Subgroups — C₇₁×#23185 #31844057

Order 18176.

1238 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₇₁	C₂	1	2	yes	maximal, complemented
3	C₇₁×#2366	C₂	1	2	yes	maximal, complemented
4	C₇₁×#1982	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1970	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1879	C₂	1	4	yes	maximal, complemented
7	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₇₁	C₂²	1	2	yes	—
10	C₂²×D₈×C₇₁	C₂²	1	2	yes	—
11	C₂×#127×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₇₁	—	2	2	no	—
13	C₂×C₄×D₄×C₇₁	C₂²	1	4	yes	—
14	#493×C₇₁	C₄	1	8	yes	complemented
15	C₄×D₈×C₇₁	—	2	8	no	—
16	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#102×C₇₁	C₂²	1	4	yes	—
20	C₂×#102×C₇₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#390×C₇₁	—	2	2	no	—
24	#390×C₇₁	C₂²	1	4	yes	—
25	#386×C₇₁	—	2	4	no	—
26	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
28	#340×C₇₁	C₂²	1	2	yes	—
29	#336×C₇₁	C₂²	1	2	yes	—
30	C₂²×D₄×C₇₁	—	2	2	no	—
31	C₂²×D₄×C₇₁	—	2	2	no	—
32	C₂²×D₄×C₇₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₇₁	—	2	4	no	—
34	C₂×D₈×C₇₁	—	2	4	no	—
35	C₂×D₈×C₇₁	—	2	4	no	—
36	C₂×D₈×C₇₁	—	4	2	no	—
37	C₂×D₈×C₇₁	—	4	2	no	—
38	C₂×D₈×C₇₁	C₂×C₄	1	8	yes	complemented
39	C₂×D₈×C₇₁	C₂×C₄	1	8	yes	complemented
40	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	#127×C₇₁	—	2	4	no	—
44	#127×C₇₁	C₂×C₄	1	8	yes	complemented
45	C₄×D₄×C₇₁	—	2	8	no	—
46	C₄×D₄×C₇₁	—	4	4	no	—
47	C₄×D₄×C₇₁	—	4	8	no	—
48	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
54	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
55	C₂×C₄²×C₇₁	—	2	2	no	—
56	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
57	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
58	#107×C₇₁	—	2	2	no	—
59	#107×C₇₁	D₄	1	4	yes	complemented
60	C₄⋊C₈×C₇₁	—	2	2	no	—
61	#102×C₇₁	—	2	8	no	—
62	#102×C₇₁	—	4	4	no	—
63	C₄×C₈×C₇₁	—	2	4	no	—
64	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	complemented
65	#95×C₇₁	—	2	2	no	—
66	#95×C₇₁	D₄	1	2	yes	—
67	C₂⁴×C₇₁	—	2	4	no	—
68	C₂×D₄×C₇₁	—	2	8	no	—
69	C₂×D₄×C₇₁	—	2	8	no	—
70	C₂×D₄×C₇₁	—	2	4	no	—
71	C₂×D₄×C₇₁	—	2	4	no	—
72	C₂×D₄×C₇₁	—	2	4	no	—
73	C₂×D₄×C₇₁	—	2	4	no	—
74	C₂×D₄×C₇₁	—	2	4	no	—
75	C₂×D₄×C₇₁	—	4	4	no	—
76	C₂×D₄×C₇₁	—	4	4	no	—
77	C₂×D₄×C₇₁	—	8	2	no	—
78	C₂×D₄×C₇₁	—	8	2	no	—
79	C₂×D₄×C₇₁	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₇₁	—	2	2	no	—
81	C₂²×C₄×C₇₁	—	2	2	no	—
82	C₂²×C₄×C₇₁	—	4	8	no	—
83	C₂²×C₄×C₇₁	—	4	8	no	—
84	C₂²×C₄×C₇₁	—	8	4	no	—
85	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
90	D₈×C₇₁	—	2	16	no	—
91	D₈×C₇₁	—	4	8	no	—
92	C₂×C₈×C₇₁	—	2	1	no	—
93	C₂×C₈×C₇₁	—	2	1	no	—
94	C₂×C₈×C₇₁	—	2	2	no	—
95	C₂×C₈×C₇₁	—	2	2	no	—
96	C₂×C₈×C₇₁	—	2	2	no	—
97	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₇₁	#41	1	2	yes	—
101	C₄⋊C₄×C₇₁	—	2	2	no	—
102	C₄⋊C₄×C₇₁	—	2	2	no	—
103	C₄⋊C₄×C₇₁	—	2	2	no	—
104	C₄⋊C₄×C₇₁	—	2	2	no	—
105	C₄⋊C₄×C₇₁	D₈	1	4	yes	complemented
106	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₇₁	—	4	8	no	—
108	C₂²⋊C₄×C₇₁	—	4	8	no	—
109	C₂²⋊C₄×C₇₁	—	8	4	no	—
