Subgroups — C₇₁×#61559 #31882431

Order 18176.

2854 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3294	C₂	1	8	yes	maximal, complemented
4	D₄×D₈×C₇₁	C₂	1	8	yes	maximal, complemented
5	C₇₁×#3205	C₂	1	2	yes	maximal, complemented
6	C₇₁×#3186	C₂	1	4	yes	maximal, complemented
7	C₇₁×#3100	C₂	1	2	yes	maximal, complemented
8	C₂×#447×C₇₁	C₂	1	4	yes	maximal, complemented
9	C₇₁×#2972	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₇₁	—	2	1	no	—
11	C₂³×D₄×C₇₁	—	2	1	no	—
12	C₂³×D₄×C₇₁	C₂²	1	2	yes	complemented
13	C₂²×D₈×C₇₁	C₂²	1	8	yes	complemented
14	#547×C₇₁	C₂²	1	4	yes	complemented
15	#546×C₇₁	—	2	4	no	—
16	D₄²×C₇₁	—	2	4	no	—
17	D₄²×C₇₁	C₂²	1	8	yes	complemented
18	D₄²×C₇₁	C₂²	1	4	yes	complemented
19	#537×C₇₁	—	2	1	no	—
20	#534×C₇₁	—	2	2	no	—
21	#532×C₇₁	C₂²	1	2	yes	complemented
22	C₂×#127×C₇₁	C₂²	1	1	yes	characteristic, complemented
23	C₂×#121×C₇₁	—	2	1	no	—
24	C₂×#121×C₇₁	C₂²	1	2	yes	complemented
25	C₂×#121×C₇₁	C₂²	1	4	yes	complemented
26	C₂×#120×C₇₁	—	2	2	no	—
27	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
29	#493×C₇₁	C₂²	1	4	yes	complemented
30	#486×C₇₁	C₂²	1	4	yes	complemented
31	#480×C₇₁	C₂²	1	4	yes	complemented
32	#480×C₇₁	C₂²	1	8	yes	complemented
33	#474×C₇₁	C₂²	1	4	yes	complemented
34	#466×C₇₁	C₂²	1	16	yes	complemented
35	#459×C₇₁	C₂²	1	4	yes	complemented
36	#459×C₇₁	C₂²	1	8	yes	complemented
37	#447×C₇₁	C₂²	1	4	yes	complemented
38	#447×C₇₁	C₂²	1	8	yes	complemented
39	#447×C₇₁	C₂²	1	16	yes	complemented
40	#447×C₇₁	C₂²	1	8	yes	complemented
41	C₄×D₈×C₇₁	C₂²	1	8	yes	complemented
42	C₈×D₄×C₇₁	C₂²	1	8	yes	complemented
43	#431×C₇₁	C₂²	1	2	yes	complemented
44	C₂×#102×C₇₁	C₂²	1	8	yes	complemented
45	C₂×C₂²⋊C₈×C₇₁	C₂²	1	4	yes	complemented
46	C₂⁵×C₇₁	—	2	2	no	—
47	H(5,𝔽₂)×C₇₁	—	4	8	no	—
48	C₂×#41×C₇₁	—	2	4	no	—
49	C₂×#41×C₇₁	—	2	2	no	—
50	C₂²×D₄×C₇₁	—	2	2	no	—
51	C₂²×D₄×C₇₁	—	2	4	no	—
52	C₂²×D₄×C₇₁	—	2	8	no	—
53	C₂²×D₄×C₇₁	—	2	4	no	—
54	C₂²×D₄×C₇₁	—	2	2	no	—
55	C₂²×D₄×C₇₁	—	2	1	no	—
56	C₂²×D₄×C₇₁	—	2	4	no	—
57	C₂²×D₄×C₇₁	—	2	1	no	—
58	C₂²×D₄×C₇₁	—	4	8	no	—
59	C₂²×D₄×C₇₁	—	4	8	no	—
60	C₂²×D₄×C₇₁	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₇₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₇₁	C₂³	1	8	yes	—
63	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₇₁	C₂³	1	4	yes	complemented
65	C₂³×C₄×C₇₁	—	2	1	no	—
66	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
