Subgroups — C₇₁×#24286 #31845158

Order 18176.

1322 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#3054	C₂	1	2	yes	maximal, complemented
3	C₇₁×#3025	C₂	1	2	yes	maximal, complemented
4	C₇₁×#2930	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#2053	C₂	1	2	yes	maximal, complemented
6	C₇₁×#2010	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1976	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1812	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#1794	C₂	1	2	yes	maximal, complemented
10	#585×C₇₁	C₂²	1	2	yes	—
11	#578×C₇₁	—	2	2	no	—
12	#577×C₇₁	—	2	4	no	—
13	#576×C₇₁	—	2	2	no	—
14	#575×C₇₁	—	2	2	no	—
15	#575×C₇₁	C₂²	1	2	yes	—
16	C₂×#135×C₇₁	—	2	2	no	—
17	C₂×#135×C₇₁	C₂²	1	2	yes	—
18	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₇₁	C₂²	1	1	yes	characteristic, complemented
20	#525×C₇₁	C₂²	1	1	yes	characteristic
21	#525×C₇₁	C₂²	1	4	yes	complemented
22	#470×C₇₁	—	2	2	no	—
23	#469×C₇₁	—	2	2	no	—
24	#468×C₇₁	—	2	4	no	—
25	#458×C₇₁	—	2	4	no	—
26	B(4,𝔽₂)×C₇₁	—	2	4	no	—
27	#436×C₇₁	—	2	2	no	—
28	#435×C₇₁	—	2	2	no	—
29	#427×C₇₁	C₂²	1	2	yes	complemented
30	#424×C₇₁	C₂²	1	1	yes	characteristic, complemented
31	#421×C₇₁	C₂²	1	4	yes	complemented
32	#416×C₇₁	C₂²	1	4	yes	complemented
33	#410×C₇₁	C₂²	1	2	yes	—
34	#410×C₇₁	C₂²	1	4	yes	complemented
35	#408×C₇₁	C₂²	1	1	yes	characteristic
36	#407×C₇₁	C₂²	1	1	yes	characteristic, complemented
37	#405×C₇₁	C₂²	1	1	yes	characteristic, complemented
38	H(3,ℤ/4)×C₇₁	C₂²	1	2	yes	complemented
39	2₋¹⁺⁴×C₇₁	—	2	4	no	—
40	2₋¹⁺⁴×C₇₁	—	2	2	no	—
41	H(5,𝔽₂)×C₇₁	—	2	4	no	—
42	H(5,𝔽₂)×C₇₁	—	2	4	no	—
43	H(5,𝔽₂)×C₇₁	—	2	2	no	—
44	C₂×#41×C₇₁	—	2	4	no	—
45	C₂×#41×C₇₁	—	4	4	no	—
46	C₂×#41×C₇₁	C₂³	1	4	yes	—
47	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
49	#137×C₇₁	—	2	8	no	—
50	#137×C₇₁	—	2	4	no	—
51	AGL(1,ℤ/8)×C₇₁	—	2	8	no	—
52	AGL(1,ℤ/8)×C₇₁	—	2	4	no	—
53	#135×C₇₁	—	2	4	no	—
54	#135×C₇₁	—	2	4	no	—
55	#135×C₇₁	—	2	2	no	—
56	#135×C₇₁	—	2	4	no	—
57	#135×C₇₁	—	2	2	no	—
58	#135×C₇₁	—	2	2	no	—
59	#135×C₇₁	—	2	2	no	—
60	C₂×Q₁₆×C₇₁	—	2	2	no	—
61	C₂×Q₁₆×C₇₁	D₄	1	2	yes	complemented
62	C₂×SD₁₆×C₇₁	—	2	4	no	—
63	C₂×SD₁₆×C₇₁	D₄	1	4	yes	complemented
64	C₂×D₈×C₇₁	—	2	2	no	—
65	C₂×D₈×C₇₁	D₄	1	2	yes	complemented
66	#131×C₇₁	—	2	2	no	—
67	#131×C₇₁	—	2	2	no	—
68	C₂×M₁₆×C₇₁	—	2	1	no	—
69	C₂×M₁₆×C₇₁	—	2	1	no	—
70	C₂×M₁₆×C₇₁	—	2	2	no	—
71	C₂×M₁₆×C₇₁	—	2	2	no	—
72	C₂×M₁₆×C₇₁	D₄	1	2	yes	complemented
