	no	—
195	Q₈×C₇₁	—	2	3	no	—
196	Q₈×C₇₁	—	2	3	no	—
197	Q₈×C₇₁	—	4	6	no	—
198	Q₈×C₇₁	—	4	2	no	—
199	Q₈×C₇₁	—	8	3	no	—
200	Q₈×C₇₁	C₂×SD₁₆	1	4	yes	complemented
201	Q₈×C₇₁	H(5,𝔽₂)	1	2	yes	complemented
202	D₄×C₇₁	—	2	6	no	—
203	D₄×C₇₁	—	2	6	no	—
204	D₄×C₇₁	—	4	6	no	—
205	D₄×C₇₁	—	4	12	no	—
206	D₄×C₇₁	—	4	12	no	—
207	D₄×C₇₁	—	4	6	no	—
208	D₄×C₇₁	—	8	6	no	—
209	D₄×C₇₁	—	8	6	no	—
210	D₄×C₇₁	C₂×SD₁₆	1	12	yes	complemented
211	C₂×C₄×C₇₁	—	2	1	no	—
212	C₂×C₄×C₇₁	—	2	3	no	—
213	C₂×C₄×C₇₁	—	2	3	no	—
214	C₂×C₄×C₇₁	—	2	1	no	—
215	C₂×C₄×C₇₁	—	2	3	no	—
216	C₂×C₄×C₇₁	—	2	3	no	—
217	C₂×C₄×C₇₁	—	2	2	no	—
218	C₂×C₄×C₇₁	—	4	3	no	—
219	C₂×C₄×C₇₁	—	4	6	no	—
220	C₂×C₄×C₇₁	—	4	12	no	—
221	C₂×C₄×C₇₁	—	8	3	no	—
222	C₂×C₄×C₇₁	C₂×SD₁₆	1	12	yes	complemented
223	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
228	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
229	C₈×C₇₁	—	2	6	no	—
230	C₈×C₇₁	—	2	2	no	—
231	C₂²×C₇₁	—	2	3	no	—
232	C₂²×C₇₁	—	2	3	no	—
233	C₂²×C₇₁	—	4	6	no	—
234	C₂²×C₇₁	—	4	6	no	—
235	C₂²×C₇₁	—	4	6	no	—
236	C₂²×C₇₁	—	16	12	no	—
237	C₂²×C₇₁	C₂²×SD₁₆	1	6	yes	—
238	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₇₁	—	2	1	no	—
240	C₄×C₇₁	—	2	3	no	—
241	C₄×C₇₁	—	2	3	no	—
242	C₄×C₇₁	—	2	2	no	—
243	C₄×C₇₁	—	4	1	no	—
244	C₄×C₇₁	—	4	3	no	—
245	C₄×C₇₁	—	4	3	no	—
246	C₄×C₇₁	—	4	2	no	—
247	C₄×C₇₁	C₂²×SD₁₆	1	6	yes	—
248	C₄×C₇₁	C₂²×SD₁₆	1	2	yes	—
249	C₄×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
250	C₄×C₇₁	#585	1	1	yes	characteristic
251	#61951	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₇₁	—	2	6	no	—
253	C₂×C₇₁	—	8	6	no	—
254	C₂×C₇₁	#3488	1	1	yes	characteristic
255	C₂×C₇₁	C₂³×SD₁₆	1	1	yes	characteristic
256	C₂×C₇₁	#3585	1	1	yes	characteristic
257	#3498	C₂×C₇₁	1	1	yes	characteristic
258	Q₈×#41	C₂×C₇₁	1	1	yes	characteristic, complemented
259	#3392	C₂×C₇₁	1	2	yes	complemented
260	#3362	C₂×C₇₁	1	2	yes	complemented
261	#3295	C₂×C₇₁	1	6	yes	complemented
262	D₄×SD₁₆	C₂×C₇₁	1	6	yes	complemented
263	#3246	C₂×C₇₁	1	3	yes	complemented
264	#3208	C₂×C₇₁	1	3	yes	complemented
265	#3103	C₂×C₇₁	1	3	yes	complemented
266	C₂×#460	C₂×C₇₁	1	3	yes	complemented
267	#2988	C₂×C₇₁	1	1	yes	characteristic, complemented
268	C₇₁	#61951	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×H(5,𝔽₂)	—	2	1	no	—
270	C₂²×SD₁₆	C₂²×C₇₁	1	6	yes	complemented
271	Q₈²	C₂²×C₇₁	1	1	yes	characteristic, complemented
272	#557	—	2	1	no	—
273	#557	C₂²×C₇₁	1	1	yes	characteristic, complemented
274	#554	—	2	3	no	—
275	#554	C₂²×C₇₁	1	3	yes	complemented
