Subgroups — C₇₁×#12937 #31833809

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#357×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#357×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₇₁	—	2	1	no	—
10	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	C₄×D₈×C₇₁	—	2	2	no	—
13	C₄×D₈×C₇₁	—	2	2	no	—
14	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#102×C₇₁	—	2	1	no	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	#389×C₇₁	—	2	1	no	—
20	#367×C₇₁	—	2	2	no	—
21	#357×C₇₁	—	2	2	no	—
22	#357×C₇₁	C₄	1	4	yes	complemented
23	#340×C₇₁	—	2	1	no	—
24	C₂²×D₄×C₇₁	—	2	1	no	—
25	C₂³×C₄×C₇₁	—	4	1	no	—
26	C₂×D₈×C₇₁	—	2	1	no	—
27	C₂×D₈×C₇₁	—	2	1	no	—
28	C₂×D₈×C₇₁	—	2	1	no	—
29	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×D₈×C₇₁	D₄	1	2	yes	—
31	C₂×D₈×C₇₁	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₄×D₄×C₇₁	—	2	4	no	—
35	C₄×D₄×C₇₁	—	4	2	no	—
36	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₁	—	2	1	no	—
43	C₂×C₁₆×C₇₁	—	2	1	no	—
44	C₂×C₁₆×C₇₁	—	2	1	no	—
45	C₂×C₁₆×C₇₁	—	2	1	no	—
46	C₂×C₁₆×C₇₁	—	2	1	no	—
47	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
48	#107×C₇₁	—	2	1	no	—
49	#107×C₇₁	—	2	1	no	—
50	#107×C₇₁	—	2	1	no	—
51	#107×C₇₁	—	2	1	no	—
52	#107×C₇₁	C₂×C₄	1	2	yes	complemented
53	#107×C₇₁	D₄	1	2	yes	—
54	#102×C₇₁	—	2	4	no	—
55	C₄×C₈×C₇₁	—	2	1	no	—
56	C₄×C₈×C₇₁	—	2	1	no	—
57	#95×C₇₁	—	4	1	no	—
58	C₂⁴×C₇₁	—	4	1	no	—
59	C₂×D₄×C₇₁	—	2	2	no	—
60	C₂×D₄×C₇₁	—	2	2	no	—
61	C₂×D₄×C₇₁	—	2	2	no	—
62	C₂×D₄×C₇₁	—	4	1	no	—
63	C₂×D₄×C₇₁	—	4	1	no	—
64	C₂×D₄×C₇₁	—	4	1	no	—
65	C₂²×C₄×C₇₁	—	4	4	no	—
66	C₂²×C₄×C₇₁	—	4	1	no	—
67	C₂²×C₄×C₇₁	—	4	1	no	—
68	C₂²×C₄×C₇₁	—	4	1	no	—
69	C₂²×C₄×C₇₁	—	8	2	no	—
70	C₂²×C₄×C₇₁	—	8	2	no	—
71	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
74	D₈×C₇₁	—	2	2	no	—
75	D₈×C₇₁	—	2	2	no	—
76	D₈×C₇₁	—	2	2	no	—
77	D₈×C₇₁	C₄⋊C₄	1	4	yes	complemented
78	C₂×C₈×C₇₁	—	2	1	no	—
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	1	no	—
81	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
87	C₄⋊C₄×C₇₁	—	2	2	no	—
88	C₄⋊C₄×C₇₁	—	2	2	no	—
89	C₄⋊C₄×C₇₁	—	4	1	no	—
90	C₄⋊C₄×C₇₁	—	4	1	no	—
91	C₄⋊C₄×C₇₁	—	4	1	no	—
92	C₄⋊C₄×C₇₁	—	4	1	no	—
93	C₄⋊C₄×C₇₁	—	8	2	no	—
94	C₂²⋊C₄×C₇₁	—	4	4	no	—
95	C₄²×C₇₁	—	2	1	no	—
96	C₄²×C₇₁	—	2	1	no	—
97	C₁₆×C₇₁	—	2	2	no	—
98	C₁₆×C₇₁	—	4	1	no	—
99	C₂³×C₇₁	—	4	2	no	—
100	C₂³×C₇₁	—	4	2	no	—
101	C₂³×C₇₁	—	4	2	no	—
102	C₂³×C₇₁	—	8	1	no	—
103	C₂³×C₇₁	—	8	1	no	—
104	C₂³×C₇₁	—	8	1	no	—
105	C₂³×C₇₁	—	8	1	no	—
106	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
107	D₄×C₇₁	—	2	4	no	—
108	D₄×C₇₁	—	4	2	no	—
109	D₄×C₇₁	—	4	2	no	—
110	D₄×C₇₁	—	4	2	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	8	8	no	—
119	C₂×C₄×C₇₁	—	8	1	no	—
120	C₂×C₄×C₇₁	—	8	1	no	—
121	C₂×C₄×C₇₁	—	8	1	no	—
122	C₂×C₄×C₇₁	—	8	1	no	—
123	C₂×C₄×C₇₁	—	8	1	no	—
124	C₂×C₄×C₇₁	—	8	1	no	—
125	C₂×C₄×C₇₁	—	8	1	no	—
126	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
