Subgroups — C₇₁×#19908 #31840780

Order 18176.

692 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#3332	C₂	1	2	yes	maximal, complemented
3	C₇₁×#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1701	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#1642	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1625	C₂	1	2	yes	maximal, complemented
10	C₇₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₇₁	—	2	1	no	—
13	#550×C₇₁	C₂²	1	2	yes	—
14	#547×C₇₁	C₂²	1	2	yes	—
15	#494×C₇₁	C₂²	1	1	yes	characteristic
16	#493×C₇₁	—	2	1	no	—
17	#493×C₇₁	C₂²	1	1	yes	characteristic
18	#486×C₇₁	C₂²	1	2	yes	—
19	#486×C₇₁	C₂²	1	2	yes	—
20	#481×C₇₁	—	2	2	no	—
21	#475×C₇₁	C₂²	1	2	yes	—
22	#469×C₇₁	—	2	2	no	—
23	#467×C₇₁	—	2	1	no	—
24	#461×C₇₁	C₂²	1	2	yes	—
25	#459×C₇₁	—	2	2	no	—
26	#459×C₇₁	C₂²	1	2	yes	—
27	#459×C₇₁	C₂²	1	2	yes	—
28	#452×C₇₁	—	2	2	no	—
29	#449×C₇₁	—	2	2	no	—
30	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₇₁	—	2	1	no	—
33	C₄×D₈×C₇₁	C₂²	1	1	yes	characteristic
34	#436×C₇₁	C₂²	1	2	yes	—
35	C₈×D₄×C₇₁	C₂²	1	1	yes	characteristic
36	#417×C₇₁	—	2	2	no	—
37	#335×C₇₁	C₂²	1	1	yes	characteristic
38	#334×C₇₁	C₂²	1	1	yes	characteristic
39	#332×C₇₁	C₂²	1	2	yes	—
40	#331×C₇₁	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₁	C₂²	1	2	yes	—
44	C₈²×C₇₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₇₁	—	2	2	no	—
46	C₂×#41×C₇₁	—	2	2	no	—
47	#135×C₇₁	—	4	4	no	—
48	C₂×Q₁₆×C₇₁	—	2	1	no	—
49	C₂×Q₁₆×C₇₁	—	2	1	no	—
50	C₂×SD₁₆×C₇₁	—	2	2	no	—
51	C₂×SD₁₆×C₇₁	D₄	1	1	yes	characteristic
52	C₂×D₈×C₇₁	—	2	2	no	—
53	C₂×D₈×C₇₁	—	2	2	no	—
54	C₂×D₈×C₇₁	—	2	1	no	—
55	C₂×D₈×C₇₁	—	2	1	no	—
56	C₂×D₈×C₇₁	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₇₁	—	2	2	no	—
58	C₂²×C₈×C₇₁	—	2	1	no	—
59	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
60	#127×C₇₁	—	2	2	no	—
61	#127×C₇₁	C₂³	1	2	yes	—
62	#123×C₇₁	—	4	2	no	—
63	C₂²⋊Q₈×C₇₁	—	2	2	no	—
64	#121×C₇₁	—	2	2	no	—
65	#121×C₇₁	—	2	2	no	—
66	#121×C₇₁	—	4	2	no	—
67	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₇₁	—	2	2	no	—
69	C₄×D₄×C₇₁	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₇₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
72	#107×C₇₁	D₄	1	1	yes	characteristic
73	#106×C₇₁	—	2	2	no	—
74	#106×C₇₁	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₇₁	—	2	1	no	—
76	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₁	C₂³	1	2	yes	—
78	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
79	#103×C₇₁	—	2	2	no	—
80	#103×C₇₁	—	2	2	no	—
81	#103×C₇₁	D₄	1	1	yes	characteristic
82	#102×C₇₁	—	2	2	no	—
83	#102×C₇₁	—	2	2	no	—
84	#102×C₇₁	—	2	2	no	—
85	#102×C₇₁	—	2	2	no	—
86	#102×C₇₁	—	4	2	no	—
87	#102×C₇₁	D₄	1	2	yes	—
88	#102×C₇₁	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₇₁	—	2	1	no	—
90	C₂²⋊C₈×C₇₁	—	2	2	no	—
91	C₄×C₈×C₇₁	—	2	1	no	—
92	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
95	#41×C₇₁	—	4	4	no	—
96	#41×C₇₁	—	4	4	no	—
97	#41×C₇₁	—	8	2	no	—
98	#41×C₇₁	—	8	2	no	—
99	C₂×Q₈×C₇₁	—	2	2	no	—
100	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄×C₇₁	—	2	2	no	—