110	C₄²×C₇₁	—	2	2	no	—
111	C₄²×C₇₁	—	2	2	no	—
112	C₄²×C₇₁	—	4	4	no	—
113	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
114	C₄²×C₇₁	#41	1	2	yes	—
115	C₂³×C₇₁	—	2	8	no	—
116	C₂³×C₇₁	—	4	4	no	—
117	C₂³×C₇₁	—	4	4	no	—
118	C₂³×C₇₁	—	4	4	no	—
119	C₂³×C₇₁	—	4	4	no	—
120	C₂³×C₇₁	—	4	4	no	—
121	C₂³×C₇₁	—	4	4	no	—
122	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
123	D₄×C₇₁	—	2	8	no	—
124	D₄×C₇₁	—	2	8	no	—
125	D₄×C₇₁	—	4	4	no	—
126	D₄×C₇₁	—	4	4	no	—
127	D₄×C₇₁	—	4	16	no	—
128	D₄×C₇₁	—	8	8	no	—
129	C₂×C₄×C₇₁	—	2	2	no	—
130	C₂×C₄×C₇₁	—	2	2	no	—
131	C₂×C₄×C₇₁	—	2	1	no	—
132	C₂×C₄×C₇₁	—	2	1	no	—
133	C₂×C₄×C₇₁	—	2	2	no	—
134	C₂×C₄×C₇₁	—	2	4	no	—
135	C₂×C₄×C₇₁	—	2	2	no	—
136	C₂×C₄×C₇₁	—	2	2	no	—
137	C₂×C₄×C₇₁	—	4	2	no	—
138	C₂×C₄×C₇₁	—	4	2	no	—
139	C₂×C₄×C₇₁	—	4	2	no	—
140	C₂×C₄×C₇₁	—	4	2	no	—
141	C₂×C₄×C₇₁	—	4	2	no	—
142	C₂×C₄×C₇₁	—	8	16	no	—
143	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
146	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₇₁	#135	1	2	yes	—
148	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
154	C₈×C₇₁	—	2	2	no	—
155	C₈×C₇₁	—	4	2	no	—
156	C₈×C₇₁	C₄×D₄	1	4	yes	complemented
157	C₂²×C₇₁	—	4	8	no	—
158	C₂²×C₇₁	—	4	8	no	—
159	C₂²×C₇₁	—	4	8	no	—
160	C₂²×C₇₁	—	8	4	no	—
161	C₂²×C₇₁	—	8	4	no	—
162	C₂²×C₇₁	—	8	4	no	—
163	C₂²×C₇₁	—	8	4	no	—
164	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₇₁	D₄²	1	1	yes	characteristic
166	C₂²×C₇₁	#550	1	1	yes	characteristic
167	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
168	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
169	C₂²×C₇₁	#576	1	1	yes	characteristic
170	C₂²×C₇₁	#576	1	1	yes	characteristic
171	C₄×C₇₁	—	2	2	no	—
172	C₄×C₇₁	—	2	4	no	—
173	C₄×C₇₁	—	4	4	no	—
174	C₄×C₇₁	—	4	2	no	—
175	C₄×C₇₁	—	8	2	no	—
176	C₄×C₇₁	C₄×D₈	1	4	yes	complemented
177	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₇₁	#518	1	1	yes	characteristic
179	C₄×C₇₁	#550	1	1	yes	characteristic
180	C₄×C₇₁	#559	1	1	yes	characteristic
181	#23185	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂×C₇₁	—	8	8	no	—
183	C₂×C₇₁	#2366	1	1	yes	characteristic
184	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
185	C₂×C₇₁	#2948	1	1	yes	characteristic
186	C₂×C₇₁	D₄×D₈	1	1	yes	characteristic
187	C₂×C₇₁	#3287	1	1	yes	characteristic
188	C₂×C₇₁	#3391	1	1	yes	characteristic
189	C₂×C₇₁	#3396	1	1	yes	characteristic
190	C₂×#493	C₂×C₇₁	1	1	yes	characteristic
191	C₂×C₄×D₈	C₂×C₇₁	1	2	yes	complemented
192	#2366	C₂×C₇₁	1	2	yes	complemented
193	#1982	C₂×C₇₁	1	2	yes	complemented
194	#1970	C₂×C₇₁	1	2	yes	complemented
195	#1879	C₂×C₇₁	1	4	yes	complemented
196	C₄×#107	C₂×C₇₁	1	1	yes	characteristic, complemented
197	C₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic, complemented
198	C₇₁	#23185	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×D₈	C₂²×C₇₁	1	2	yes	—
200	C₂²×D₈	C₂²×C₇₁	1	2	yes	—
201	C₂×#127	C₂²×C₇₁	1	2	yes	—
202	C₂×C₄×D₄	—	2	2	no	—
203	C₂×C₄×D₄	C₂²×C₇₁	1	4	yes	—
204	#493	C₄×C₇₁	1	8	yes	complemented
205	C₄×D₈	—	2	8	no	—
206	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
