Subgroups — C₇₁×#11206 #31832078

Order 18176.

670 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	C₇₁×#2905	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2317	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₇₁	C₂	1	2	yes	maximal
4	C₇₁×#1742	C₂	1	1	yes	maximal, characteristic
5	C₂×#343×C₇₁	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
8	C₂×#117×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
11	#424×C₇₁	—	2	4	no	—
12	#423×C₇₁	—	2	2	no	—
13	#389×C₇₁	—	2	2	no	—
14	#385×C₇₁	C₄	1	4	yes	—
15	#384×C₇₁	—	2	1	no	—
16	#382×C₇₁	—	2	1	no	—
17	#344×C₇₁	—	2	4	no	—
18	#343×C₇₁	—	2	4	no	—
19	#343×C₇₁	—	2	1	no	—
20	#343×C₇₁	—	2	1	no	—
21	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₇₁	—	2	2	no	—
24	C₂×M₁₆×C₇₁	—	2	2	no	—
25	C₂×M₁₆×C₇₁	—	2	2	no	—
26	C₂×M₁₆×C₇₁	—	2	2	no	—
27	C₂×M₁₆×C₇₁	—	2	2	no	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂×M₁₆×C₇₁	—	2	2	no	—
31	C₂×M₁₆×C₇₁	—	2	2	no	—
32	C₂×M₁₆×C₇₁	—	2	2	no	—
33	C₂×M₁₆×C₇₁	D₄	1	4	yes	—
34	C₂×M₁₆×C₇₁	Q₈	1	4	yes	—
35	C₂²×C₈×C₇₁	—	2	2	no	—
36	C₂²×C₈×C₇₁	—	2	2	no	—
37	#117×C₇₁	—	2	2	no	—
38	#117×C₇₁	—	2	1	no	—
39	#117×C₇₁	—	2	1	no	—
40	#117×C₇₁	—	2	2	no	—
41	#117×C₇₁	D₄	1	2	yes	—
42	#117×C₇₁	Q₈	1	2	yes	—
43	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
49	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₁	—	2	1	no	—
54	C₂×C₄²×C₇₁	D₄	1	2	yes	—
55	C₄⋊C₈×C₇₁	—	2	8	no	—
56	#95×C₇₁	—	2	4	no	—
57	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	2	2	no	—
60	C₂²×C₄×C₇₁	—	2	4	no	—
61	C₂²×C₄×C₇₁	—	8	2	no	—
62	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	#41	1	2	yes	—
73	C₂²×C₄×C₇₁	#41	1	2	yes	—
74	M₁₆×C₇₁	—	2	4	no	—
75	M₁₆×C₇₁	—	2	4	no	—
76	M₁₆×C₇₁	—	2	4	no	—
77	M₁₆×C₇₁	—	2	4	no	—
78	M₁₆×C₇₁	—	4	2	no	—
79	M₁₆×C₇₁	—	4	2	no	—
80	M₁₆×C₇₁	—	4	2	no	—
81	M₁₆×C₇₁	—	4	2	no	—
82	C₂×C₈×C₇₁	—	2	4	no	—
83	C₂×C₈×C₇₁	—	2	4	no	—
84	C₂×C₈×C₇₁	—	4	2	no	—
85	C₂×C₈×C₇₁	—	4	2	no	—
86	C₂×C₈×C₇₁	—	4	2	no	—
87	C₂×C₈×C₇₁	—	4	2	no	—
88	C₄⋊C₄×C₇₁	—	2	2	no	—
89	C₄⋊C₄×C₇₁	—	2	2	no	—
90	C₄⋊C₄×C₇₁	—	2	2	no	—
91	C₄⋊C₄×C₇₁	—	2	2	no	—
92	C₄⋊C₄×C₇₁	—	4	1	no	—
93	C₄⋊C₄×C₇₁	—	4	2	no	—
94	C₄⋊C₄×C₇₁	—	4	1	no	—
95	C₄⋊C₄×C₇₁	—	8	4	no	—
96	C₂²⋊C₄×C₇₁	—	2	2	no	—
97	C₂²⋊C₄×C₇₁	—	2	4	no	—
98	C₂²⋊C₄×C₇₁	C₄⋊C₄	1	4	yes	—
99	C₄²×C₇₁	—	2	2	no	—
100	C₄²×C₇₁	—	2	2	no	—
101	C₄²×C₇₁	—	2	2	no	—
102	C₄²×C₇₁	—	2	2	no	—
103	C₂³×C₇₁	—	4	1	no	—
104	C₂³×C₇₁	—	4	1	no	—
105	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₇₁	#117	1	1	yes	characteristic
108	C₂³×C₇₁	C₄×D₄	1	2	yes	—
109	C₂³×C₇₁	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₇₁	—	2	4	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	2	no	—
113	C₂×C₄×C₇₁	—	4	2	no	—
114	C₂×C₄×C₇₁	—	4	2	no	—
115	C₂×C₄×C₇₁	—	4	2	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	4	4	no	—
118	C₂×C₄×C₇₁	—	4	2	no	—
119	C₂×C₄×C₇₁	—	8	4	no	—
120	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#121	1	2	yes	—
122	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
124	C₂×C₄×C₇₁	#121	1	2	yes	—
125	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇₁	#123	1	2	yes	—
131	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
132	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
133	C₂×C₄×C₇₁	#125	1	2	yes	—
134	C₂×C₄×C₇₁	#125	1	2	yes	—
135	C₂×C₄×C₇₁	#126	1	2	yes	—
136	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
138	C₈×C₇₁	—	4	4	no	—
139	C₈×C₇₁	—	4	4	no	—
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	—	2	2	no	—
142	C₂²×C₇₁	—	2	1	no	—
143	C₂²×C₇₁	—	2	2	no	—
144	C₂²×C₇₁	—	4	1	no	—
145	C₂²×C₇₁	—	4	1	no	—
146	C₂²×C₇₁	—	4	2	no	—
147	C₂²×C₇₁	#380	1	1	yes	characteristic
148	C₂²×C₇₁	#386	1	1	yes	characteristic
149	C₂²×C₇₁	#387	1	2	yes	—
150	C₂²×C₇₁	#389	1	2	yes	—
151	C₂²×C₇₁	#391	1	1	yes	characteristic
152	C₄×C₇₁	—	4	4	no	—
153	C₄×C₇₁	—	4	2	no	—
154	C₄×C₇₁	—	4	2	no	—
155	C₄×C₇₁	—	8	2	no	—
156	C₄×C₇₁	#394	1	1	yes	characteristic
157	C₄×C₇₁	#395	1	1	yes	characteristic
158	C₄×C₇₁	#396	1	2	yes	—
159	C₄×C₇₁	#398	1	1	yes	characteristic
160	C₄×C₇₁	#398	1	2	yes	—
161	C₄×C₇₁	#400	1	1	yes	characteristic