67	C₂×D₈×C₇₁	—	4	16	no	—
68	C₂×D₈×C₇₁	C₂³	1	8	yes	—
69	C₂×D₈×C₇₁	C₂³	1	16	yes	complemented
70	C₂²×C₈×C₇₁	C₂³	1	8	yes	complemented
71	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	complemented
72	#127×C₇₁	—	2	2	no	—
73	#127×C₇₁	C₂³	1	4	yes	—
74	#127×C₇₁	C₂³	1	2	yes	complemented
75	#124×C₇₁	—	2	2	no	—
76	#123×C₇₁	—	2	8	no	—
77	C₂²⋊Q₈×C₇₁	—	2	4	no	—
78	#121×C₇₁	—	2	4	no	—
79	#121×C₇₁	—	2	4	no	—
80	#121×C₇₁	—	2	4	no	—
81	#121×C₇₁	—	2	2	no	—
82	#121×C₇₁	—	2	8	no	—
83	#121×C₇₁	—	2	4	no	—
84	#121×C₇₁	C₂³	1	4	yes	—
85	#121×C₇₁	C₂³	1	8	yes	—
86	#121×C₇₁	C₂³	1	8	yes	complemented
87	#120×C₇₁	—	2	16	no	—
88	#120×C₇₁	—	2	8	no	—
89	C₄×D₄×C₇₁	—	2	2	no	—
90	C₄×D₄×C₇₁	—	2	2	no	—
91	C₄×D₄×C₇₁	D₄	1	4	yes	—
92	C₄×D₄×C₇₁	C₂³	1	4	yes	—
93	C₄×D₄×C₇₁	C₂³	1	4	yes	—
94	C₄×D₄×C₇₁	C₂³	1	4	yes	complemented
95	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	complemented
98	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
101	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
102	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
103	#107×C₇₁	C₂³	1	8	yes	complemented
104	C₄⋊C₈×C₇₁	C₂³	1	4	yes	complemented
105	#102×C₇₁	C₂³	1	8	yes	—
106	#102×C₇₁	C₂³	1	16	yes	complemented
107	#102×C₇₁	C₂³	1	8	yes	complemented
108	C₂²⋊C₈×C₇₁	C₂³	1	4	yes	—
109	C₂²⋊C₈×C₇₁	C₂³	1	8	yes	complemented
110	C₄×C₈×C₇₁	C₂³	1	4	yes	complemented
111	C₂⁴×C₇₁	—	2	1	no	—
112	C₂⁴×C₇₁	—	2	2	no	—
113	C₂⁴×C₇₁	—	2	8	no	—
114	C₂⁴×C₇₁	—	2	4	no	—
115	C₂⁴×C₇₁	—	4	8	no	—
116	C₂⁴×C₇₁	—	4	8	no	—
117	C₂⁴×C₇₁	—	8	4	no	—
118	C₂⁴×C₇₁	C₂×D₄	1	2	yes	—
119	#41×C₇₁	—	4	16	no	—
120	#41×C₇₁	—	4	8	no	—
121	C₂×Q₈×C₇₁	—	2	2	no	—
122	C₂×D₄×C₇₁	—	2	4	no	—
123	C₂×D₄×C₇₁	—	2	16	no	—
124	C₂×D₄×C₇₁	—	2	8	no	—
125	C₂×D₄×C₇₁	—	2	8	no	—
126	C₂×D₄×C₇₁	—	2	4	no	—
127	C₂×D₄×C₇₁	—	2	1	no	—
128	C₂×D₄×C₇₁	—	2	8	no	—
129	C₂×D₄×C₇₁	—	2	2	no	—
130	C₂×D₄×C₇₁	—	2	2	no	—
131	C₂×D₄×C₇₁	—	2	4	no	—
132	C₂×D₄×C₇₁	—	2	2	no	—
133	C₂×D₄×C₇₁	—	2	4	no	—
134	C₂×D₄×C₇₁	—	2	4	no	—
135	C₂×D₄×C₇₁	—	4	8	no	—
136	C₂×D₄×C₇₁	—	4	4	no	—
137	C₂×D₄×C₇₁	—	4	8	no	—
138	C₂×D₄×C₇₁	—	4	16	no	—
139	C₂×D₄×C₇₁	—	4	8	no	—
140	C₂×D₄×C₇₁	—	4	8	no	—
141	C₂×D₄×C₇₁	—	4	16	no	—
142	C₂×D₄×C₇₁	—	4	32	no	—
143	C₂×D₄×C₇₁	—	4	16	no	—
144	C₂×D₄×C₇₁	—	4	4	no	—
145	C₂×D₄×C₇₁	—	8	2	no	—
146	C₂×D₄×C₇₁	D₈	1	8	yes	complemented
147	C₂×D₄×C₇₁	C₂×D₄	1	4	yes	—
148	C₂×D₄×C₇₁	C₂⁴	1	4	yes	—