276	#550	—	2	1	no	—
277	#550	C₂²×C₇₁	1	2	yes	—
278	D₄×Q₈	C₂²×C₇₁	1	3	yes	complemented
279	D₄²	—	2	3	no	—
280	D₄²	C₂²×C₇₁	1	6	yes	—
281	#535	—	2	3	no	—
282	#533	—	2	3	no	—
283	#533	C₂²×C₇₁	1	3	yes	complemented
284	#532	C₂²×C₇₁	1	3	yes	—
285	C₂×#127	C₂²×C₇₁	1	3	yes	—
286	C₄×#41	—	2	1	no	—
287	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
288	C₂×C₄×Q₈	C₂²×C₇₁	1	3	yes	complemented
289	#492	C₂²×C₇₁	1	6	yes	complemented
290	#486	C₂²×C₇₁	1	6	yes	complemented
291	#481	C₂²×C₇₁	1	6	yes	complemented
292	#476	C₂²×C₇₁	1	6	yes	complemented
293	#475	C₂²×C₇₁	1	1	yes	characteristic, complemented
294	#475	C₂²×C₇₁	1	3	yes	complemented
295	#465	C₂²×C₇₁	1	12	yes	complemented
296	#461	C₂²×C₇₁	1	6	yes	complemented
297	#460	C₂²×C₇₁	1	3	yes	complemented
298	#460	C₂²×C₇₁	1	3	yes	complemented
299	#460	C₂²×C₇₁	1	6	yes	complemented
300	#460	C₂²×C₇₁	1	6	yes	complemented
301	#460	C₂²×C₇₁	1	6	yes	complemented
302	#450	C₂²×C₇₁	1	12	yes	complemented
303	#448	C₂²×C₇₁	1	6	yes	complemented
304	C₈×Q₈	C₂²×C₇₁	1	2	yes	complemented
305	C₄×SD₁₆	C₂²×C₇₁	1	6	yes	complemented
306	C₄×SD₁₆	C₂²×C₇₁	1	6	yes	complemented
307	C₈×D₄	C₂²×C₇₁	1	6	yes	complemented
308	#431	C₂²×C₇₁	1	3	yes	complemented
309	C₂×C₄⋊C₈	C₂²×C₇₁	1	3	yes	complemented
310	C₂×#102	C₂²×C₇₁	1	6	yes	complemented
311	H(5,𝔽₂)	—	4	8	no	—
312	C₂×#41	—	2	2	no	—
313	C₂×#41	—	2	3	no	—
314	C₂×#41	D₄×C₇₁	1	1	yes	characteristic
315	C₂²×Q₈	C₂³×C₇₁	1	3	yes	—
316	C₂²×D₄	—	2	6	no	—
317	C₂²×D₄	—	2	3	no	—
318	C₂²×D₄	—	2	3	no	—
319	C₂²×D₄	C₂³×C₇₁	1	6	yes	—
320	C₂×SD₁₆	—	4	12	no	—
321	C₂×SD₁₆	C₂³×C₇₁	1	6	yes	—
322	C₂×SD₁₆	C₂³×C₇₁	1	12	yes	—
323	C₂×SD₁₆	C₂³×C₇₁	1	6	yes	—
324	C₂²×C₈	C₂³×C₇₁	1	6	yes	—
325	C₄⋊Q₈	—	2	3	no	—
326	C₄⋊Q₈	—	2	3	no	—
327	C₄⋊Q₈	C₂³×C₇₁	1	3	yes	—
328	C₄⋊Q₈	C₂³×C₇₁	1	1	yes	characteristic
329	C₄⋊Q₈	C₂³×C₇₁	1	3	yes	—
330	#127	—	2	3	no	—
331	#127	—	2	3	no	—
332	#127	C₂³×C₇₁	1	3	yes	—
333	#127	C₂³×C₇₁	1	6	yes	—
334	#125	—	2	3	no	—
335	#125	—	2	3	no	—
336	#125	—	2	3	no	—
337	#124	—	2	3	no	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	6	no	—
340	C₂²⋊Q₈	—	2	6	no	—
341	C₂²⋊Q₈	C₂³×C₇₁	1	6	yes	—
342	#121	—	2	3	no	—
343	#121	—	2	6	no	—
344	#121	—	2	6	no	—
345	#121	C₂³×C₇₁	1	6	yes	—
346	#120	—	2	12	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	—	2	1	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	—	2	3	no	—
351	C₄×Q₈	—	2	3	no	—
352	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
353	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
