Subgroups — C₇₁×#9783 #31830655

Order 18176.

606 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1964	C₂	1	2	yes	maximal, complemented
3	C₇₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1519	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₇₁	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×#121×C₇₁	C₂²	1	2	yes	—
9	#466×C₇₁	—	2	1	no	—
10	#466×C₇₁	—	2	1	no	—
11	#466×C₇₁	—	2	1	no	—
12	#466×C₇₁	—	2	1	no	—
13	C₂×#107×C₇₁	—	2	1	no	—
14	C₂×#107×C₇₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#102×C₇₁	C₄	1	2	yes	complemented
16	C₂×#99×C₇₁	—	2	4	no	—
17	C₂×C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₁	—	2	1	no	—
19	#380×C₇₁	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
21	#340×C₇₁	—	2	1	no	—
22	#336×C₇₁	—	2	1	no	—
23	#327×C₇₁	—	2	4	no	—
24	C₂²×D₄×C₇₁	—	2	2	no	—
25	C₂²×D₄×C₇₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×D₈×C₇₁	—	2	1	no	—
28	C₂×D₈×C₇₁	—	2	1	no	—
29	C₂×D₈×C₇₁	—	4	2	no	—
30	C₂²×C₈×C₇₁	—	2	1	no	—
31	C₂²×C₈×C₇₁	—	4	2	no	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
38	#121×C₇₁	—	2	2	no	—
39	#121×C₇₁	—	2	2	no	—
40	#121×C₇₁	—	2	2	no	—
41	#121×C₇₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
47	C₂×C₄²×C₇₁	—	2	1	no	—
48	#107×C₇₁	—	2	1	no	—
49	#107×C₇₁	—	2	1	no	—
50	#107×C₇₁	—	4	2	no	—
51	#102×C₇₁	—	2	2	no	—
52	#102×C₇₁	—	2	2	no	—
53	#99×C₇₁	—	4	8	no	—
54	C₂²⋊C₈×C₇₁	—	2	1	no	—
55	C₂²⋊C₈×C₇₁	D₄	1	2	yes	—
56	C₄×C₈×C₇₁	—	4	2	no	—
57	#95×C₇₁	—	2	1	no	—
58	#95×C₇₁	—	2	2	no	—
59	C₂⁴×C₇₁	—	2	2	no	—
60	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₇₁	—	2	2	no	—
62	C₂×D₄×C₇₁	—	2	2	no	—
63	C₂×D₄×C₇₁	—	4	2	no	—
64	C₂×D₄×C₇₁	—	4	2	no	—
65	C₂×D₄×C₇₁	—	4	2	no	—
66	C₂×D₄×C₇₁	—	4	2	no	—
67	C₂×D₄×C₇₁	—	4	4	no	—
68	C₂×D₄×C₇₁	—	4	4	no	—
69	C₂²×C₄×C₇₁	—	2	2	no	—
70	C₂²×C₄×C₇₁	—	2	2	no	—
71	C₂²×C₄×C₇₁	—	4	1	no	—
72	C₂²×C₄×C₇₁	—	4	1	no	—
73	C₂²×C₄×C₇₁	—	4	2	no	—
74	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
79	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
80	D₈×C₇₁	—	4	2	no	—
81	D₈×C₇₁	—	4	2	no	—
82	C₂×C₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	—	2	2	no	—
84	C₂×C₈×C₇₁	—	2	2	no	—
85	C₂×C₈×C₇₁	—	2	2	no	—
86	C₂×C₈×C₇₁	—	4	1	no	—
87	C₂×C₈×C₇₁	—	4	1	no	—
88	C₂×C₈×C₇₁	—	4	1	no	—
89	C₂×C₈×C₇₁	—	4	1	no	—
90	C₂×C₈×C₇₁	—	4	1	no	—
91	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₇₁	—	2	2	no	—
96	C₄⋊C₄×C₇₁	—	2	2	no	—
97	C₄⋊C₄×C₇₁	—	4	2	no	—
98	C₄⋊C₄×C₇₁	—	4	2	no	—
99	C₂²⋊C₄×C₇₁	—	2	2	no	—
100	C₂²⋊C₄×C₇₁	—	4	2	no	—
101	C₂²⋊C₄×C₇₁	—	4	2	no	—
102	C₂²⋊C₄×C₇₁	—	4	4	no	—
103	C₂²⋊C₄×C₇₁	D₈	1	4	yes	complemented
104	C₄²×C₇₁	—	4	1	no	—
105	C₄²×C₇₁	—	4	1	no	—
106	C₂³×C₇₁	—	4	1	no	—
107	C₂³×C₇₁	—	4	1	no	—
108	C₂³×C₇₁	—	4	2	no	—
109	C₂³×C₇₁	—	4	4	no	—
110	C₂³×C₇₁	—	8	2	no	—
111	C₂³×C₇₁	—	8	2	no	—
112	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
113	C₂³×C₇₁	#121	1	1	yes	characteristic
114	C₂³×C₇₁	#127	1	1	yes	characteristic
115	C₂³×C₇₁	C₂×D₈	1	2	yes	—