149	C₂×D₄×C₇₁	C₂⁴	1	4	yes	—
150	C₂×D₄×C₇₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₇₁	—	2	1	no	—
152	C₂²×C₄×C₇₁	—	2	4	no	—
153	C₂²×C₄×C₇₁	—	2	4	no	—
154	C₂²×C₄×C₇₁	—	2	4	no	—
155	C₂²×C₄×C₇₁	—	4	4	no	—
156	C₂²×C₄×C₇₁	—	4	4	no	—
157	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
163	D₈×C₇₁	—	2	16	no	—
164	D₈×C₇₁	—	4	16	no	—
165	C₂×C₈×C₇₁	—	4	8	no	—
166	C₂×C₈×C₇₁	C₂⁴	1	8	yes	—
167	C₄⋊C₄×C₇₁	—	2	1	no	—
168	C₄⋊C₄×C₇₁	—	2	4	no	—
169	C₄⋊C₄×C₇₁	—	2	2	no	—
170	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
171	C₄⋊C₄×C₇₁	C₂⁴	1	4	yes	—
172	C₄⋊C₄×C₇₁	C₂⁴	1	2	yes	—
173	C₂²⋊C₄×C₇₁	—	2	4	no	—
174	C₂²⋊C₄×C₇₁	—	2	4	no	—
175	C₂²⋊C₄×C₇₁	—	2	8	no	—
176	C₂²⋊C₄×C₇₁	—	2	8	no	—
177	C₂²⋊C₄×C₇₁	—	2	4	no	—
178	C₂²⋊C₄×C₇₁	C₂×D₄	1	8	yes	—
179	C₄²×C₇₁	C₂×D₄	1	2	yes	—
180	C₄²×C₇₁	C₂⁴	1	2	yes	—
181	C₂³×C₇₁	—	2	4	no	—
182	C₂³×C₇₁	—	2	2	no	—
183	C₂³×C₇₁	—	2	4	no	—
184	C₂³×C₇₁	—	2	4	no	—
185	C₂³×C₇₁	—	4	2	no	—
186	C₂³×C₇₁	—	4	2	no	—
187	C₂³×C₇₁	—	4	16	no	—
188	C₂³×C₇₁	—	4	2	no	—
189	C₂³×C₇₁	—	4	4	no	—
190	C₂³×C₇₁	—	4	4	no	—
191	C₂³×C₇₁	—	4	2	no	—
192	C₂³×C₇₁	—	4	16	no	—
193	C₂³×C₇₁	—	4	4	no	—
194	C₂³×C₇₁	—	4	8	no	—
195	C₂³×C₇₁	—	8	8	no	—
196	C₂³×C₇₁	—	8	4	no	—
197	C₂³×C₇₁	—	8	4	no	—
198	C₂³×C₇₁	—	8	32	no	—
199	C₂³×C₇₁	C₂×D₈	1	8	yes	complemented
200	C₂³×C₇₁	C₂²×D₄	1	4	yes	—
201	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₇₁	—	4	4	no	—
203	D₄×C₇₁	—	2	4	no	—
204	D₄×C₇₁	—	2	2	no	—
205	D₄×C₇₁	—	2	8	no	—
206	D₄×C₇₁	—	2	8	no	—
207	D₄×C₇₁	—	4	16	no	—
208	D₄×C₇₁	—	4	4	no	—
209	D₄×C₇₁	—	4	16	no	—
210	D₄×C₇₁	—	4	8	no	—
211	D₄×C₇₁	—	4	16	no	—
212	D₄×C₇₁	—	4	8	no	—
213	D₄×C₇₁	—	4	4	no	—
214	D₄×C₇₁	—	8	2	no	—
215	D₄×C₇₁	—	8	4	no	—
216	D₄×C₇₁	—	8	4	no	—
217	D₄×C₇₁	C₂×D₈	1	16	yes	complemented
218	D₄×C₇₁	H(5,𝔽₂)	1	4	yes	complemented
219	C₂×C₄×C₇₁	—	2	8	no	—
220	C₂×C₄×C₇₁	—	2	2	no	—
221	C₂×C₄×C₇₁	—	2	2	no	—
222	C₂×C₄×C₇₁	—	2	4	no	—
223	C₂×C₄×C₇₁	—	4	4	no	—
224	C₂×C₄×C₇₁	—	4	2	no	—
225	C₂×C₄×C₇₁	—	4	1	no	—
226	C₂×C₄×C₇₁	—	4	2	no	—
227	C₂×C₄×C₇₁	—	4	8	no	—
228	C₂×C₄×C₇₁	—	4	8	no	—
229	C₂×C₄×C₇₁	—	4	8	no	—
230	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	complemented
231	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₇₁	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
236	C₈×C₇₁	—	2	8	no	—
237	C₂²×C₇₁	—	2	1	no	—
238	C₂²×C₇₁	—	2	4	no	—
239	C₂²×C₇₁	—	2	1	no	—
240	C₂²×C₇₁	—	2	4	no	—
