Subgroups — C₇₁×#3056 #22585618

Order 9088.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#525×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#470×C₇₁	C₂	1	2	yes	maximal, complemented
6	#468×C₇₁	C₂	1	2	yes	maximal, complemented
7	#467×C₇₁	C₂	1	2	yes	maximal, complemented
8	#465×C₇₁	C₂	1	2	yes	maximal, complemented
9	#428×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#419×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
15	#137×C₇₁	—	2	4	no	—
16	C₂×Q₁₆×C₇₁	C₂²	1	2	yes	complemented
17	C₂×SD₁₆×C₇₁	C₂²	1	2	yes	complemented
18	C₂×M₁₆×C₇₁	—	2	4	no	—
19	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
21	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
23	#123×C₇₁	—	2	1	no	—
24	C₂²⋊Q₈×C₇₁	—	2	1	no	—
25	C₂²⋊Q₈×C₇₁	—	2	1	no	—
26	C₂²⋊Q₈×C₇₁	C₂²	1	2	yes	—
27	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂²⋊Q₈×C₇₁	C₂²	1	2	yes	—
29	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
30	#121×C₇₁	C₂²	1	1	yes	characteristic
31	#121×C₇₁	C₂²	1	1	yes	characteristic
32	#120×C₇₁	—	2	1	no	—
33	C₄×D₄×C₇₁	—	2	1	no	—
34	C₄×D₄×C₇₁	—	2	1	no	—
35	#117×C₇₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
39	#107×C₇₁	C₂²	1	2	yes	complemented
40	#106×C₇₁	C₂²	1	2	yes	complemented
41	#103×C₇₁	C₂²	1	2	yes	complemented
42	#103×C₇₁	C₂²	1	2	yes	complemented
43	#103×C₇₁	C₂²	1	2	yes	complemented
44	#102×C₇₁	C₂²	1	2	yes	complemented
45	C₂⁴×C₇₁	D₄	1	1	yes	characteristic
46	#41×C₇₁	—	2	2	no	—
47	#41×C₇₁	—	2	2	no	—
48	C₂×Q₈×C₇₁	—	2	1	no	—
49	C₂×Q₈×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	—	2	2	no	—
51	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
52	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₇₁	—	2	1	no	—
54	C₂×D₄×C₇₁	—	2	1	no	—
55	C₂×D₄×C₇₁	—	2	1	no	—
56	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	2	1	no	—
60	C₂²×C₄×C₇₁	—	2	1	no	—
61	C₂²×C₄×C₇₁	—	2	1	no	—
62	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
68	Q₁₆×C₇₁	—	2	4	no	—
69	SD₁₆×C₇₁	—	2	4	no	—
70	M₁₆×C₇₁	—	2	2	no	—
71	M₁₆×C₇₁	—	2	4	no	—
72	M₁₆×C₇₁	D₄	1	4	yes	complemented
73	C₂×C₈×C₇₁	—	2	4	no	—
74	C₂×C₈×C₇₁	C₂³	1	2	yes	—
75	C₂×C₈×C₇₁	C₂³	1	2	yes	—
76	C₄⋊C₄×C₇₁	—	2	2	no	—
77	C₄⋊C₄×C₇₁	—	2	2	no	—
78	C₄⋊C₄×C₇₁	—	2	1	no	—
79	C₄⋊C₄×C₇₁	—	2	1	no	—
80	C₄⋊C₄×C₇₁	—	2	1	no	—
81	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
82	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₇₁	—	2	2	no	—
85	C₂²⋊C₄×C₇₁	—	2	2	no	—
86	C₂²⋊C₄×C₇₁	—	2	1	no	—
87	C₂²⋊C₄×C₇₁	—	2	1	no	—
88	C₂²⋊C₄×C₇₁	—	2	1	no	—
89	C₂²⋊C₄×C₇₁	—	2	1	no	—
90	C₂²⋊C₄×C₇₁	—	2	1	no	—
91	C₄²×C₇₁	—	2	1	no	—
92	C₂³×C₇₁	—	2	2	no	—
93	C₂³×C₇₁	—	2	2	no	—
94	C₂³×C₇₁	—	2	1	no	—
95	C₂³×C₇₁	—	4	1	no	—
96	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
99	Q₈×C₇₁	—	2	2	no	—
100	Q₈×C₇₁	—	2	1	no	—
101	Q₈×C₇₁	—	2	2	no	—
102	Q₈×C₇₁	—	2	1	no	—
103	Q₈×C₇₁	—	4	2	no	—
104	D₄×C₇₁	—	2	1	no	—
105	D₄×C₇₁	—	2	1	no	—
106	D₄×C₇₁	—	4	2	no	—
107	D₄×C₇₁	—	4	2	no	—
108	D₄×C₇₁	—	4	2	no	—
109	C₂×C₄×C₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	—	2	2	no	—
111	C₂×C₄×C₇₁	—	2	2	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	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	2	no	—