207	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
208	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
209	C₂×#102	C₂²×C₇₁	1	4	yes	—
210	C₂×#102	C₂²×C₇₁	1	4	yes	—
211	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
212	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
213	#390	—	2	2	no	—
214	#390	C₂²×C₇₁	1	4	yes	—
215	#386	—	2	4	no	—
216	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
217	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
218	#340	C₂²×C₇₁	1	2	yes	—
219	#336	C₂²×C₇₁	1	2	yes	—
220	C₂²×D₄	—	2	2	no	—
221	C₂²×D₄	—	2	2	no	—
222	C₂²×D₄	C₂³×C₇₁	1	4	yes	—
223	C₂³×C₄	—	2	4	no	—
224	C₂×D₈	—	2	4	no	—
225	C₂×D₈	—	2	4	no	—
226	C₂×D₈	—	4	2	no	—
227	C₂×D₈	—	4	2	no	—
228	C₂×D₈	C₂×C₄×C₇₁	1	8	yes	complemented
229	C₂×D₈	C₂×C₄×C₇₁	1	8	yes	complemented
230	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
231	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
233	#127	—	2	4	no	—
234	#127	C₂×C₄×C₇₁	1	8	yes	complemented
235	C₄×D₄	—	2	8	no	—
236	C₄×D₄	—	4	4	no	—
237	C₄×D₄	—	4	8	no	—
238	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
239	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
240	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
241	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
242	C₂×C₂²⋊C₄	—	2	4	no	—
243	C₂×C₂²⋊C₄	—	2	4	no	—
244	C₂×C₂²⋊C₄	—	2	4	no	—
245	C₂×C₄²	—	2	2	no	—
246	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
247	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
248	#107	—	2	2	no	—
249	#107	D₄×C₇₁	1	4	yes	complemented
250	C₄⋊C₈	—	2	2	no	—
251	#102	—	2	8	no	—
252	#102	—	4	4	no	—
253	C₄×C₈	—	2	4	no	—
254	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	complemented
255	#95	—	2	2	no	—
256	#95	D₄×C₇₁	1	2	yes	—
257	C₂⁴	—	2	4	no	—
258	C₂×D₄	—	2	8	no	—
259	C₂×D₄	—	2	8	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	4	no	—
267	C₂×D₄	—	8	2	no	—
268	C₂×D₄	—	8	2	no	—
269	C₂×D₄	C₂²×C₄×C₇₁	1	8	yes	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	4	8	no	—
273	C₂²×C₄	—	4	8	no	—
274	C₂²×C₄	—	8	4	no	—
275	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
279	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
280	D₈	—	2	16	no	—
281	D₈	—	4	8	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
288	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
290	C₂×C₈	#41×C₇₁	1	2	yes	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	D₈×C₇₁	1	4	yes	complemented
296	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
297	C₂²⋊C₄	—	4	8	no	—
298	C₂²⋊C₄	—	4	8	no	—
299	C₂²⋊C₄	—	8	4	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	4	no	—
303	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
304	C₄²	#41×C₇₁	1	2	yes	—
305	C₂³	—	2	8	no	—
306	C₂³	—	4	4	no	—
307	C₂³	—	4	4	no	—
308	C₂³	—	4	4	no	—
309	C₂³	—	4	4	no	—
310	C₂³	—	4	4	no	—
311	C₂³	—	4	4	no	—
312	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
313	D₄	—	2	8	no	—
314	D₄	—	2	8	no	—