241	C₂²×C₇₁	—	2	4	no	—
242	C₂²×C₇₁	—	4	8	no	—
243	C₂²×C₇₁	—	4	8	no	—
244	C₂²×C₇₁	—	4	2	no	—
245	C₂²×C₇₁	—	4	2	no	—
246	C₂²×C₇₁	—	4	4	no	—
247	C₂²×C₇₁	—	4	4	no	—
248	C₂²×C₇₁	—	4	4	no	—
249	C₂²×C₇₁	—	4	4	no	—
250	C₂²×C₇₁	—	4	4	no	—
251	C₂²×C₇₁	—	4	4	no	—
252	C₂²×C₇₁	—	8	32	no	—
253	C₂²×C₇₁	—	8	8	no	—
254	C₂²×C₇₁	—	16	8	no	—
255	C₂²×C₇₁	C₂²×D₈	1	2	yes	—
256	C₂²×C₇₁	C₂²×D₈	1	8	yes	—
257	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₇₁	—	2	2	no	—
259	C₄×C₇₁	—	2	1	no	—
260	C₄×C₇₁	—	2	4	no	—
261	C₄×C₇₁	—	4	2	no	—
262	C₄×C₇₁	—	4	4	no	—
263	C₄×C₇₁	C₂²×D₈	1	4	yes	—
264	C₄×C₇₁	C₂×H(5,𝔽₂)	1	2	yes	—
265	#61559	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₇₁	—	2	2	no	—
267	C₂×C₇₁	—	2	8	no	—
268	C₂×C₇₁	—	4	4	no	—
269	C₂×C₇₁	—	8	4	no	—
270	C₂×C₇₁	—	8	4	no	—
271	C₂×C₇₁	#3484	1	1	yes	characteristic
272	C₂×C₇₁	C₂³×D₈	1	1	yes	characteristic
273	C₂×C₇₁	#3585	1	1	yes	characteristic
274	#3496	C₂×C₇₁	1	1	yes	characteristic, complemented
275	C₂×D₄²	C₂×C₇₁	1	1	yes	characteristic, complemented
276	#3294	C₂×C₇₁	1	8	yes	complemented
277	D₄×D₈	C₂×C₇₁	1	8	yes	complemented
278	#3205	C₂×C₇₁	1	2	yes	complemented
279	#3186	C₂×C₇₁	1	4	yes	complemented
280	#3100	C₂×C₇₁	1	2	yes	complemented
281	C₂×#447	C₂×C₇₁	1	4	yes	complemented
282	#2972	C₂×C₇₁	1	1	yes	characteristic, complemented
283	C₇₁	#61559	1	1	yes	characteristic, direct product factor, Sylow
284	C₂×H(5,𝔽₂)	—	2	1	no	—
285	C₂³×D₄	—	2	1	no	—
286	C₂³×D₄	C₂²×C₇₁	1	2	yes	complemented
287	C₂²×D₈	C₂²×C₇₁	1	8	yes	complemented
288	#547	C₂²×C₇₁	1	4	yes	complemented
289	#546	—	2	4	no	—
290	D₄²	—	2	4	no	—
291	D₄²	C₂²×C₇₁	1	8	yes	complemented
292	D₄²	C₂²×C₇₁	1	4	yes	complemented
293	#537	—	2	1	no	—
294	#534	—	2	2	no	—
295	#532	C₂²×C₇₁	1	2	yes	complemented
296	C₂×#127	C₂²×C₇₁	1	1	yes	characteristic, complemented
297	C₂×#121	—	2	1	no	—
298	C₂×#121	C₂²×C₇₁	1	2	yes	complemented
299	C₂×#121	C₂²×C₇₁	1	4	yes	complemented
300	C₂×#120	—	2	2	no	—
301	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
302	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic, complemented
303	#493	C₂²×C₇₁	1	4	yes	complemented
304	#486	C₂²×C₇₁	1	4	yes	complemented
305	#480	C₂²×C₇₁	1	4	yes	complemented
306	#480	C₂²×C₇₁	1	8	yes	complemented
307	#474	C₂²×C₇₁	1	4	yes	complemented
308	#466	C₂²×C₇₁	1	16	yes	complemented
309	#459	C₂²×C₇₁	1	4	yes	complemented