102	C₂×D₄×C₇₁	—	2	2	no	—
103	C₂×D₄×C₇₁	—	2	2	no	—
104	C₂×D₄×C₇₁	—	4	2	no	—
105	C₂×D₄×C₇₁	—	4	2	no	—
106	C₂×D₄×C₇₁	—	4	2	no	—
107	C₂×D₄×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₇₁	—	2	2	no	—
110	C₂²×C₄×C₇₁	—	2	1	no	—
111	C₂²×C₄×C₇₁	—	4	2	no	—
112	C₂²×C₄×C₇₁	—	4	2	no	—
113	Q₁₆×C₇₁	—	4	2	no	—
114	Q₁₆×C₇₁	—	8	1	no	—
115	SD₁₆×C₇₁	—	2	2	no	—
116	SD₁₆×C₇₁	—	4	4	no	—
117	D₈×C₇₁	—	2	2	no	—
118	D₈×C₇₁	—	4	2	no	—
119	D₈×C₇₁	—	8	1	no	—
120	D₈×C₇₁	—	8	2	no	—
121	D₈×C₇₁	—	8	2	no	—
122	M₁₆×C₇₁	—	8	2	no	—
123	C₂×C₈×C₇₁	—	2	1	no	—
124	C₂×C₈×C₇₁	—	2	2	no	—
125	C₂×C₈×C₇₁	—	2	1	no	—
126	C₂×C₈×C₇₁	—	2	1	no	—
127	C₂×C₈×C₇₁	—	4	2	no	—
128	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₁	—	2	1	no	—
133	C₄⋊C₄×C₇₁	—	2	2	no	—
134	C₄⋊C₄×C₇₁	—	2	2	no	—
135	C₄⋊C₄×C₇₁	—	4	2	no	—
136	C₄⋊C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₇₁	—	2	2	no	—
140	C₂²⋊C₄×C₇₁	—	2	1	no	—
141	C₂²⋊C₄×C₇₁	—	4	2	no	—
142	C₂²⋊C₄×C₇₁	—	4	2	no	—
143	C₄²×C₇₁	—	2	1	no	—
144	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
145	C₂³×C₇₁	—	2	1	no	—
146	C₂³×C₇₁	—	2	2	no	—
147	C₂³×C₇₁	—	4	2	no	—
148	Q₈×C₇₁	—	2	1	no	—
149	Q₈×C₇₁	—	4	2	no	—
150	Q₈×C₇₁	—	4	2	no	—
151	Q₈×C₇₁	#135	1	2	yes	complemented
152	D₄×C₇₁	—	2	1	no	—
153	D₄×C₇₁	—	2	2	no	—
154	D₄×C₇₁	—	2	1	no	—
155	D₄×C₇₁	—	2	2	no	—
156	D₄×C₇₁	—	4	2	no	—
157	D₄×C₇₁	—	4	2	no	—
158	D₄×C₇₁	—	4	2	no	—
159	D₄×C₇₁	—	4	2	no	—
160	D₄×C₇₁	—	8	4	no	—
161	D₄×C₇₁	—	8	2	no	—
162	D₄×C₇₁	—	8	4	no	—
163	D₄×C₇₁	—	16	2	no	—
164	C₂×C₄×C₇₁	—	2	1	no	—
165	C₂×C₄×C₇₁	—	2	1	no	—
166	C₂×C₄×C₇₁	—	2	2	no	—
167	C₂×C₄×C₇₁	—	2	1	no	—
168	C₂×C₄×C₇₁	—	4	2	no	—
169	C₂×C₄×C₇₁	—	4	1	no	—
170	C₂×C₄×C₇₁	—	4	2	no	—
171	C₂×C₄×C₇₁	—	4	1	no	—
172	C₂×C₄×C₇₁	—	4	2	no	—
173	C₂×C₄×C₇₁	—	8	4	no	—
174	C₂×C₄×C₇₁	—	8	4	no	—
175	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
178	C₈×C₇₁	—	2	1	no	—
179	C₈×C₇₁	—	2	1	no	—
180	C₈×C₇₁	—	2	1	no	—
181	C₈×C₇₁	—	4	1	no	—
182	C₈×C₇₁	—	4	1	no	—
183	C₈×C₇₁	—	4	2	no	—
184	C₈×C₇₁	—	4	1	no	—
185	C₈×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
186	C₂²×C₇₁	—	4	1	no	—
187	C₂²×C₇₁	—	4	2	no	—
188	C₂²×C₇₁	—	4	2	no	—
189	C₂²×C₇₁	—	4	1	no	—
190	C₂²×C₇₁	—	4	2	no	—
191	C₂²×C₇₁	—	4	1	no	—
192	C₂²×C₇₁	—	8	2	no	—
193	C₂²×C₇₁	—	8	2	no	—
194	C₂²×C₇₁	—	8	2	no	—
195	C₂²×C₇₁	D₄²	1	1	yes	characteristic
196	C₄×C₇₁	—	2	2	no	—
197	C₄×C₇₁	—	2	1	no	—
198	C₄×C₇₁	—	4	1	no	—
199	C₄×C₇₁	—	4	2	no	—
200	C₄×C₇₁	—	4	1	no	—
201	C₄×C₇₁	—	8	2	no	—
202	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
203	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₄×C₇₁	#576	1	1	yes	characteristic
205	C₄×C₇₁	#576	1	1	yes	characteristic
206	#19908	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₇₁	—	8	1	no	—
208	C₂×C₇₁	—	8	2	no	—
209	C₂×C₇₁	—	16	2	no	—
210	C₂×C₇₁	#3274	1	1	yes	characteristic
211	C₂×C₇₁	#3280	1	1	yes	characteristic
212	C₂×C₇₁	#3290	1	1	yes	characteristic
213	#3391	C₂×C₇₁	1	1	yes	characteristic, complemented
214	#3332	C₂×C₇₁	1	2	yes	complemented