73	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₇₁	D₄	1	2	yes	complemented
76	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	#123×C₇₁	—	2	4	no	—
79	#123×C₇₁	D₄	1	2	yes	—
80	C₂²⋊Q₈×C₇₁	—	2	1	no	—
81	C₂²⋊Q₈×C₇₁	—	2	4	no	—
82	#121×C₇₁	—	2	1	no	—
83	#121×C₇₁	—	2	4	no	—
84	#120×C₇₁	—	2	4	no	—
85	#120×C₇₁	D₄	1	2	yes	—
86	C₄×D₄×C₇₁	—	2	2	no	—
87	C₄×D₄×C₇₁	—	4	4	no	—
88	#117×C₇₁	C₂³	1	4	yes	—
89	#117×C₇₁	C₂³	1	1	yes	characteristic
90	#107×C₇₁	—	2	2	no	—
91	#106×C₇₁	—	2	2	no	—
92	C₄⋊C₈×C₇₁	—	2	2	no	—
93	#104×C₇₁	—	4	4	no	—
94	#103×C₇₁	—	2	4	no	—
95	#102×C₇₁	—	2	4	no	—
96	#99×C₇₁	—	2	4	no	—
97	#99×C₇₁	—	2	8	no	—
98	C₂²⋊C₈×C₇₁	—	2	2	no	—
99	C₂²⋊C₈×C₇₁	—	2	2	no	—
100	#97×C₇₁	—	2	1	no	—
101	C₄×C₈×C₇₁	—	2	1	no	—
102	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
103	#41×C₇₁	—	2	2	no	—
104	#41×C₇₁	—	2	2	no	—
105	#41×C₇₁	—	2	8	no	—
106	#41×C₇₁	—	2	4	no	—
107	#41×C₇₁	—	2	4	no	—
108	#41×C₇₁	—	2	4	no	—
109	#41×C₇₁	—	2	4	no	—
110	#41×C₇₁	—	4	8	no	—
111	#41×C₇₁	—	4	4	no	—
112	#41×C₇₁	—	4	4	no	—
113	#41×C₇₁	—	4	4	no	—
114	#41×C₇₁	—	4	4	no	—
115	C₂×Q₈×C₇₁	—	2	4	no	—
116	C₂×Q₈×C₇₁	—	4	4	no	—
117	C₂×Q₈×C₇₁	C₂×D₄	1	4	yes	—
118	C₂×Q₈×C₇₁	#41	1	1	yes	characteristic
119	C₂×D₄×C₇₁	—	2	4	no	—
120	C₂×D₄×C₇₁	—	2	4	no	—
121	C₂×D₄×C₇₁	—	2	2	no	—
122	C₂×D₄×C₇₁	—	4	4	no	—
123	C₂×D₄×C₇₁	—	4	8	no	—
124	C₂×D₄×C₇₁	—	4	4	no	—
125	C₂×D₄×C₇₁	—	4	4	no	—
126	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₇₁	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₇₁	#41	1	1	yes	characteristic
129	C₂²×C₄×C₇₁	—	2	1	no	—
130	C₂²×C₄×C₇₁	—	2	2	no	—
131	C₂²×C₄×C₇₁	—	2	1	no	—
132	C₂²×C₄×C₇₁	—	2	2	no	—
133	C₂²×C₄×C₇₁	—	2	4	no	—
134	C₂²×C₄×C₇₁	—	4	4	no	—
135	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₇₁	—	2	4	no	—
139	Q₁₆×C₇₁	—	2	2	no	—
140	Q₁₆×C₇₁	—	2	2	no	—
141	SD₁₆×C₇₁	—	2	8	no	—
142	SD₁₆×C₇₁	—	2	4	no	—
143	SD₁₆×C₇₁	—	2	4	no	—
144	D₈×C₇₁	—	2	4	no	—
145	D₈×C₇₁	—	2	2	no	—
146	D₈×C₇₁	—	2	2	no	—
147	M₁₆×C₇₁	—	2	2	no	—
148	M₁₆×C₇₁	—	2	4	no	—
149	M₁₆×C₇₁	—	2	2	no	—
150	M₁₆×C₇₁	—	4	1	no	—
151	M₁₆×C₇₁	—	4	1	no	—
152	M₁₆×C₇₁	—	4	2	no	—
153	C₂×C₈×C₇₁	—	2	2	no	—
154	C₂×C₈×C₇₁	—	2	2	no	—
155	C₂×C₈×C₇₁	—	2	1	no	—
156	C₂×C₈×C₇₁	—	2	1	no	—
157	C₂×C₈×C₇₁	—	2	2	no	—
158	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₇₁	—	2	2	no	—
164	C₄⋊C₄×C₇₁	—	2	1	no	—
165	C₄⋊C₄×C₇₁	—	2	4	no	—
166	C₄⋊C₄×C₇₁	—	4	4	no	—
167	C₂²⋊C₄×C₇₁	—	2	2	no	—
168	C₂²⋊C₄×C₇₁	—	2	2	no	—