310	#459	C₂²×C₇₁	1	8	yes	complemented
311	#447	C₂²×C₇₁	1	4	yes	complemented
312	#447	C₂²×C₇₁	1	8	yes	complemented
313	#447	C₂²×C₇₁	1	16	yes	complemented
314	#447	C₂²×C₇₁	1	8	yes	complemented
315	C₄×D₈	C₂²×C₇₁	1	8	yes	complemented
316	C₈×D₄	C₂²×C₇₁	1	8	yes	complemented
317	#431	C₂²×C₇₁	1	2	yes	complemented
318	C₂×#102	C₂²×C₇₁	1	8	yes	complemented
319	C₂×C₂²⋊C₈	C₂²×C₇₁	1	4	yes	complemented
320	C₂⁵	—	2	2	no	—
321	H(5,𝔽₂)	—	4	8	no	—
322	C₂×#41	—	2	4	no	—
323	C₂×#41	—	2	2	no	—
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	—	2	8	no	—
327	C₂²×D₄	—	2	4	no	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	4	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	4	8	no	—
333	C₂²×D₄	—	4	8	no	—
334	C₂²×D₄	D₄×C₇₁	1	1	yes	characteristic
335	C₂²×D₄	C₂³×C₇₁	1	1	yes	characteristic
336	C₂²×D₄	C₂³×C₇₁	1	8	yes	—
337	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
338	C₂²×D₄	C₂³×C₇₁	1	4	yes	complemented
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
341	C₂×D₈	—	4	16	no	—
342	C₂×D₈	C₂³×C₇₁	1	8	yes	—
343	C₂×D₈	C₂³×C₇₁	1	16	yes	complemented
344	C₂²×C₈	C₂³×C₇₁	1	8	yes	complemented
345	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	complemented
346	#127	—	2	2	no	—
347	#127	C₂³×C₇₁	1	4	yes	—
348	#127	C₂³×C₇₁	1	2	yes	complemented
349	#124	—	2	2	no	—
350	#123	—	2	8	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	#121	—	2	4	no	—
353	#121	—	2	4	no	—
354	#121	—	2	4	no	—
355	#121	—	2	2	no	—
356	#121	—	2	8	no	—
357	#121	—	2	4	no	—
358	#121	C₂³×C₇₁	1	4	yes	—
359	#121	C₂³×C₇₁	1	8	yes	—
360	#121	C₂³×C₇₁	1	8	yes	complemented
361	#120	—	2	16	no	—
362	#120	—	2	8	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	D₄×C₇₁	1	4	yes	—
366	C₄×D₄	C₂³×C₇₁	1	4	yes	—
367	C₄×D₄	C₂³×C₇₁	1	4	yes	—
368	C₄×D₄	C₂³×C₇₁	1	4	yes	complemented
369	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	complemented
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	4	no	—
375	C₂×C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
376	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
377	#107	C₂³×C₇₁	1	8	yes	complemented
378	C₄⋊C₈	C₂³×C₇₁	1	4	yes	complemented
379	#102	C₂³×C₇₁	1	8	yes	—
380	#102	C₂³×C₇₁	1	16	yes	complemented
381	#102	C₂³×C₇₁	1	8	yes	complemented
382	C₂²⋊C₈	C₂³×C₇₁	1	4	yes	—
383	C₂²⋊C₈	C₂³×C₇₁	1	8	yes	complemented
384	C₄×C₈	C₂³×C₇₁	1	4	yes	complemented
