Subgroups — C₇₁×#11225 #31832097

Order 18176.

686 conjugacy classes of subgroups.

The subgroup lattice has 344 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	C₇₁×#1740	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1736	C₂	1	2	yes	maximal
5	C₂×H(3,ℤ/4)×C₇₁	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
7	C₂×#117×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	—	2	2	no	—
15	#384×C₇₁	C₄	1	2	yes	complemented
16	#382×C₇₁	C₄	1	2	yes	complemented
17	#342×C₇₁	—	2	2	no	—
18	#340×C₇₁	—	2	4	no	—
19	H(3,ℤ/4)×C₇₁	—	2	4	no	—
20	C₂³×C₄×C₇₁	—	2	1	no	—
21	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₇₁	—	2	2	no	—
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	C₂²×C₈×C₇₁	D₄	1	4	yes	—
29	#117×C₇₁	—	2	2	no	—
30	#117×C₇₁	—	2	1	no	—
31	#117×C₇₁	—	2	1	no	—
32	#117×C₇₁	—	2	2	no	—
33	#117×C₇₁	—	2	2	no	—
34	#117×C₇₁	—	2	2	no	—
35	#117×C₇₁	—	2	2	no	—
36	#117×C₇₁	D₄	1	2	yes	—
37	#117×C₇₁	Q₈	1	4	yes	—
38	#117×C₇₁	Q₈	1	2	yes	—
39	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
51	C₂×C₄²×C₇₁	—	2	2	no	—
52	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₁	D₄	1	2	yes	—
54	C₄⋊C₈×C₇₁	—	2	8	no	—
55	#95×C₇₁	—	2	4	no	—
56	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	4	2	no	—
60	C₂²×C₄×C₇₁	—	4	2	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	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
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	—	4	2	no	—
75	M₁₆×C₇₁	—	4	2	no	—
76	M₁₆×C₇₁	—	4	2	no	—
77	M₁₆×C₇₁	—	4	2	no	—
78	C₂×C₈×C₇₁	—	2	4	no	—
79	C₂×C₈×C₇₁	—	2	4	no	—
80	C₂×C₈×C₇₁	—	2	4	no	—
81	C₄⋊C₄×C₇₁	—	2	2	no	—
82	C₄⋊C₄×C₇₁	—	2	2	no	—
83	C₄⋊C₄×C₇₁	—	2	2	no	—
84	C₄⋊C₄×C₇₁	—	2	2	no	—
85	C₄⋊C₄×C₇₁	—	2	4	no	—
86	C₄⋊C₄×C₇₁	—	2	4	no	—
87	C₄⋊C₄×C₇₁	—	4	1	no	—
88	C₄⋊C₄×C₇₁	—	4	2	no	—
89	C₄⋊C₄×C₇₁	—	4	1	no	—
90	C₄⋊C₄×C₇₁	—	8	4	no	—
91	C₂²⋊C₄×C₇₁	—	2	2	no	—
92	C₂²⋊C₄×C₇₁	—	2	4	no	—
93	C₂²⋊C₄×C₇₁	—	2	2	no	—
94	C₂²⋊C₄×C₇₁	—	2	4	no	—
95	C₂²⋊C₄×C₇₁	—	4	2	no	—
96	C₄²×C₇₁	—	2	2	no	—
97	C₄²×C₇₁	—	2	2	no	—
98	C₄²×C₇₁	—	2	2	no	—
99	C₄²×C₇₁	—	2	4	no	—
100	C₄²×C₇₁	—	4	2	no	—
101	C₄²×C₇₁	C₄⋊C₄	1	4	yes	—
102	C₂³×C₇₁	—	4	1	no	—
103	C₂³×C₇₁	—	4	1	no	—
104	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₇₁	#121	1	2	yes	—
106	C₂³×C₇₁	#121	1	1	yes	characteristic
107	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₇₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₁	—	2	4	no	—
110	C₂×C₄×C₇₁	—	2	2	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	4	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	4	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	4	4	no	—
119	C₂×C₄×C₇₁	—	8	2	no	—
120	C₂×C₄×C₇₁	—	8	2	no	—
121	C₂×C₄×C₇₁	—	8	4	no	—
122	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
127	C₂×C₄×C₇₁	#121	1	2	yes	—
128	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₇₁	#123	1	2	yes	—
133	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
134	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
135	C₂×C₄×C₇₁	#125	1	2	yes	—
136	C₂×C₄×C₇₁	#125	1	2	yes	—
137	C₂×C₄×C₇₁	#126	1	2	yes	—
138	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
140	C₈×C₇₁	—	4	4	no	—
141	C₂²×C₇₁	—	2	1	no	—
142	C₂²×C₇₁	—	2	2	no	—
143	C₂²×C₇₁	—	2	1	no	—
144	C₂²×C₇₁	—	2	2	no	—
145	C₂²×C₇₁	—	4	1	no	—
146	C₂²×C₇₁	—	4	1	no	—
147	C₂²×C₇₁	—	4	2	no	—
148	C₂²×C₇₁	#380	1	1	yes	characteristic
149	C₂²×C₇₁	#386	1	1	yes	characteristic
150	C₂²×C₇₁	#391	1	1	yes	characteristic
151	C₂²×C₇₁	#396	1	2	yes	—
152	C₂²×C₇₁	#398	1	2	yes	—
153	C₄×C₇₁	—	4	4	no	—
154	C₄×C₇₁	—	4	2	no	—
155	C₄×C₇₁	—	4	2	no	—
156	C₄×C₇₁	—	8	4	no	—
157	C₄×C₇₁	—	8	2	no	—
158	C₄×C₇₁	#387	1	2	yes	—
159	C₄×C₇₁	#389	1	2	yes	—
160	C₄×C₇₁	#394	1	1	yes	characteristic
161	C₄×C₇₁	#395	1	1	yes	characteristic
162	C₄×C₇₁	#398	1	1	yes	characteristic
163	C₄×C₇₁	#400	1	1	yes	characteristic
164	#11225	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₇₁	—	2	1	no	—
166	C₂×C₇₁	—	2	2	no	—
167	C₂×C₇₁	—	2	1	no	—
168	C₂×C₇₁	—	2	2	no	—