127	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
131	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
133	C₈×C₇₁	—	4	1	no	—
134	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
136	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₇₁	#123	1	1	yes	characteristic
138	C₂²×C₇₁	—	4	4	no	—
139	C₂²×C₇₁	—	8	2	no	—
140	C₂²×C₇₁	—	8	2	no	—
141	C₂²×C₇₁	—	8	2	no	—
142	C₂²×C₇₁	—	8	2	no	—
143	C₂²×C₇₁	—	8	2	no	—
144	C₂²×C₇₁	—	8	2	no	—
145	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₇₁	#447	1	1	yes	characteristic
147	C₂²×C₇₁	#474	1	1	yes	characteristic
148	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
149	C₂²×C₇₁	#508	1	1	yes	characteristic
150	C₂²×C₇₁	#509	1	1	yes	characteristic
151	C₂²×C₇₁	#509	1	1	yes	characteristic
152	C₄×C₇₁	—	2	2	no	—
153	C₄×C₇₁	—	4	1	no	—
154	C₄×C₇₁	—	8	2	no	—
155	C₄×C₇₁	—	8	2	no	—
156	C₄×C₇₁	—	16	1	no	—
157	C₄×C₇₁	#385	1	1	yes	characteristic
158	C₄×C₇₁	#440	1	1	yes	characteristic
159	C₄×C₇₁	#452	1	1	yes	characteristic
160	C₄×C₇₁	#478	1	1	yes	characteristic
161	#12937	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₇₁	—	8	4	no	—
163	C₂×C₇₁	#1862	1	1	yes	characteristic
164	C₂×C₇₁	C₄×SD₃₂	1	1	yes	characteristic
165	C₂×C₇₁	#2177	1	1	yes	characteristic
166	C₂×C₇₁	#2186	1	1	yes	characteristic
167	C₂×C₇₁	#2187	1	1	yes	characteristic
168	C₂×C₇₁	#2226	1	1	yes	characteristic
169	C₂×C₇₁	#2228	1	1	yes	characteristic
170	C₂×C₄×D₈	C₂×C₇₁	1	1	yes	characteristic
171	C₂×#367	C₂×C₇₁	1	1	yes	characteristic, complemented
172	C₂×#357	C₂×C₇₁	1	1	yes	characteristic
173	C₂×#357	C₂×C₇₁	1	1	yes	characteristic
174	#1942	C₂×C₇₁	1	1	yes	characteristic, complemented
175	#1380	C₂×C₇₁	1	1	yes	characteristic, complemented
176	#1374	C₂×C₇₁	1	1	yes	characteristic, complemented
177	C₇₁	#12937	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×D₈	C₂²×C₇₁	1	1	yes	characteristic
179	C₂×C₄×D₄	—	2	1	no	—
180	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
181	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
182	C₄×D₈	—	2	2	no	—
183	C₄×D₈	—	2	2	no	—
184	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
185	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
186	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
187	C₂×#102	—	2	1	no	—
188	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
189	#389	—	2	1	no	—
190	#367	—	2	2	no	—
191	#357	—	2	2	no	—
192	#357	C₄×C₇₁	1	4	yes	complemented
193	#340	—	2	1	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂³×C₄	—	4	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	C₂×C₄×C₇₁	1	2	yes	—
200	C₂×D₈	D₄×C₇₁	1	2	yes	—
201	C₂×D₈	Q₈×C₇₁	1	2	yes	—
202	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
204	C₄×D₄	—	2	4	no	—
205	C₄×D₄	—	4	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₂²⋊C₄	—	4	1	no	—
211	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	C₂×C₄×C₇₁	1	2	yes	complemented
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#107	C₂×C₄×C₇₁	1	2	yes	complemented