354	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
355	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
356	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
357	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	D₄×C₇₁	1	3	yes	—
362	C₄×D₄	C₂³×C₇₁	1	6	yes	—
363	C₄×D₄	C₂³×C₇₁	1	3	yes	—
364	#117	—	2	3	no	—
365	#117	D₄×C₇₁	1	3	yes	—
366	C₂×C₄⋊C₄	—	2	3	no	—
367	C₂×C₄⋊C₄	C₂³×C₇₁	1	3	yes	—
368	C₂×C₄²	—	2	3	no	—
369	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
370	#106	C₂³×C₇₁	1	2	yes	—
371	#106	C₂³×C₇₁	1	6	yes	—
372	C₄⋊C₈	C₂³×C₇₁	1	3	yes	—
373	C₄⋊C₈	C₂³×C₇₁	1	6	yes	—
374	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
375	#103	C₂³×C₇₁	1	2	yes	—
376	#103	C₂³×C₇₁	1	6	yes	—
377	#102	C₂³×C₇₁	1	6	yes	—
378	#102	C₂³×C₇₁	1	12	yes	—
379	#102	C₂³×C₇₁	1	6	yes	—
380	C₂²⋊C₈	C₂³×C₇₁	1	6	yes	—
381	C₄×C₈	C₂³×C₇₁	1	6	yes	—
382	C₂⁴	—	2	6	no	—
383	#41	—	4	8	no	—
384	#41	—	4	12	no	—
385	#41	SD₁₆×C₇₁	1	8	yes	complemented
386	C₂×Q₈	—	2	3	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	3	no	—
389	C₂×Q₈	—	2	6	no	—
390	C₂×Q₈	—	4	3	no	—
391	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
392	C₂×Q₈	C₂⁴×C₇₁	1	3	yes	—
393	C₂×Q₈	C₂⁴×C₇₁	1	1	yes	characteristic
394	C₂×D₄	—	2	6	no	—
395	C₂×D₄	—	2	3	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	3	no	—
398	C₂×D₄	—	2	6	no	—
399	C₂×D₄	—	2	6	no	—
400	C₂×D₄	—	4	6	no	—
401	C₂×D₄	—	4	6	no	—
402	C₂×D₄	—	4	24	no	—
403	C₂×D₄	—	4	12	no	—
404	C₂×D₄	—	8	6	no	—
405	C₂×D₄	C₂×D₄×C₇₁	1	3	yes	—
406	C₂×D₄	C₂⁴×C₇₁	1	6	yes	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
411	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
412	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
413	SD₁₆	—	2	12	no	—
414	SD₁₆	—	4	12	no	—
415	SD₁₆	—	4	6	no	—
416	C₂×C₈	—	4	6	no	—
417	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
418	C₂×C₈	C₂⁴×C₇₁	1	6	yes	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	3	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	3	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	3	no	—
428	C₄⋊C₄	—	2	6	no	—
429	C₄⋊C₄	C₂×D₄×C₇₁	1	3	yes	—
430	C₄⋊C₄	C₂×D₄×C₇₁	1	3	yes	—
431	C₄⋊C₄	C₂⁴×C₇₁	1	1	yes	characteristic
432	C₄⋊C₄	C₂⁴×C₇₁	1	3	yes	—
433	C₄⋊C₄	C₂⁴×C₇₁	1	2	yes	—
434	C₂²⋊C₄	—	2	6	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₂²⋊C₄	—	2	6	no	—
437	C₂²⋊C₄	C₂×D₄×C₇₁	1	6	yes	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	3	no	—
440	C₄²	—	2	3	no	—
441	C₄²	—	2	3	no	—
442	C₄²	C₂×D₄×C₇₁	1	3	yes	—