385	C₂⁴	—	2	1	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	2	8	no	—
388	C₂⁴	—	2	4	no	—
389	C₂⁴	—	4	8	no	—
390	C₂⁴	—	4	8	no	—
391	C₂⁴	—	8	4	no	—
392	C₂⁴	C₂×D₄×C₇₁	1	2	yes	—
393	#41	—	4	16	no	—
394	#41	—	4	8	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	16	no	—
398	C₂×D₄	—	2	8	no	—
399	C₂×D₄	—	2	8	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	4	8	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	8	no	—
412	C₂×D₄	—	4	16	no	—
413	C₂×D₄	—	4	8	no	—
414	C₂×D₄	—	4	8	no	—
415	C₂×D₄	—	4	16	no	—
416	C₂×D₄	—	4	32	no	—
417	C₂×D₄	—	4	16	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	—	8	2	no	—
420	C₂×D₄	D₈×C₇₁	1	8	yes	complemented
421	C₂×D₄	C₂×D₄×C₇₁	1	4	yes	—
422	C₂×D₄	C₂⁴×C₇₁	1	4	yes	—
423	C₂×D₄	C₂⁴×C₇₁	1	4	yes	—
424	C₂×D₄	C₂⁴×C₇₁	1	2	yes	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	4	no	—
431	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
433	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
434	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
437	D₈	—	2	16	no	—
438	D₈	—	4	16	no	—
439	C₂×C₈	—	4	8	no	—
440	C₂×C₈	C₂⁴×C₇₁	1	8	yes	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₇₁	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₇₁	1	2	yes	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	C₂×D₄×C₇₁	1	8	yes	—
453	C₄²	C₂×D₄×C₇₁	1	2	yes	—
454	C₄²	C₂⁴×C₇₁	1	2	yes	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	16	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	4	no	—
464	C₂³	—	4	4	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	16	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	8	no	—
469	C₂³	—	8	8	no	—
470	C₂³	—	8	4	no	—
471	C₂³	—	8	4	no	—
472	C₂³	—	8	32	no	—
473	C₂³	C₂×D₈×C₇₁	1	8	yes	complemented
474	C₂³	C₂²×D₄×C₇₁	1	4	yes	—
475	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
476	Q₈	—	4	4	no	—
477	D₄	—	2	4	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	8	no	—
480	D₄	—	2	8	no	—
481	D₄	—	4	16	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	16	no	—
484	D₄	—	4	8	no	—
485	D₄	—	4	16	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	4	no	—
488	D₄	—	8	4	no	—
489	D₄	—	8	4	no	—
490	D₄	—	8	2	no	—
491	D₄	C₂×D₈×C₇₁	1	16	yes	complemented
492	D₄	H(5,𝔽₂)×C₇₁	1	4	yes	complemented
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	—	4	8	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	C₂×D₈×C₇₁	1	4	yes	complemented