223	#107	D₄×C₇₁	1	2	yes	—
224	#102	—	2	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	4	1	no	—
228	C₂⁴	—	4	1	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	8	2	no	—
240	C₂²×C₄	—	8	2	no	—
241	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
242	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	C₄⋊C₄×C₇₁	1	4	yes	complemented
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
254	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
255	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
256	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
277	D₄	—	2	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	8	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	SD₃₂×C₇₁	1	2	yes	—
297	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
298	C₂×C₄	#120×C₇₁	1	1	yes	characteristic
299	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
300	C₂×C₄	#135×C₇₁	1	1	yes	characteristic
301	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
302	C₂×C₄	#137×C₇₁	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
305	C₈	C₄×D₄×C₇₁	1	1	yes	characteristic
306	C₈	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
307	C₈	#123×C₇₁	1	1	yes	characteristic
308	C₂²	—	4	4	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₄×D₈×C₇₁	1	1	yes	characteristic
316	C₂²	#447×C₇₁	1	1	yes	characteristic
317	C₂²	#474×C₇₁	1	1	yes	characteristic
318	C₂²	C₂×SD₃₂×C₇₁	1	1	yes	characteristic
319	C₂²	#508×C₇₁	1	1	yes	characteristic
320	C₂²	#509×C₇₁	1	1	yes	characteristic
321	C₂²	#509×C₇₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	1	no	—
327	C₄	#385×C₇₁	1	1	yes	characteristic
328	C₄	#440×C₇₁	1	1	yes	characteristic
329	C₄	#452×C₇₁	1	1	yes	characteristic
330	C₄	#478×C₇₁	1	1	yes	characteristic
331	C₂	—	8	4	no	—
332	C₂	C₇₁×#1862	1	1	yes	characteristic
333	C₂	C₄×SD₃₂×C₇₁	1	1	yes	characteristic
334	C₂	C₇₁×#2177	1	1	yes	characteristic
335	C₂	C₇₁×#2186	1	1	yes	characteristic
336	C₂	C₇₁×#2187	1	1	yes	characteristic
337	C₂	C₇₁×#2226	1	1	yes	characteristic
338	C₂	C₇₁×#2228	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₂²×D₈×C₇₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#367×C₇₁	C₂	semidirect
5. C₇₁×#1942	C₂	semidirect
6. C₇₁×#1380	C₂	semidirect
7. C₇₁×#1374	C₂	semidirect
22. #357×C₇₁	C₄	semidirect
47. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
52. #107×C₇₁	C₂×C₄	semidirect
77. D₈×C₇₁	C₄⋊C₄	semidirect
161. #12937	C₇₁	direct (N × H)
171. C₂×#367	C₂×C₇₁	semidirect
174. #1942	C₂×C₇₁	semidirect
175. #1380	C₂×C₇₁	semidirect
176. #1374	C₂×C₇₁	semidirect
177. C₇₁	#12937	direct (N × H)
192. #357	C₄×C₇₁	semidirect
217. C₂×C₁₆	C₂×C₄×C₇₁	semidirect
222. #107	C₂×C₄×C₇₁	semidirect
247. D₈	C₄⋊C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₇₁	2
C₂×#367×C₇₁	2
C₂×#357×C₇₁	2
C₇₁×#1942	2
C₇₁×#1380	2
C₇₁×#1374	2
#12937	71

Subgroups — C71×#12937 #31833809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12937 #31833809