116	C₂³×C₇₁	#135	1	2	yes	—
117	D₄×C₇₁	—	4	2	no	—
118	D₄×C₇₁	—	4	2	no	—
119	D₄×C₇₁	—	8	2	no	—
120	D₄×C₇₁	—	8	4	no	—
121	D₄×C₇₁	—	8	2	no	—
122	D₄×C₇₁	—	8	2	no	—
123	C₂×C₄×C₇₁	—	2	1	no	—
124	C₂×C₄×C₇₁	—	2	1	no	—
125	C₂×C₄×C₇₁	—	2	1	no	—
126	C₂×C₄×C₇₁	—	2	1	no	—
127	C₂×C₄×C₇₁	—	2	1	no	—
128	C₂×C₄×C₇₁	—	2	1	no	—
129	C₂×C₄×C₇₁	—	2	4	no	—
130	C₂×C₄×C₇₁	—	4	1	no	—
131	C₂×C₄×C₇₁	—	4	1	no	—
132	C₂×C₄×C₇₁	—	4	1	no	—
133	C₂×C₄×C₇₁	—	4	2	no	—
134	C₂×C₄×C₇₁	—	4	2	no	—
135	C₂×C₄×C₇₁	—	4	2	no	—
136	C₂×C₄×C₇₁	—	4	4	no	—
137	C₂×C₄×C₇₁	—	8	2	no	—
138	C₂×C₄×C₇₁	—	8	2	no	—
139	C₂×C₄×C₇₁	—	8	2	no	—
140	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
144	C₈×C₇₁	—	2	1	no	—
145	C₈×C₇₁	—	2	1	no	—
146	C₈×C₇₁	—	2	1	no	—
147	C₈×C₇₁	—	4	2	no	—
148	C₈×C₇₁	#99	1	2	yes	complemented
149	C₂²×C₇₁	—	2	1	no	—
150	C₂²×C₇₁	—	2	1	no	—
151	C₂²×C₇₁	—	2	1	no	—
152	C₂²×C₇₁	—	2	1	no	—
153	C₂²×C₇₁	—	2	1	no	—
154	C₂²×C₇₁	—	2	1	no	—
155	C₂²×C₇₁	—	4	2	no	—
156	C₂²×C₇₁	—	4	2	no	—
157	C₂²×C₇₁	—	8	2	no	—
158	C₂²×C₇₁	—	8	2	no	—
159	C₂²×C₇₁	—	8	2	no	—
160	C₂²×C₇₁	—	16	4	no	—
161	C₂²×C₇₁	#390	1	1	yes	characteristic
162	C₂²×C₇₁	C₄×D₈	1	2	yes	—
163	C₂²×C₇₁	#466	1	2	yes	—
164	C₂²×C₇₁	#493	1	1	yes	characteristic
165	C₂²×C₇₁	#495	1	1	yes	characteristic
166	C₄×C₇₁	—	2	1	no	—
167	C₄×C₇₁	—	2	1	no	—
168	C₄×C₇₁	—	2	1	no	—
169	C₄×C₇₁	—	4	4	no	—
170	C₄×C₇₁	—	8	2	no	—
171	C₄×C₇₁	—	16	2	no	—
172	C₄×C₇₁	C₂×#99	1	1	yes	characteristic
173	C₄×C₇₁	#410	1	1	yes	characteristic
174	#9783	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₇₁	—	2	2	no	—
176	C₂×C₇₁	—	2	1	no	—
177	C₂×C₇₁	—	2	2	no	—
178	C₂×C₇₁	—	2	1	no	—
179	C₂×C₇₁	—	16	2	no	—
180	C₂×C₇₁	#1964	1	1	yes	characteristic
181	C₂×C₇₁	#1970	1	1	yes	characteristic
182	C₂×C₇₁	#1976	1	1	yes	characteristic
183	C₂×#466	C₂×C₇₁	1	1	yes	characteristic
184	#1964	C₂×C₇₁	1	2	yes	complemented
185	#1808	C₂×C₇₁	1	1	yes	characteristic, complemented
186	C₈×C₂²⋊C₄	C₂×C₇₁	1	1	yes	characteristic, complemented
187	#1519	C₂×C₇₁	1	2	yes	complemented
188	C₇₁	#9783	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×D₈	C₄×C₇₁	1	1	yes	characteristic, complemented
190	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
191	C₂×#121	C₂²×C₇₁	1	2	yes	—
192	#466	—	2	1	no	—
193	#466	—	2	1	no	—
194	#466	—	2	1	no	—
195	#466	—	2	1	no	—
196	C₂×#107	—	2	1	no	—
197	C₂×#107	C₄×C₇₁	1	1	yes	characteristic, complemented
198	C₂×#102	C₄×C₇₁	1	2	yes	complemented
199	C₂×#99	—	2	4	no	—
200	C₂×C₂²⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
201	C₂×C₄×C₈	—	2	1	no	—
202	#380	C₂²×C₇₁	1	2	yes	—
203	C₄×C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
204	#340	—	2	1	no	—
205	#336	—	2	1	no	—
206	#327	—	2	4	no	—
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	C₂×C₄×C₇₁	1	2	yes	—
209	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	4	2	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	4	2	no	—
215	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
216	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