443	C₄²	C₂×D₄×C₇₁	1	3	yes	—
444	C₄²	C₂⁴×C₇₁	1	3	yes	—
445	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
446	C₂³	—	2	6	no	—
447	C₂³	—	4	12	no	—
448	C₂³	—	8	6	no	—
449	C₂³	—	8	6	no	—
450	C₂³	C₂²×D₄×C₇₁	1	3	yes	—
451	Q₈	—	2	1	no	—
452	Q₈	—	2	3	no	—
453	Q₈	—	2	3	no	—
454	Q₈	—	4	6	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	8	3	no	—
457	Q₈	C₂×SD₁₆×C₇₁	1	4	yes	complemented
458	Q₈	H(5,𝔽₂)×C₇₁	1	2	yes	complemented
459	D₄	—	2	6	no	—
460	D₄	—	2	6	no	—
461	D₄	—	4	6	no	—
462	D₄	—	4	12	no	—
463	D₄	—	4	12	no	—
464	D₄	—	4	6	no	—
465	D₄	—	8	6	no	—
466	D₄	—	8	6	no	—
467	D₄	C₂×SD₁₆×C₇₁	1	12	yes	complemented
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	3	no	—
470	C₂×C₄	—	2	3	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	3	no	—
476	C₂×C₄	—	4	6	no	—
477	C₂×C₄	—	4	12	no	—
478	C₂×C₄	—	8	3	no	—
479	C₂×C₄	C₂×SD₁₆×C₇₁	1	12	yes	complemented
480	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
482	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
483	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
485	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
486	C₈	—	2	6	no	—
487	C₈	—	2	2	no	—
488	C₂²	—	2	3	no	—
489	C₂²	—	2	3	no	—
490	C₂²	—	4	6	no	—
491	C₂²	—	4	6	no	—
492	C₂²	—	4	6	no	—
493	C₂²	—	16	12	no	—
494	C₂²	C₂²×SD₁₆×C₇₁	1	6	yes	—
495	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
496	C₄	—	2	1	no	—
497	C₄	—	2	3	no	—
498	C₄	—	2	3	no	—
499	C₄	—	2	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	3	no	—
503	C₄	—	4	3	no	—
504	C₄	C₂²×SD₁₆×C₇₁	1	6	yes	—
505	C₄	C₂²×SD₁₆×C₇₁	1	2	yes	—
506	C₄	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
507	C₄	#585×C₇₁	1	1	yes	characteristic
508	C₂	—	2	6	no	—
509	C₂	—	8	6	no	—
510	C₂	C₇₁×#3488	1	1	yes	characteristic
511	C₂	C₂³×SD₁₆×C₇₁	1	1	yes	characteristic
512	C₂	C₇₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
prime radical	C₇₁×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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. Q₈×#41×C₇₁	C₂	semidirect
3. C₇₁×#3392	C₂	semidirect
4. C₇₁×#3362	C₂	semidirect
5. C₇₁×#3295	C₂	semidirect
6. D₄×SD₁₆×C₇₁	C₂	semidirect
7. C₇₁×#3246	C₂	semidirect
8. C₇₁×#3208	C₂	semidirect
9. C₇₁×#3103	C₂	semidirect
10. C₂×#460×C₇₁	C₂	semidirect
11. C₇₁×#2988	C₂	semidirect
13. C₂²×SD₁₆×C₇₁	C₂²	semidirect
14. Q₈²×C₇₁	C₂²	semidirect
16. #557×C₇₁	C₂²	semidirect
18. #554×C₇₁	C₂²	semidirect
21. D₄×Q₈×C₇₁	C₂²	semidirect
26. #533×C₇₁	C₂²	semidirect
31. C₂×C₄×Q₈×C₇₁	C₂²	semidirect