121	C₂×C₄×C₇₁	—	4	2	no	—
122	C₂×C₄×C₇₁	—	4	2	no	—
123	C₂×C₄×C₇₁	—	4	2	no	—
124	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₇₁	#41	1	2	yes	—
129	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
130	C₈×C₇₁	—	2	2	no	—
131	C₈×C₇₁	C₂×D₄	1	4	yes	—
132	C₂²×C₇₁	—	2	1	no	—
133	C₂²×C₇₁	—	2	1	no	—
134	C₂²×C₇₁	—	2	1	no	—
135	C₂²×C₇₁	—	2	1	no	—
136	C₂²×C₇₁	—	2	1	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	—	4	4	no	—
140	C₂²×C₇₁	—	4	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	#121	1	2	yes	—
144	C₂²×C₇₁	#137	1	2	yes	complemented
145	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	—	4	2	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	2	no	—
152	C₄×C₇₁	—	4	1	no	—
153	C₄×C₇₁	#121	1	2	yes	—
154	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₇₁	—	2	2	no	—
157	C₂×C₇₁	—	2	2	no	—
158	C₂×C₇₁	—	4	1	no	—
159	C₂×C₇₁	—	8	1	no	—
160	C₂×C₇₁	C₂×#121	1	1	yes	characteristic
161	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
162	C₂×C₇₁	#575	1	1	yes	characteristic
163	#3056	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₇₁	#3056	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×#137	C₂×C₇₁	1	1	yes	characteristic, complemented
166	C₂²×M₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
167	#525	C₂×C₇₁	1	1	yes	characteristic
168	C₂×C₂²⋊Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
169	#470	C₂×C₇₁	1	2	yes	complemented
170	#468	C₂×C₇₁	1	2	yes	complemented
171	#467	C₂×C₇₁	1	2	yes	complemented
172	#465	C₂×C₇₁	1	2	yes	complemented
173	#428	C₂×C₇₁	1	1	yes	characteristic, complemented
174	#419	C₂×C₇₁	1	1	yes	characteristic, complemented
175	#417	C₂×C₇₁	1	1	yes	characteristic, complemented
176	C₂×#41	C₂²×C₇₁	1	1	yes	characteristic
177	C₂²×Q₈	C₂²×C₇₁	1	1	yes	characteristic
178	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
179	#137	—	2	4	no	—
180	C₂×Q₁₆	C₂²×C₇₁	1	2	yes	complemented
181	C₂×SD₁₆	C₂²×C₇₁	1	2	yes	complemented
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
184	C₂×M₁₆	C₂²×C₇₁	1	2	yes	—
185	C₂×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
186	C₂²×C₈	C₂²×C₇₁	1	2	yes	—
187	#123	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	—	2	1	no	—
190	C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
191	C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
192	C₂²⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
193	C₂²⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
194	#121	C₂²×C₇₁	1	1	yes	characteristic
195	#121	C₂²×C₇₁	1	1	yes	characteristic
196	#120	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	#117	C₂²×C₇₁	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	#107	C₂²×C₇₁	1	2	yes	complemented
204	#106	C₂²×C₇₁	1	2	yes	complemented
205	#103	C₂²×C₇₁	1	2	yes	complemented
206	#103	C₂²×C₇₁	1	2	yes	complemented
207	#103	C₂²×C₇₁	1	2	yes	complemented
208	#102	C₂²×C₇₁	1	2	yes	complemented
209	C₂⁴	D₄×C₇₁	1	1	yes	characteristic
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	C₂³×C₇₁	1	2	yes	—