169	C₂²⋊C₄×C₇₁	—	2	4	no	—
170	C₂²⋊C₄×C₇₁	—	4	4	no	—
171	C₂²⋊C₄×C₇₁	—	4	4	no	—
172	C₂²⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
173	C₄²×C₇₁	—	2	1	no	—
174	C₄²×C₇₁	—	2	4	no	—
175	C₂³×C₇₁	—	2	1	no	—
176	C₂³×C₇₁	—	2	2	no	—
177	C₂³×C₇₁	—	2	4	no	—
178	C₂³×C₇₁	—	4	4	no	—
179	C₂³×C₇₁	—	8	1	no	—
180	C₂³×C₇₁	#121	1	2	yes	—
181	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₇₁	—	2	2	no	—
183	Q₈×C₇₁	—	2	4	no	—
184	Q₈×C₇₁	—	2	4	no	—
185	Q₈×C₇₁	—	4	4	no	—
186	D₄×C₇₁	—	2	4	no	—
187	D₄×C₇₁	—	2	4	no	—
188	D₄×C₇₁	—	2	2	no	—
189	D₄×C₇₁	—	2	4	no	—
190	D₄×C₇₁	—	4	4	no	—
191	D₄×C₇₁	—	4	8	no	—
192	D₄×C₇₁	—	4	4	no	—
193	D₄×C₇₁	—	4	4	no	—
194	D₄×C₇₁	—	8	2	no	—
195	D₄×C₇₁	—	8	8	no	—
196	C₂×C₄×C₇₁	—	2	2	no	—
197	C₂×C₄×C₇₁	—	2	2	no	—
198	C₂×C₄×C₇₁	—	2	2	no	—
199	C₂×C₄×C₇₁	—	2	1	no	—
200	C₂×C₄×C₇₁	—	2	2	no	—
201	C₂×C₄×C₇₁	—	2	4	no	—
202	C₂×C₄×C₇₁	—	4	8	no	—
203	C₂×C₄×C₇₁	—	4	4	no	—
204	C₂×C₄×C₇₁	—	4	1	no	—
205	C₂×C₄×C₇₁	—	4	1	no	—
206	C₂×C₄×C₇₁	—	4	4	no	—
207	C₂×C₄×C₇₁	—	4	2	no	—
208	C₂×C₄×C₇₁	—	4	4	no	—
209	C₂×C₄×C₇₁	—	4	4	no	—
210	C₂×C₄×C₇₁	#121	1	2	yes	—
211	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
215	C₈×C₇₁	—	2	2	no	—
216	C₈×C₇₁	—	4	1	no	—
217	C₈×C₇₁	—	4	1	no	—
218	C₈×C₇₁	#120	1	1	yes	characteristic
219	C₈×C₇₁	#120	1	1	yes	characteristic
220	C₈×C₇₁	#120	1	2	yes	—
221	C₂²×C₇₁	—	2	2	no	—
222	C₂²×C₇₁	—	4	1	no	—
223	C₂²×C₇₁	—	4	2	no	—
224	C₂²×C₇₁	—	4	1	no	—
225	C₂²×C₇₁	—	4	4	no	—
226	C₂²×C₇₁	—	4	1	no	—
227	C₂²×C₇₁	—	4	4	no	—
228	C₂²×C₇₁	—	4	1	no	—
229	C₂²×C₇₁	—	8	4	no	—
230	C₂²×C₇₁	—	16	4	no	—
231	C₂²×C₇₁	C₂×#121	1	1	yes	characteristic
232	C₂²×C₇₁	D₄²	1	2	yes	—
233	C₄×C₇₁	—	2	2	no	—
234	C₄×C₇₁	—	4	2	no	—
235	C₄×C₇₁	—	4	1	no	—
236	C₄×C₇₁	—	4	4	no	—
237	C₄×C₇₁	—	8	4	no	—
238	C₄×C₇₁	C₂×#120	1	1	yes	characteristic
239	C₄×C₇₁	#535	1	1	yes	characteristic
240	C₄×C₇₁	#546	1	2	yes	—
241	#24286	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂×C₇₁	—	2	1	no	—
243	C₂×C₇₁	—	2	2	no	—
244	C₂×C₇₁	—	4	2	no	—
245	C₂×C₇₁	—	8	1	no	—
246	C₂×C₇₁	—	8	4	no	—
247	C₂×C₇₁	#2408	1	1	yes	characteristic
248	#3584	C₂×C₇₁	1	1	yes	characteristic, complemented
249	#3054	C₂×C₇₁	1	2	yes	complemented
250	#3025	C₂×C₇₁	1	2	yes	complemented
251	#2930	C₂×C₇₁	1	1	yes	characteristic, complemented
252	#2053	C₂×C₇₁	1	2	yes	complemented
253	#2010	C₂×C₇₁	1	2	yes	complemented
254	#1976	C₂×C₇₁	1	2	yes	complemented
255	#1812	C₂×C₇₁	1	1	yes	characteristic, complemented
