Subgroups — C₇₁×#3190 #22585752

Order 9088.

420 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#525×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#485×C₇₁	C₂	1	2	yes	maximal, complemented
4	#483×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#480×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#467×C₇₁	C₂	1	2	yes	maximal, complemented
7	#465×C₇₁	C₂	1	2	yes	maximal, complemented
8	#452×C₇₁	C₂	1	2	yes	maximal, complemented
9	#451×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#448×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₈×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	C₂×Q₁₆×C₇₁	C₂²	1	2	yes	complemented
16	C₂×SD₁₆×C₇₁	C₂²	1	2	yes	complemented
17	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
18	#123×C₇₁	—	2	1	no	—
19	C₂²⋊Q₈×C₇₁	—	2	1	no	—
20	C₂²⋊Q₈×C₇₁	—	2	1	no	—
21	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₂²⋊Q₈×C₇₁	C₂²	1	2	yes	—
23	C₂²⋊Q₈×C₇₁	C₂²	1	2	yes	—
24	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
25	#121×C₇₁	C₂²	1	1	yes	characteristic
26	#121×C₇₁	C₂²	1	1	yes	characteristic
27	#120×C₇₁	—	2	1	no	—
28	C₄×D₄×C₇₁	—	2	1	no	—
29	C₄×D₄×C₇₁	—	2	1	no	—
30	#117×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
34	#107×C₇₁	C₂²	1	2	yes	complemented
35	#106×C₇₁	C₂²	1	2	yes	complemented
36	#103×C₇₁	C₂²	1	2	yes	complemented
37	#103×C₇₁	C₂²	1	2	yes	complemented
38	#103×C₇₁	C₂²	1	2	yes	complemented
39	#102×C₇₁	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
43	C₂⁴×C₇₁	D₄	1	1	yes	characteristic
44	#41×C₇₁	—	4	2	no	—
45	C₂×Q₈×C₇₁	—	2	1	no	—
46	C₂×Q₈×C₇₁	—	4	2	no	—
47	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₇₁	—	2	1	no	—
50	C₂×D₄×C₇₁	—	2	1	no	—
51	C₂×D₄×C₇₁	—	2	1	no	—
52	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	—	2	1	no	—
56	C₂²×C₄×C₇₁	—	4	2	no	—
57	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
63	Q₁₆×C₇₁	—	4	2	no	—
64	SD₁₆×C₇₁	—	4	2	no	—
65	C₂×C₈×C₇₁	—	4	2	no	—
66	C₂×C₈×C₇₁	C₂³	1	2	yes	—
67	C₂×C₈×C₇₁	C₂³	1	2	yes	—
68	C₄⋊C₄×C₇₁	—	2	2	no	—
69	C₄⋊C₄×C₇₁	—	2	2	no	—
70	C₄⋊C₄×C₇₁	—	2	1	no	—
71	C₄⋊C₄×C₇₁	—	2	1	no	—
72	C₄⋊C₄×C₇₁	—	2	1	no	—
73	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
74	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₇₁	—	2	2	no	—
77	C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₂²⋊C₄×C₇₁	—	2	1	no	—
79	C₂²⋊C₄×C₇₁	—	2	1	no	—
80	C₂²⋊C₄×C₇₁	—	2	1	no	—
81	C₂²⋊C₄×C₇₁	—	2	1	no	—
82	C₂²⋊C₄×C₇₁	—	2	1	no	—
83	C₄²×C₇₁	—	2	1	no	—
84	C₂³×C₇₁	—	2	2	no	—
85	C₂³×C₇₁	—	2	2	no	—
86	C₂³×C₇₁	—	2	1	no	—
87	C₂³×C₇₁	—	4	1	no	—
88	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₇₁	—	2	2	no	—
92	Q₈×C₇₁	—	2	2	no	—
93	Q₈×C₇₁	—	2	2	no	—
94	Q₈×C₇₁	—	4	2	no	—
95	D₄×C₇₁	—	2	2	no	—
96	D₄×C₇₁	—	4	2	no	—
97	D₄×C₇₁	—	4	2	no	—
98	D₄×C₇₁	—	4	2	no	—