162	#11206	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₇₁	—	2	1	no	—
164	C₂×C₇₁	—	2	2	no	—
165	C₂×C₇₁	—	2	2	no	—
166	C₂×C₇₁	—	2	1	no	—
167	C₂×C₇₁	#1441	1	1	yes	characteristic
168	C₂×C₇₁	#1914	1	1	yes	characteristic
169	C₂×C₇₁	#1915	1	1	yes	characteristic
170	#2905	C₂×C₇₁	1	1	yes	characteristic
171	#2317	C₂×C₇₁	1	1	yes	characteristic
172	C₂×#344	C₂×C₇₁	1	2	yes	—
173	#1742	C₂×C₇₁	1	1	yes	characteristic
174	C₂×#343	C₂×C₇₁	1	2	yes	—
175	C₇₁	#11206	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
177	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
178	C₂×#117	C₂²×C₇₁	1	1	yes	characteristic
179	C₂×#117	C₂²×C₇₁	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
181	#424	—	2	4	no	—
182	#423	—	2	2	no	—
183	#389	—	2	2	no	—
184	#385	C₄×C₇₁	1	4	yes	—
185	#384	—	2	1	no	—
186	#382	—	2	1	no	—
187	#344	—	2	4	no	—
188	#343	—	2	4	no	—
189	#343	—	2	1	no	—
190	#343	—	2	1	no	—
191	C₂³×C₄	C₂×C₄×C₇₁	1	2	yes	—
192	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	D₄×C₇₁	1	4	yes	—
204	C₂×M₁₆	Q₈×C₇₁	1	4	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	—	2	1	no	—
209	#117	—	2	1	no	—
210	#117	—	2	2	no	—
211	#117	D₄×C₇₁	1	2	yes	—
212	#117	Q₈×C₇₁	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
220	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
222	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	D₄×C₇₁	1	2	yes	—
225	C₄⋊C₈	—	2	8	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂²×C₄×C₇₁	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₇₁	1	2	yes	—
239	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
240	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
241	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
242	C₂²×C₄	#41×C₇₁	1	2	yes	—
243	C₂²×C₄	#41×C₇₁	1	2	yes	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	4	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	8	4	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	C₄⋊C₄×C₇₁	1	4	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
277	C₂³	#117×C₇₁	1	1	yes	characteristic
278	C₂³	C₄×D₄×C₇₁	1	2	yes	—
279	C₂³	C₄×Q₈×C₇₁	1	2	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	#120×C₇₁	1	1	yes	characteristic
291	C₂×C₄	#121×C₇₁	1	2	yes	—
292	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
293	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
294	C₂×C₄	#121×C₇₁	1	2	yes	—
295	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
296	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
298	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
299	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
300	C₂×C₄	#123×C₇₁	1	2	yes	—
301	C₂×C₄	#123×C₇₁	1	1	yes	characteristic
302	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
303	C₂×C₄	#125×C₇₁	1	2	yes	—
304	C₂×C₄	#125×C₇₁	1	2	yes	—
305	C₂×C₄	#126×C₇₁	1	2	yes	—
306	C₂×C₄	C₄⋊Q₈×C₇₁	1	2	yes	—
307	C₂×C₄	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	4	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	#380×C₇₁	1	1	yes	characteristic
318	C₂²	#386×C₇₁	1	1	yes	characteristic
319	C₂²	#387×C₇₁	1	2	yes	—
320	C₂²	#389×C₇₁	1	2	yes	—
321	C₂²	#391×C₇₁	1	1	yes	characteristic
322	C₄	—	4	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#394×C₇₁	1	1	yes	characteristic
327	C₄	#395×C₇₁	1	1	yes	characteristic
328	C₄	#396×C₇₁	1	2	yes	—
329	C₄	#398×C₇₁	1	1	yes	characteristic
330	C₄	#398×C₇₁	1	2	yes	—
331	C₄	#400×C₇₁	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	C₇₁×#1441	1	1	yes	characteristic
337	C₂	C₇₁×#1914	1	1	yes	characteristic
338	C₂	C₇₁×#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#386
commutator	C₂⁴	C₂²×C₄×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
prime radical	C₂⁴×C₇₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ 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
162. #11206	C₇₁	direct (N × H)
175. C₇₁	#11206	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₁×#2905	2
C₇₁×#2317	2
C₂×#344×C₇₁	2
C₇₁×#1742	2
C₂×#343×C₇₁	2
#11206	71

Subgroups — C71×#11206 #31832078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#11206 #31832078