256	#1794	C₂×C₇₁	1	2	yes	complemented
257	C₇₁	#24286	1	1	yes	characteristic, direct product factor, Sylow
258	#585	C₂²×C₇₁	1	2	yes	—
259	#578	—	2	2	no	—
260	#577	—	2	4	no	—
261	#576	—	2	2	no	—
262	#575	—	2	2	no	—
263	#575	C₂²×C₇₁	1	2	yes	—
264	C₂×#135	—	2	2	no	—
265	C₂×#135	C₂²×C₇₁	1	2	yes	—
266	C₂×#131	C₂²×C₇₁	1	1	yes	characteristic
267	C₂²×M₁₆	C₂²×C₇₁	1	1	yes	characteristic, complemented
268	#525	C₂²×C₇₁	1	1	yes	characteristic
269	#525	C₂²×C₇₁	1	4	yes	complemented
270	#470	—	2	2	no	—
271	#469	—	2	2	no	—
272	#468	—	2	4	no	—
273	#458	—	2	4	no	—
274	B(4,𝔽₂)	—	2	4	no	—
275	#436	—	2	2	no	—
276	#435	—	2	2	no	—
277	#427	C₂²×C₇₁	1	2	yes	complemented
278	#424	C₂²×C₇₁	1	1	yes	characteristic, complemented
279	#421	C₂²×C₇₁	1	4	yes	complemented
280	#416	C₂²×C₇₁	1	4	yes	complemented
281	#410	C₂²×C₇₁	1	2	yes	—
282	#410	C₂²×C₇₁	1	4	yes	complemented
283	#408	C₂²×C₇₁	1	1	yes	characteristic
284	#407	C₂²×C₇₁	1	1	yes	characteristic, complemented
285	#405	C₂²×C₇₁	1	1	yes	characteristic, complemented
286	H(3,ℤ/4)	C₂²×C₇₁	1	2	yes	complemented
287	2₋¹⁺⁴	—	2	4	no	—
288	2₋¹⁺⁴	—	2	2	no	—
289	H(5,𝔽₂)	—	2	4	no	—
290	H(5,𝔽₂)	—	2	4	no	—
291	H(5,𝔽₂)	—	2	2	no	—
292	C₂×#41	—	2	4	no	—
293	C₂×#41	—	4	4	no	—
294	C₂×#41	C₂³×C₇₁	1	4	yes	—
295	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
297	#137	—	2	8	no	—
298	#137	—	2	4	no	—
299	AGL(1,ℤ/8)	—	2	8	no	—
300	AGL(1,ℤ/8)	—	2	4	no	—
301	#135	—	2	4	no	—
302	#135	—	2	4	no	—
303	#135	—	2	2	no	—
304	#135	—	2	4	no	—
305	#135	—	2	2	no	—
306	#135	—	2	2	no	—
307	#135	—	2	2	no	—
308	C₂×Q₁₆	—	2	2	no	—
309	C₂×Q₁₆	D₄×C₇₁	1	2	yes	complemented
310	C₂×SD₁₆	—	2	4	no	—
311	C₂×SD₁₆	D₄×C₇₁	1	4	yes	complemented
312	C₂×D₈	—	2	2	no	—
313	C₂×D₈	D₄×C₇₁	1	2	yes	complemented
314	#131	—	2	2	no	—
315	#131	—	2	2	no	—
316	C₂×M₁₆	—	2	1	no	—
317	C₂×M₁₆	—	2	1	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	2	no	—
320	C₂×M₁₆	D₄×C₇₁	1	2	yes	complemented
321	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
322	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
323	C₂²×C₈	D₄×C₇₁	1	2	yes	complemented
324	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
326	#123	—	2	4	no	—
327	#123	D₄×C₇₁	1	2	yes	—
328	C₂²⋊Q₈	—	2	1	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	#121	—	2	1	no	—
331	#121	—	2	4	no	—
332	#120	—	2	4	no	—
333	#120	D₄×C₇₁	1	2	yes	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	4	4	no	—