315	D₄	—	4	16	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	8	8	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	16	no	—
333	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
334	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
335	C₂×C₄	C₂×D₈×C₇₁	1	4	yes	—
336	C₂×C₄	C₂×D₈×C₇₁	1	2	yes	—
337	C₂×C₄	#135×C₇₁	1	2	yes	—
338	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
340	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
342	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
343	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
344	C₈	—	2	2	no	—
345	C₈	—	4	2	no	—
346	C₈	C₄×D₄×C₇₁	1	4	yes	complemented
347	C₂²	—	4	8	no	—
348	C₂²	—	4	8	no	—
349	C₂²	—	4	8	no	—
350	C₂²	—	8	4	no	—
351	C₂²	—	8	4	no	—
352	C₂²	—	8	4	no	—
353	C₂²	—	8	4	no	—
354	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
355	C₂²	D₄²×C₇₁	1	1	yes	characteristic
356	C₂²	#550×C₇₁	1	1	yes	characteristic
357	C₂²	C₂²×D₈×C₇₁	1	1	yes	characteristic
358	C₂²	C₂×#135×C₇₁	1	1	yes	characteristic
359	C₂²	#576×C₇₁	1	1	yes	characteristic
360	C₂²	#576×C₇₁	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	2	4	no	—
363	C₄	—	4	4	no	—
364	C₄	—	4	2	no	—
365	C₄	—	8	2	no	—
366	C₄	C₄×D₈×C₇₁	1	4	yes	complemented
367	C₄	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
368	C₄	#518×C₇₁	1	1	yes	characteristic
369	C₄	#550×C₇₁	1	1	yes	characteristic
370	C₄	#559×C₇₁	1	1	yes	characteristic
371	C₂	—	8	8	no	—
372	C₂	C₇₁×#2366	1	1	yes	characteristic
373	C₂	C₂×C₄×D₈×C₇₁	1	1	yes	characteristic
374	C₂	C₇₁×#2948	1	1	yes	characteristic
375	C₂	D₄×D₈×C₇₁	1	1	yes	characteristic
376	C₂	C₇₁×#3287	1	1	yes	characteristic
377	C₂	C₇₁×#3391	1	1	yes	characteristic
378	C₂	C₇₁×#3396	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×#493×C₇₁	C₂
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×C₄×D₈×C₇₁	C₂	semidirect
3. C₇₁×#2366	C₂	semidirect
4. C₇₁×#1982	C₂	semidirect
5. C₇₁×#1970	C₂	semidirect
6. C₇₁×#1879	C₂	semidirect
7. C₄×#107×C₇₁	C₂	semidirect
8. C₈×C₄⋊C₄×C₇₁	C₂	semidirect
14. #493×C₇₁	C₄	semidirect
38. C₂×D₈×C₇₁	C₂×C₄	semidirect
39. C₂×D₈×C₇₁	C₂×C₄	semidirect
44. #127×C₇₁	C₂×C₄	semidirect
59. #107×C₇₁	D₄	semidirect
64. C₄×C₈×C₇₁	C₂×C₄	semidirect
105. C₄⋊C₄×C₇₁	D₈	semidirect
156. C₈×C₇₁	C₄×D₄	semidirect
176. C₄×C₇₁	C₄×D₈	semidirect
181. #23185	C₇₁	direct (N × H)
191. C₂×C₄×D₈	C₂×C₇₁	semidirect
192. #2366	C₂×C₇₁	semidirect
193. #1982	C₂×C₇₁	semidirect
194. #1970	C₂×C₇₁	semidirect
195. #1879	C₂×C₇₁	semidirect
196. C₄×#107	C₂×C₇₁	semidirect
197. C₈×C₄⋊C₄	C₂×C₇₁	semidirect
198. C₇₁	#23185	direct (N × H)
204. #493	C₄×C₇₁	semidirect
228. C₂×D₈	C₂×C₄×C₇₁	semidirect
229. C₂×D₈	C₂×C₄×C₇₁	semidirect
234. #127	C₂×C₄×C₇₁	semidirect
249. #107	D₄×C₇₁	semidirect
254. C₄×C₈	C₂×C₄×C₇₁	semidirect
295. C₄⋊C₄	D₈×C₇₁	semidirect
346. C₈	C₄×D₄×C₇₁	semidirect
366. C₄	C₄×D₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₇₁	2
C₂×C₄×D₈×C₇₁	2
C₇₁×#2366	2
C₇₁×#1982	2
C₇₁×#1970	2
C₇₁×#1879	2
C₄×#107×C₇₁	2
C₈×C₄⋊C₄×C₇₁	2
#23185	71

Subgroups — C71×#23185 #31844057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23185 #31844057