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	2	2	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	2	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	—	4	1	no	—
108	C₂×C₄×C₇₁	—	4	1	no	—
109	C₂×C₄×C₇₁	—	4	1	no	—
110	C₂×C₄×C₇₁	—	4	2	no	—
111	C₂×C₄×C₇₁	—	4	2	no	—
112	C₂×C₄×C₇₁	—	4	2	no	—
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
117	C₈×C₇₁	—	2	2	no	—
118	C₈×C₇₁	—	2	2	no	—
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	—	2	1	no	—
121	C₂²×C₇₁	—	2	1	no	—
122	C₂²×C₇₁	—	2	1	no	—
123	C₂²×C₇₁	—	2	1	no	—
124	C₂²×C₇₁	—	2	1	no	—
125	C₂²×C₇₁	—	2	1	no	—
126	C₂²×C₇₁	—	4	4	no	—
127	C₂²×C₇₁	—	4	1	no	—
128	C₂²×C₇₁	—	4	1	no	—
129	C₂²×C₇₁	—	4	1	no	—
130	C₂²×C₇₁	#135	1	2	yes	—
131	C₂²×C₇₁	#137	1	2	yes	—
132	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	2	1	no	—
136	C₄×C₇₁	—	4	1	no	—
137	C₄×C₇₁	—	4	2	no	—
138	C₄×C₇₁	—	4	2	no	—
139	C₄×C₇₁	—	4	1	no	—
140	C₄×C₇₁	—	4	1	no	—
141	C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
142	C₂×C₇₁	—	2	2	no	—
143	C₂×C₇₁	—	2	2	no	—
144	C₂×C₇₁	—	4	1	no	—
145	C₂×C₇₁	—	8	1	no	—
146	C₂×C₇₁	#534	1	1	yes	characteristic
147	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
148	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
149	#3190	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₇₁	#3190	1	1	yes	characteristic, direct product factor, Sylow
151	#525	C₂×C₇₁	1	1	yes	characteristic
152	C₂×C₂²⋊Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
153	#485	C₂×C₇₁	1	2	yes	complemented
154	#483	C₂×C₇₁	1	1	yes	characteristic, complemented
155	#480	C₂×C₇₁	1	1	yes	characteristic, complemented
156	#467	C₂×C₇₁	1	2	yes	complemented
157	#465	C₂×C₇₁	1	2	yes	complemented
158	#452	C₂×C₇₁	1	2	yes	complemented
159	#451	C₂×C₇₁	1	1	yes	characteristic, complemented
160	#448	C₂×C₇₁	1	1	yes	characteristic, complemented
161	C₂×C₂²⋊C₈	C₂×C₇₁	1	1	yes	characteristic, complemented
162	C₂×#41	C₂²×C₇₁	1	1	yes	characteristic
163	C₂²×Q₈	C₂²×C₇₁	1	1	yes	characteristic
164	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
165	C₂×Q₁₆	C₂²×C₇₁	1	2	yes	complemented
166	C₂×SD₁₆	C₂²×C₇₁	1	2	yes	complemented
167	C₂²×C₈	C₂²×C₇₁	1	2	yes	—
168	#123	—	2	1	no	—
169	C₂²⋊Q₈	—	2	1	no	—
170	C₂²⋊Q₈	—	2	1	no	—
171	C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
172	C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
173	C₂²⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
174	C₂²⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
175	#121	C₂²×C₇₁	1	1	yes	characteristic
176	#121	C₂²×C₇₁	1	1	yes	characteristic
177	#120	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	#117	C₂²×C₇₁	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	#107	C₂²×C₇₁	1	2	yes	complemented
185	#106	C₂²×C₇₁	1	2	yes	complemented
186	#103	C₂²×C₇₁	1	2	yes	complemented
187	#103	C₂²×C₇₁	1	2	yes	complemented
188	#103	C₂²×C₇₁	1	2	yes	complemented
189	#102	C₂²×C₇₁	1	2	yes	complemented