217	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
218	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
219	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
230	C₂×C₄²	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	—	2	1	no	—
233	#107	—	4	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	#99	—	4	8	no	—
237	C₂²⋊C₈	—	2	1	no	—
238	C₂²⋊C₈	D₄×C₇₁	1	2	yes	—
239	C₄×C₈	—	4	2	no	—
240	#95	—	2	1	no	—
241	#95	—	2	2	no	—
242	C₂⁴	—	2	2	no	—
243	C₂⁴	C₂×D₄×C₇₁	1	1	yes	characteristic
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
261	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
262	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
263	D₈	—	4	2	no	—
264	D₈	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
275	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
276	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	D₈×C₇₁	1	4	yes	complemented
287	C₄²	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	C₄×D₄×C₇₁	1	1	yes	characteristic
296	C₂³	#121×C₇₁	1	1	yes	characteristic
297	C₂³	#127×C₇₁	1	1	yes	characteristic
298	C₂³	C₂×D₈×C₇₁	1	2	yes	—
299	C₂³	#135×C₇₁	1	2	yes	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
324	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
325	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
326	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	4	2	no	—
331	C₈	#99×C₇₁	1	2	yes	complemented
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	16	4	no	—
344	C₂²	#390×C₇₁	1	1	yes	characteristic
345	C₂²	C₄×D₈×C₇₁	1	2	yes	—
346	C₂²	#466×C₇₁	1	2	yes	—
347	C₂²	#493×C₇₁	1	1	yes	characteristic
348	C₂²	#495×C₇₁	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	2	no	—
354	C₄	—	16	2	no	—
355	C₄	C₂×#99×C₇₁	1	1	yes	characteristic
356	C₄	#410×C₇₁	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	2	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	C₇₁×#1964	1	1	yes	characteristic
363	C₂	C₇₁×#1970	1	1	yes	characteristic
364	C₂	C₇₁×#1976	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
prime radical	C₂×#466×C₇₁	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#1964	C₂	semidirect
3. C₇₁×#1808	C₂	semidirect
4. C₈×C₂²⋊C₄×C₇₁	C₂	semidirect
5. C₇₁×#1519	C₂	semidirect
6. C₂²×D₈×C₇₁	C₄	semidirect
14. C₂×#107×C₇₁	C₄	semidirect
15. C₂×#102×C₇₁	C₄	semidirect
103. C₂²⋊C₄×C₇₁	D₈	semidirect
148. C₈×C₇₁	#99	semidirect
174. #9783	C₇₁	direct (N × H)
184. #1964	C₂×C₇₁	semidirect
185. #1808	C₂×C₇₁	semidirect
186. C₈×C₂²⋊C₄	C₂×C₇₁	semidirect
187. #1519	C₂×C₇₁	semidirect
188. C₇₁	#9783	direct (N × H)
189. C₂²×D₈	C₄×C₇₁	semidirect
197. C₂×#107	C₄×C₇₁	semidirect
198. C₂×#102	C₄×C₇₁	semidirect
286. C₂²⋊C₄	D₈×C₇₁	semidirect
331. C₈	#99×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#466×C₇₁	2
C₇₁×#1964	2
C₇₁×#1808	2
C₈×C₂²⋊C₄×C₇₁	2
C₇₁×#1519	2
#9783	71

Subgroups — C71×#9783 #31830655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#9783 #31830655