215	#3283	C₂×C₇₁	1	1	yes	characteristic, complemented
216	#1709	C₂×C₇₁	1	1	yes	characteristic, complemented
217	#1701	C₂×C₇₁	1	2	yes	complemented
218	#1670	C₂×C₇₁	1	1	yes	characteristic, complemented
219	#1667	C₂×C₇₁	1	1	yes	characteristic, complemented
220	#1642	C₂×C₇₁	1	2	yes	complemented
221	#1625	C₂×C₇₁	1	2	yes	complemented
222	#1589	C₂×C₇₁	1	1	yes	characteristic, complemented
223	C₈×SD₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
224	C₇₁	#19908	1	1	yes	characteristic, direct product factor, Sylow
225	C₂×#135	—	2	1	no	—
226	#550	C₂²×C₇₁	1	2	yes	—
227	#547	C₂²×C₇₁	1	2	yes	—
228	#494	C₂²×C₇₁	1	1	yes	characteristic
229	#493	—	2	1	no	—
230	#493	C₂²×C₇₁	1	1	yes	characteristic
231	#486	C₂²×C₇₁	1	2	yes	—
232	#486	C₂²×C₇₁	1	2	yes	—
233	#481	—	2	2	no	—
234	#475	C₂²×C₇₁	1	2	yes	—
235	#469	—	2	2	no	—
236	#467	—	2	1	no	—
237	#461	C₂²×C₇₁	1	2	yes	—
238	#459	—	2	2	no	—
239	#459	C₂²×C₇₁	1	2	yes	—
240	#459	C₂²×C₇₁	1	2	yes	—
241	#452	—	2	2	no	—
242	#449	—	2	2	no	—
243	C₈×Q₈	C₂²×C₇₁	1	1	yes	characteristic
244	C₄×SD₁₆	C₂²×C₇₁	1	1	yes	characteristic
245	C₄×D₈	—	2	1	no	—
246	C₄×D₈	C₂²×C₇₁	1	1	yes	characteristic
247	#436	C₂²×C₇₁	1	2	yes	—
248	C₈×D₄	C₂²×C₇₁	1	1	yes	characteristic
249	#417	—	2	2	no	—
250	#335	C₂²×C₇₁	1	1	yes	characteristic
251	#334	C₂²×C₇₁	1	1	yes	characteristic
252	#332	C₂²×C₇₁	1	2	yes	—
253	#331	C₂²×C₇₁	1	2	yes	—
254	Q₈⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
255	D₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
256	D₄⋊C₈	C₂²×C₇₁	1	2	yes	—
257	C₈²	C₂²×C₇₁	1	1	yes	characteristic
258	C₂×#41	—	2	2	no	—
259	C₂×#41	—	2	2	no	—
260	#135	—	4	4	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	2	1	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×SD₁₆	D₄×C₇₁	1	1	yes	characteristic
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	—	2	2	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	—	2	1	no	—
269	C₂×D₈	D₄×C₇₁	1	1	yes	characteristic
270	C₂×M₁₆	—	2	2	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
273	#127	—	2	2	no	—
274	#127	C₂³×C₇₁	1	2	yes	—
275	#123	—	4	2	no	—
276	C₂²⋊Q₈	—	2	2	no	—
277	#121	—	2	2	no	—
278	#121	—	2	2	no	—
279	#121	—	4	2	no	—
280	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	C₂³×C₇₁	1	1	yes	characteristic
283	C₄×D₄	C₂³×C₇₁	1	2	yes	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	#107	D₄×C₇₁	1	1	yes	characteristic
286	#106	—	2	2	no	—
287	#106	D₄×C₇₁	1	1	yes	characteristic
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	C₂³×C₇₁	1	2	yes	—
290	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
291	C₄⋊C₈	C₂³×C₇₁	1	1	yes	characteristic
292	#103	—	2	2	no	—
293	#103	—	2	2	no	—
294	#103	D₄×C₇₁	1	1	yes	characteristic
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	4	2	no	—
300	#102	D₄×C₇₁	1	2	yes	—
301	#102	D₄×C₇₁	1	1	yes	characteristic
302	C₂²⋊C₈	—	2	1	no	—
303	C₂²⋊C₈	—	2	2	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
306	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
308	#41	—	4	4	no	—
309	#41	—	4	4	no	—
310	#41	—	8	2	no	—