216	C₂×Q₈	C₂³×C₇₁	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₂³×C₇₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
227	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
228	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
231	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	SD₁₆	—	2	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	D₄×C₇₁	1	4	yes	complemented
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂³×C₇₁	1	2	yes	—
239	C₂×C₈	C₂³×C₇₁	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
246	C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
247	C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
261	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
262	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	2	2	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	4	2	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×C₇₁	1	2	yes	—
292	C₂×C₄	#41×C₇₁	1	2	yes	—
293	C₂×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₂×D₄×C₇₁	1	4	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	#121×C₇₁	1	2	yes	—
308	C₂²	#137×C₇₁	1	2	yes	complemented
309	C₂²	C₂²×D₄×C₇₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	#121×C₇₁	1	2	yes	—
318	C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
319	C₄	C₂×#41×C₇₁	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	1	no	—
324	C₂	C₂×#121×C₇₁	1	1	yes	characteristic
325	C₂	C₂×#137×C₇₁	1	1	yes	characteristic
326	C₂	#575×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₁
prime radical	#525×C₇₁	C₂
LCS	G ▷ C₂×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
1. C₂×#137×C₇₁	C₂	semidirect
2. C₂²×M₁₆×C₇₁	C₂	semidirect
4. C₂×C₂²⋊Q₈×C₇₁	C₂	semidirect
5. #470×C₇₁	C₂	semidirect
6. #468×C₇₁	C₂	semidirect
7. #467×C₇₁	C₂	semidirect
8. #465×C₇₁	C₂	semidirect
9. #428×C₇₁	C₂	semidirect
10. #419×C₇₁	C₂	semidirect
11. #417×C₇₁	C₂	semidirect
16. C₂×Q₁₆×C₇₁	C₂²	semidirect
17. C₂×SD₁₆×C₇₁	C₂²	semidirect
39. #107×C₇₁	C₂²	semidirect
40. #106×C₇₁	C₂²	semidirect
41. #103×C₇₁	C₂²	semidirect
42. #103×C₇₁	C₂²	semidirect
43. #103×C₇₁	C₂²	semidirect
44. #102×C₇₁	C₂²	semidirect
72. M₁₆×C₇₁	D₄	semidirect
144. C₂²×C₇₁	#137	semidirect
163. #3056	C₇₁	direct (N × H)
164. C₇₁	#3056	direct (N × H)
165. C₂×#137	C₂×C₇₁	semidirect
166. C₂²×M₁₆	C₂×C₇₁	semidirect
168. C₂×C₂²⋊Q₈	C₂×C₇₁	semidirect
169. #470	C₂×C₇₁	semidirect
170. #468	C₂×C₇₁	semidirect
171. #467	C₂×C₇₁	semidirect
172. #465	C₂×C₇₁	semidirect
173. #428	C₂×C₇₁	semidirect
174. #419	C₂×C₇₁	semidirect
175. #417	C₂×C₇₁	semidirect
180. C₂×Q₁₆	C₂²×C₇₁	semidirect
181. C₂×SD₁₆	C₂²×C₇₁	semidirect
203. #107	C₂²×C₇₁	semidirect
204. #106	C₂²×C₇₁	semidirect
205. #103	C₂²×C₇₁	semidirect
206. #103	C₂²×C₇₁	semidirect
207. #103	C₂²×C₇₁	semidirect
208. #102	C₂²×C₇₁	semidirect
236. M₁₆	D₄×C₇₁	semidirect
308. C₂²	#137×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₇₁	2
C₂²×M₁₆×C₇₁	2
#525×C₇₁	2
C₂×C₂²⋊Q₈×C₇₁	2
#470×C₇₁	2
#468×C₇₁	2
#467×C₇₁	2
#465×C₇₁	2
#428×C₇₁	2
#419×C₇₁	2
#417×C₇₁	2
#3056	71

Subgroups — C71×#3056 #22585618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#3056 #22585618