169	C₂×C₇₁	#1441	1	1	yes	characteristic
170	C₂×C₇₁	#1909	1	1	yes	characteristic
171	C₂×C₇₁	#1910	1	1	yes	characteristic
172	#2905	C₂×C₇₁	1	1	yes	characteristic
173	#2317	C₂×C₇₁	1	1	yes	characteristic
174	#1740	C₂×C₇₁	1	1	yes	characteristic
175	#1736	C₂×C₇₁	1	2	yes	—
176	C₂×H(3,ℤ/4)	C₂×C₇₁	1	2	yes	—
177	C₇₁	#11225	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
179	C₂×#117	C₂²×C₇₁	1	2	yes	—
180	C₂×#117	C₂²×C₇₁	1	1	yes	characteristic
181	C₂×#117	C₂²×C₇₁	1	1	yes	characteristic
182	C₂²×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
183	#424	—	2	4	no	—
184	#423	—	2	2	no	—
185	#389	—	2	2	no	—
186	#385	—	2	2	no	—
187	#384	C₄×C₇₁	1	2	yes	complemented
188	#382	C₄×C₇₁	1	2	yes	complemented
189	#342	—	2	2	no	—
190	#340	—	2	4	no	—
191	H(3,ℤ/4)	—	2	4	no	—
192	C₂³×C₄	—	2	1	no	—
193	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
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	C₂²×C₈	D₄×C₇₁	1	4	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	1	no	—
203	#117	—	2	1	no	—
204	#117	—	2	2	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	D₄×C₇₁	1	2	yes	—
209	#117	Q₈×C₇₁	1	4	yes	—
210	#117	Q₈×C₇₁	1	2	yes	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
225	C₂×C₄²	D₄×C₇₁	1	2	yes	—
226	C₄⋊C₈	—	2	8	no	—
227	#95	—	2	4	no	—
228	C₂⁴	C₂×D₄×C₇₁	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	8	2	no	—
234	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
235	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₇₁	1	2	yes	—
242	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
243	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
244	C₂²×C₄	#41×C₇₁	1	2	yes	—
245	C₂²×C₄	#41×C₇₁	1	2	yes	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	8	4	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	4	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₄⋊C₄×C₇₁	1	4	yes	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
277	C₂³	#121×C₇₁	1	2	yes	—
278	C₂³	#121×C₇₁	1	1	yes	characteristic
279	C₂³	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
280	C₂³	C₂²⋊Q₈×C₇₁	1	2	yes	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
295	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
297	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
298	C₂×C₄	#120×C₇₁	1	1	yes	characteristic
299	C₂×C₄	#121×C₇₁	1	2	yes	—
300	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
301	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
302	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
304	C₂×C₄	#123×C₇₁	1	2	yes	—
305	C₂×C₄	#123×C₇₁	1	1	yes	characteristic
306	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
307	C₂×C₄	#125×C₇₁	1	2	yes	—
308	C₂×C₄	#125×C₇₁	1	2	yes	—
309	C₂×C₄	#126×C₇₁	1	2	yes	—
310	C₂×C₄	C₄⋊Q₈×C₇₁	1	2	yes	—
311	C₂×C₄	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	#380×C₇₁	1	1	yes	characteristic
321	C₂²	#386×C₇₁	1	1	yes	characteristic
322	C₂²	#391×C₇₁	1	1	yes	characteristic
323	C₂²	#396×C₇₁	1	2	yes	—
324	C₂²	#398×C₇₁	1	2	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	2	no	—
330	C₄	#387×C₇₁	1	2	yes	—
331	C₄	#389×C₇₁	1	2	yes	—
332	C₄	#394×C₇₁	1	1	yes	characteristic
333	C₄	#395×C₇₁	1	1	yes	characteristic
334	C₄	#398×C₇₁	1	1	yes	characteristic
335	C₄	#400×C₇₁	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	C₇₁×#1441	1	1	yes	characteristic
341	C₂	C₇₁×#1909	1	1	yes	characteristic
342	C₂	C₇₁×#1910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
prime radical	C₂⁴×C₇₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ 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
15. #384×C₇₁	C₄	semidirect
16. #382×C₇₁	C₄	semidirect
164. #11225	C₇₁	direct (N × H)
177. C₇₁	#11225	direct (N × H)
187. #384	C₄×C₇₁	semidirect
188. #382	C₄×C₇₁	semidirect

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
C₇₁×#1740	2
C₇₁×#1736	2
C₂×H(3,ℤ/4)×C₇₁	2
#11225	71

Subgroups — C71×#11225 #31832097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#11225 #31832097