505	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
507	C₂×C₄	C₂²×D₄×C₇₁	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
509	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
510	C₈	—	2	8	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	4	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	4	8	no	—
517	C₂²	—	4	8	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	8	32	no	—
527	C₂²	—	8	8	no	—
528	C₂²	—	16	8	no	—
529	C₂²	C₂²×D₈×C₇₁	1	2	yes	—
530	C₂²	C₂²×D₈×C₇₁	1	8	yes	—
531	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
532	C₄	—	2	2	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	4	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	4	no	—
537	C₄	C₂²×D₈×C₇₁	1	4	yes	—
538	C₄	C₂×H(5,𝔽₂)×C₇₁	1	2	yes	—
539	C₂	—	2	2	no	—
540	C₂	—	2	8	no	—
541	C₂	—	4	4	no	—
542	C₂	—	8	4	no	—
543	C₂	—	8	4	no	—
544	C₂	C₇₁×#3484	1	1	yes	characteristic
545	C₂	C₂³×D₈×C₇₁	1	1	yes	characteristic
546	C₂	C₇₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄×D₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3496	C₂	semidirect
2. C₂×D₄²×C₇₁	C₂	semidirect
3. C₇₁×#3294	C₂	semidirect
4. D₄×D₈×C₇₁	C₂	semidirect
5. C₇₁×#3205	C₂	semidirect
6. C₇₁×#3186	C₂	semidirect
7. C₇₁×#3100	C₂	semidirect
8. C₂×#447×C₇₁	C₂	semidirect
9. C₇₁×#2972	C₂	semidirect
12. C₂³×D₄×C₇₁	C₂²	semidirect
13. C₂²×D₈×C₇₁	C₂²	semidirect
14. #547×C₇₁	C₂²	semidirect
17. D₄²×C₇₁	C₂²	semidirect
18. D₄²×C₇₁	C₂²	semidirect
21. #532×C₇₁	C₂²	semidirect
22. C₂×#127×C₇₁	C₂²	semidirect
24. C₂×#121×C₇₁	C₂²	semidirect
25. C₂×#121×C₇₁	C₂²	semidirect
28. C₂×C₄×D₄×C₇₁	C₂²	semidirect
29. #493×C₇₁	C₂²	semidirect
30. #486×C₇₁	C₂²	semidirect
31. #480×C₇₁	C₂²	semidirect
32. #480×C₇₁	C₂²	semidirect
33. #474×C₇₁	C₂²	semidirect
34. #466×C₇₁	C₂²	semidirect
35. #459×C₇₁	C₂²	semidirect
36. #459×C₇₁	C₂²	semidirect
37. #447×C₇₁	C₂²	semidirect
38. #447×C₇₁	C₂²	semidirect
39. #447×C₇₁	C₂²	semidirect
40. #447×C₇₁	C₂²	semidirect
41. C₄×D₈×C₇₁	C₂²	semidirect
42. C₈×D₄×C₇₁	C₂²	semidirect
43. #431×C₇₁	C₂²	semidirect
44. C₂×#102×C₇₁	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₇₁	C₂²	semidirect
64. C₂²×D₄×C₇₁	C₂³	semidirect
69. C₂×D₈×C₇₁	C₂³	semidirect
70. C₂²×C₈×C₇₁	C₂³	semidirect
71. C₄⋊Q₈×C₇₁	C₂³	semidirect
74. #127×C₇₁	C₂³	semidirect
86. #121×C₇₁	C₂³	semidirect
94. C₄×D₄×C₇₁	C₂³	semidirect