311	#41	—	8	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	C₂×D₄×C₇₁	1	1	yes	characteristic
321	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	—	4	2	no	—
326	Q₁₆	—	4	2	no	—
327	Q₁₆	—	8	1	no	—
328	SD₁₆	—	2	2	no	—
329	SD₁₆	—	4	4	no	—
330	D₈	—	2	2	no	—
331	D₈	—	4	2	no	—
332	D₈	—	8	1	no	—
333	D₈	—	8	2	no	—
334	D₈	—	8	2	no	—
335	M₁₆	—	8	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	4	2	no	—
341	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
350	C₄⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
351	C₄⋊C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	2	1	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₄²	—	2	1	no	—
357	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
358	C₂³	—	2	1	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	4	2	no	—
361	Q₈	—	2	1	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	#135×C₇₁	1	2	yes	complemented
365	D₄	—	2	1	no	—
366	D₄	—	2	2	no	—
367	D₄	—	2	1	no	—
368	D₄	—	2	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	4	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	4	no	—
376	D₄	—	16	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	8	4	no	—
387	C₂×C₄	—	8	4	no	—
388	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
390	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	4	1	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	1	no	—
398	C₈	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	D₄²×C₇₁	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	2	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	1	no	—
414	C₄	—	8	2	no	—
415	C₄	C₂×#135×C₇₁	1	1	yes	characteristic
416	C₄	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
417	C₄	#576×C₇₁	1	1	yes	characteristic
418	C₄	#576×C₇₁	1	1	yes	characteristic
419	C₂	—	8	1	no	—
420	C₂	—	8	2	no	—
421	C₂	—	16	2	no	—
422	C₂	C₇₁×#3274	1	1	yes	characteristic
423	C₂	C₇₁×#3280	1	1	yes	characteristic
424	C₂	C₇₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3391	C₂	semidirect
2. C₇₁×#3332	C₂	semidirect
3. C₇₁×#3283	C₂	semidirect
4. C₇₁×#1709	C₂	semidirect
5. C₇₁×#1701	C₂	semidirect
6. C₇₁×#1670	C₂	semidirect
7. C₇₁×#1667	C₂	semidirect
8. C₇₁×#1642	C₂	semidirect
9. C₇₁×#1625	C₂	semidirect
10. C₇₁×#1589	C₂	semidirect
11. C₈×SD₁₆×C₇₁	C₂	semidirect
151. Q₈×C₇₁	#135	semidirect
206. #19908	C₇₁	direct (N × H)
213. #3391	C₂×C₇₁	semidirect
214. #3332	C₂×C₇₁	semidirect
215. #3283	C₂×C₇₁	semidirect
216. #1709	C₂×C₇₁	semidirect
217. #1701	C₂×C₇₁	semidirect
218. #1670	C₂×C₇₁	semidirect
219. #1667	C₂×C₇₁	semidirect
220. #1642	C₂×C₇₁	semidirect
221. #1625	C₂×C₇₁	semidirect
222. #1589	C₂×C₇₁	semidirect
223. C₈×SD₁₆	C₂×C₇₁	semidirect
224. C₇₁	#19908	direct (N × H)
364. Q₈	#135×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3391	2
C₇₁×#3332	2
C₇₁×#3283	2
C₇₁×#1709	2
C₇₁×#1701	2
C₇₁×#1670	2
C₇₁×#1667	2
C₇₁×#1642	2
C₇₁×#1625	2
C₇₁×#1589	2
C₈×SD₁₆×C₇₁	2
#19908	71

Subgroups — C71×#19908 #31840780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#19908 #31840780