32. #492×C₇₁	C₂²	semidirect
33. #486×C₇₁	C₂²	semidirect
34. #481×C₇₁	C₂²	semidirect
35. #476×C₇₁	C₂²	semidirect
36. #475×C₇₁	C₂²	semidirect
37. #475×C₇₁	C₂²	semidirect
38. #465×C₇₁	C₂²	semidirect
39. #461×C₇₁	C₂²	semidirect
40. #460×C₇₁	C₂²	semidirect
41. #460×C₇₁	C₂²	semidirect
42. #460×C₇₁	C₂²	semidirect
43. #460×C₇₁	C₂²	semidirect
44. #460×C₇₁	C₂²	semidirect
45. #450×C₇₁	C₂²	semidirect
46. #448×C₇₁	C₂²	semidirect
47. C₈×Q₈×C₇₁	C₂²	semidirect
48. C₄×SD₁₆×C₇₁	C₂²	semidirect
49. C₄×SD₁₆×C₇₁	C₂²	semidirect
50. C₈×D₄×C₇₁	C₂²	semidirect
51. #431×C₇₁	C₂²	semidirect
52. C₂×C₄⋊C₈×C₇₁	C₂²	semidirect
53. C₂×#102×C₇₁	C₂²	semidirect
128. #41×C₇₁	SD₁₆	semidirect
200. Q₈×C₇₁	C₂×SD₁₆	semidirect
201. Q₈×C₇₁	H(5,𝔽₂)	semidirect
210. D₄×C₇₁	C₂×SD₁₆	semidirect
222. C₂×C₄×C₇₁	C₂×SD₁₆	semidirect
251. #61951	C₇₁	direct (N × H)
258. Q₈×#41	C₂×C₇₁	semidirect
259. #3392	C₂×C₇₁	semidirect
260. #3362	C₂×C₇₁	semidirect
261. #3295	C₂×C₇₁	semidirect
262. D₄×SD₁₆	C₂×C₇₁	semidirect
263. #3246	C₂×C₇₁	semidirect
264. #3208	C₂×C₇₁	semidirect
265. #3103	C₂×C₇₁	semidirect
266. C₂×#460	C₂×C₇₁	semidirect
267. #2988	C₂×C₇₁	semidirect
268. C₇₁	#61951	direct (N × H)
270. C₂²×SD₁₆	C₂²×C₇₁	semidirect
271. Q₈²	C₂²×C₇₁	semidirect
273. #557	C₂²×C₇₁	semidirect
275. #554	C₂²×C₇₁	semidirect
278. D₄×Q₈	C₂²×C₇₁	semidirect
283. #533	C₂²×C₇₁	semidirect
288. C₂×C₄×Q₈	C₂²×C₇₁	semidirect
289. #492	C₂²×C₇₁	semidirect
290. #486	C₂²×C₇₁	semidirect
291. #481	C₂²×C₇₁	semidirect
292. #476	C₂²×C₇₁	semidirect
293. #475	C₂²×C₇₁	semidirect
294. #475	C₂²×C₇₁	semidirect
295. #465	C₂²×C₇₁	semidirect
296. #461	C₂²×C₇₁	semidirect
297. #460	C₂²×C₇₁	semidirect
298. #460	C₂²×C₇₁	semidirect
299. #460	C₂²×C₇₁	semidirect
300. #460	C₂²×C₇₁	semidirect
301. #460	C₂²×C₇₁	semidirect
302. #450	C₂²×C₇₁	semidirect
303. #448	C₂²×C₇₁	semidirect
304. C₈×Q₈	C₂²×C₇₁	semidirect
305. C₄×SD₁₆	C₂²×C₇₁	semidirect
306. C₄×SD₁₆	C₂²×C₇₁	semidirect
307. C₈×D₄	C₂²×C₇₁	semidirect
308. #431	C₂²×C₇₁	semidirect
309. C₂×C₄⋊C₈	C₂²×C₇₁	semidirect
310. C₂×#102	C₂²×C₇₁	semidirect
385. #41	SD₁₆×C₇₁	semidirect
457. Q₈	C₂×SD₁₆×C₇₁	semidirect
458. Q₈	H(5,𝔽₂)×C₇₁	semidirect
467. D₄	C₂×SD₁₆×C₇₁	semidirect
479. C₂×C₄	C₂×SD₁₆×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3498	2
Q₈×#41×C₇₁	2
C₇₁×#3392	2
C₇₁×#3362	2
C₇₁×#3295	2
D₄×SD₁₆×C₇₁	2
C₇₁×#3246	2
C₇₁×#3208	2
C₇₁×#3103	2
C₂×#460×C₇₁	2
C₇₁×#2988	2
#61951	71

Subgroups — C71×#61951 #31882823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#61951 #31882823