336	#117	C₂³×C₇₁	1	4	yes	—
337	#117	C₂³×C₇₁	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	#106	—	2	2	no	—
340	C₄⋊C₈	—	2	2	no	—
341	#104	—	4	4	no	—
342	#103	—	2	4	no	—
343	#102	—	2	4	no	—
344	#99	—	2	4	no	—
345	#99	—	2	8	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	#97	—	2	1	no	—
349	C₄×C₈	—	2	1	no	—
350	C₂⁴	C₂×D₄×C₇₁	1	1	yes	characteristic
351	#41	—	2	2	no	—
352	#41	—	2	2	no	—
353	#41	—	2	8	no	—
354	#41	—	2	4	no	—
355	#41	—	2	4	no	—
356	#41	—	2	4	no	—
357	#41	—	2	4	no	—
358	#41	—	4	8	no	—
359	#41	—	4	4	no	—
360	#41	—	4	4	no	—
361	#41	—	4	4	no	—
362	#41	—	4	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	C₂×D₄×C₇₁	1	4	yes	—
366	C₂×Q₈	#41×C₇₁	1	1	yes	characteristic
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	8	no	—
372	C₂×D₄	—	4	4	no	—
373	C₂×D₄	—	4	4	no	—
374	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
375	C₂×D₄	C₂×D₄×C₇₁	1	4	yes	—
376	C₂×D₄	#41×C₇₁	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
386	Q₁₆	—	2	4	no	—
387	Q₁₆	—	2	2	no	—
388	Q₁₆	—	2	2	no	—
389	SD₁₆	—	2	8	no	—
390	SD₁₆	—	2	4	no	—
391	SD₁₆	—	2	4	no	—
392	D₈	—	2	4	no	—
393	D₈	—	2	2	no	—
394	D₈	—	2	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	2	4	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	4	1	no	—
399	M₁₆	—	4	1	no	—
400	M₁₆	—	4	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
407	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
408	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
409	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
410	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
421	C₄²	—	2	1	no	—
422	C₄²	—	2	4	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	8	1	no	—
428	C₂³	#121×C₇₁	1	2	yes	—
429	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
430	Q₈	—	2	2	no	—
431	Q₈	—	2	4	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	4	4	no	—
434	D₄	—	2	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	8	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	4	no	—
442	D₄	—	8	2	no	—
443	D₄	—	8	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	8	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	#121×C₇₁	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
461	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
462	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	1	no	—
466	C₈	#120×C₇₁	1	1	yes	characteristic
467	C₈	#120×C₇₁	1	1	yes	characteristic