97. C₂×C₄⋊C₄×C₇₁	C₂³	semidirect
103. #107×C₇₁	C₂³	semidirect
104. C₄⋊C₈×C₇₁	C₂³	semidirect
106. #102×C₇₁	C₂³	semidirect
107. #102×C₇₁	C₂³	semidirect
109. C₂²⋊C₈×C₇₁	C₂³	semidirect
110. C₄×C₈×C₇₁	C₂³	semidirect
146. C₂×D₄×C₇₁	D₈	semidirect
199. C₂³×C₇₁	C₂×D₈	semidirect
217. D₄×C₇₁	C₂×D₈	semidirect
218. D₄×C₇₁	H(5,𝔽₂)	semidirect
230. C₂×C₄×C₇₁	C₂×D₈	semidirect
265. #61559	C₇₁	direct (N × H)
274. #3496	C₂×C₇₁	semidirect
275. C₂×D₄²	C₂×C₇₁	semidirect
276. #3294	C₂×C₇₁	semidirect
277. D₄×D₈	C₂×C₇₁	semidirect
278. #3205	C₂×C₇₁	semidirect
279. #3186	C₂×C₇₁	semidirect
280. #3100	C₂×C₇₁	semidirect
281. C₂×#447	C₂×C₇₁	semidirect
282. #2972	C₂×C₇₁	semidirect
283. C₇₁	#61559	direct (N × H)
286. C₂³×D₄	C₂²×C₇₁	semidirect
287. C₂²×D₈	C₂²×C₇₁	semidirect
288. #547	C₂²×C₇₁	semidirect
291. D₄²	C₂²×C₇₁	semidirect
292. D₄²	C₂²×C₇₁	semidirect
295. #532	C₂²×C₇₁	semidirect
296. C₂×#127	C₂²×C₇₁	semidirect
298. C₂×#121	C₂²×C₇₁	semidirect
299. C₂×#121	C₂²×C₇₁	semidirect
302. C₂×C₄×D₄	C₂²×C₇₁	semidirect
303. #493	C₂²×C₇₁	semidirect
304. #486	C₂²×C₇₁	semidirect
305. #480	C₂²×C₇₁	semidirect
306. #480	C₂²×C₇₁	semidirect
307. #474	C₂²×C₇₁	semidirect
308. #466	C₂²×C₇₁	semidirect
309. #459	C₂²×C₇₁	semidirect
310. #459	C₂²×C₇₁	semidirect
311. #447	C₂²×C₇₁	semidirect
312. #447	C₂²×C₇₁	semidirect
313. #447	C₂²×C₇₁	semidirect
314. #447	C₂²×C₇₁	semidirect
315. C₄×D₈	C₂²×C₇₁	semidirect
316. C₈×D₄	C₂²×C₇₁	semidirect
317. #431	C₂²×C₇₁	semidirect
318. C₂×#102	C₂²×C₇₁	semidirect
319. C₂×C₂²⋊C₈	C₂²×C₇₁	semidirect
338. C₂²×D₄	C₂³×C₇₁	semidirect
343. C₂×D₈	C₂³×C₇₁	semidirect
344. C₂²×C₈	C₂³×C₇₁	semidirect
345. C₄⋊Q₈	C₂³×C₇₁	semidirect
348. #127	C₂³×C₇₁	semidirect
360. #121	C₂³×C₇₁	semidirect
368. C₄×D₄	C₂³×C₇₁	semidirect
371. C₂×C₄⋊C₄	C₂³×C₇₁	semidirect
377. #107	C₂³×C₇₁	semidirect
378. C₄⋊C₈	C₂³×C₇₁	semidirect
380. #102	C₂³×C₇₁	semidirect
381. #102	C₂³×C₇₁	semidirect
383. C₂²⋊C₈	C₂³×C₇₁	semidirect
384. C₄×C₈	C₂³×C₇₁	semidirect
420. C₂×D₄	D₈×C₇₁	semidirect
473. C₂³	C₂×D₈×C₇₁	semidirect
491. D₄	C₂×D₈×C₇₁	semidirect
492. D₄	H(5,𝔽₂)×C₇₁	semidirect
504. C₂×C₄	C₂×D₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3496	2
C₂×D₄²×C₇₁	2
C₇₁×#3294	2
D₄×D₈×C₇₁	2
C₇₁×#3205	2
C₇₁×#3186	2
C₇₁×#3100	2
C₂×#447×C₇₁	2
C₇₁×#2972	2
#61559	71

Subgroups — C71×#61559 #31882431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#61559 #31882431