190	C₂²⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
191	C₂²⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
192	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
193	C₂⁴	D₄×C₇₁	1	1	yes	characteristic
194	#41	—	4	2	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₂³×C₇₁	1	2	yes	—
198	C₂×Q₈	C₂³×C₇₁	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₂³×C₇₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
208	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
209	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
211	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
213	Q₁₆	—	4	2	no	—
214	SD₁₆	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	C₂³×C₇₁	1	2	yes	—
217	C₂×C₈	C₂³×C₇₁	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
224	C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
225	C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
239	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
240	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	4	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
266	C₂×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	#135×C₇₁	1	2	yes	—
281	C₂²	#137×C₇₁	1	2	yes	—
282	C₂²	C₂²×D₄×C₇₁	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	8	1	no	—
296	C₂	#534×C₇₁	1	1	yes	characteristic
297	C₂	C₂×#135×C₇₁	1	1	yes	characteristic
298	C₂	C₂×#137×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
2. C₂×C₂²⋊Q₈×C₇₁	C₂	semidirect
3. #485×C₇₁	C₂	semidirect
4. #483×C₇₁	C₂	semidirect
5. #480×C₇₁	C₂	semidirect
6. #467×C₇₁	C₂	semidirect
7. #465×C₇₁	C₂	semidirect
8. #452×C₇₁	C₂	semidirect
9. #451×C₇₁	C₂	semidirect
10. #448×C₇₁	C₂	semidirect
11. C₂×C₂²⋊C₈×C₇₁	C₂	semidirect
15. C₂×Q₁₆×C₇₁	C₂²	semidirect
16. C₂×SD₁₆×C₇₁	C₂²	semidirect
34. #107×C₇₁	C₂²	semidirect
35. #106×C₇₁	C₂²	semidirect
36. #103×C₇₁	C₂²	semidirect
37. #103×C₇₁	C₂²	semidirect
38. #103×C₇₁	C₂²	semidirect
39. #102×C₇₁	C₂²	semidirect
149. #3190	C₇₁	direct (N × H)
150. C₇₁	#3190	direct (N × H)
152. C₂×C₂²⋊Q₈	C₂×C₇₁	semidirect
153. #485	C₂×C₇₁	semidirect
154. #483	C₂×C₇₁	semidirect
155. #480	C₂×C₇₁	semidirect
156. #467	C₂×C₇₁	semidirect
157. #465	C₂×C₇₁	semidirect
158. #452	C₂×C₇₁	semidirect
159. #451	C₂×C₇₁	semidirect
160. #448	C₂×C₇₁	semidirect
161. C₂×C₂²⋊C₈	C₂×C₇₁	semidirect
165. C₂×Q₁₆	C₂²×C₇₁	semidirect
166. C₂×SD₁₆	C₂²×C₇₁	semidirect
184. #107	C₂²×C₇₁	semidirect
185. #106	C₂²×C₇₁	semidirect
186. #103	C₂²×C₇₁	semidirect
187. #103	C₂²×C₇₁	semidirect
188. #103	C₂²×C₇₁	semidirect
189. #102	C₂²×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
#525×C₇₁	2
C₂×C₂²⋊Q₈×C₇₁	2
#485×C₇₁	2
#483×C₇₁	2
#480×C₇₁	2
#467×C₇₁	2
#465×C₇₁	2
#452×C₇₁	2
#451×C₇₁	2
#448×C₇₁	2
C₂×C₂²⋊C₈×C₇₁	2
#3190	71

Subgroups — C71×#3190 #22585752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#3190 #22585752