468	C₈	#120×C₇₁	1	2	yes	—
469	C₂²	—	2	2	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	8	4	no	—
478	C₂²	—	16	4	no	—
479	C₂²	C₂×#121×C₇₁	1	1	yes	characteristic
480	C₂²	D₄²×C₇₁	1	2	yes	—
481	C₄	—	2	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	4	no	—
485	C₄	—	8	4	no	—
486	C₄	C₂×#120×C₇₁	1	1	yes	characteristic
487	C₄	#535×C₇₁	1	1	yes	characteristic
488	C₄	#546×C₇₁	1	2	yes	—
489	C₂	—	2	1	no	—
490	C₂	—	2	2	no	—
491	C₂	—	4	2	no	—
492	C₂	—	8	1	no	—
493	C₂	—	8	4	no	—
494	C₂	C₇₁×#2408	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₁	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂×C₇₁	#2408
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3584	C₂	semidirect
2. C₇₁×#3054	C₂	semidirect
3. C₇₁×#3025	C₂	semidirect
4. C₇₁×#2930	C₂	semidirect
5. C₇₁×#2053	C₂	semidirect
6. C₇₁×#2010	C₂	semidirect
7. C₇₁×#1976	C₂	semidirect
8. C₇₁×#1812	C₂	semidirect
9. C₇₁×#1794	C₂	semidirect
19. C₂²×M₁₆×C₇₁	C₂²	semidirect
21. #525×C₇₁	C₂²	semidirect
29. #427×C₇₁	C₂²	semidirect
30. #424×C₇₁	C₂²	semidirect
31. #421×C₇₁	C₂²	semidirect
32. #416×C₇₁	C₂²	semidirect
34. #410×C₇₁	C₂²	semidirect
36. #407×C₇₁	C₂²	semidirect
37. #405×C₇₁	C₂²	semidirect
38. H(3,ℤ/4)×C₇₁	C₂²	semidirect
61. C₂×Q₁₆×C₇₁	D₄	semidirect
63. C₂×SD₁₆×C₇₁	D₄	semidirect
65. C₂×D₈×C₇₁	D₄	semidirect
72. C₂×M₁₆×C₇₁	D₄	semidirect
75. C₂²×C₈×C₇₁	D₄	semidirect
241. #24286	C₇₁	direct (N × H)
248. #3584	C₂×C₇₁	semidirect
249. #3054	C₂×C₇₁	semidirect
250. #3025	C₂×C₇₁	semidirect
251. #2930	C₂×C₇₁	semidirect
252. #2053	C₂×C₇₁	semidirect
253. #2010	C₂×C₇₁	semidirect
254. #1976	C₂×C₇₁	semidirect
255. #1812	C₂×C₇₁	semidirect
256. #1794	C₂×C₇₁	semidirect
257. C₇₁	#24286	direct (N × H)
267. C₂²×M₁₆	C₂²×C₇₁	semidirect
269. #525	C₂²×C₇₁	semidirect
277. #427	C₂²×C₇₁	semidirect
278. #424	C₂²×C₇₁	semidirect
279. #421	C₂²×C₇₁	semidirect
280. #416	C₂²×C₇₁	semidirect
282. #410	C₂²×C₇₁	semidirect
284. #407	C₂²×C₇₁	semidirect
285. #405	C₂²×C₇₁	semidirect
286. H(3,ℤ/4)	C₂²×C₇₁	semidirect
309. C₂×Q₁₆	D₄×C₇₁	semidirect
311. C₂×SD₁₆	D₄×C₇₁	semidirect
313. C₂×D₈	D₄×C₇₁	semidirect
320. C₂×M₁₆	D₄×C₇₁	semidirect
323. C₂²×C₈	D₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3584	2
C₇₁×#3054	2
C₇₁×#3025	2
C₇₁×#2930	2
C₇₁×#2053	2
C₇₁×#2010	2
C₇₁×#1976	2
C₇₁×#1812	2
C₇₁×#1794	2
#24286	71

Subgroups — C71×#24286 #31845158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#24286 #31845158