Subgroups — C₇₁×#2976 #22585538

Order 9088.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 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	#436×C₇₁	C₂	1	2	yes	maximal, complemented
3	#435×C₇₁	C₂	1	2	yes	maximal, complemented
4	#435×C₇₁	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₇₁	C₂	1	2	yes	maximal, complemented
6	#433×C₇₁	C₂	1	2	yes	maximal, complemented
7	#408×C₇₁	C₂	1	2	yes	maximal, complemented
8	#407×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₇₁	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	C₂²	1	2	yes	complemented
18	#123×C₇₁	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
20	C₂²⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
21	#121×C₇₁	C₄	1	1	yes	characteristic
22	#121×C₇₁	C₄	1	1	yes	characteristic
23	#120×C₇₁	C₄	1	2	yes	—
24	C₄×D₄×C₇₁	C₂²	1	2	yes	—
25	C₄×D₄×C₇₁	C₂²	1	2	yes	—
26	#117×C₇₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	C₂²	1	2	yes	complemented
28	C₄⋊C₈×C₇₁	C₂²	1	2	yes	complemented
29	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
30	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
31	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
33	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	complemented
34	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	complemented
35	#97×C₇₁	C₂²	1	2	yes	complemented
36	C₄×C₈×C₇₁	C₂²	1	2	yes	complemented
37	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
38	#41×C₇₁	—	4	1	no	—
39	#41×C₇₁	—	4	1	no	—
40	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	—	4	1	no	—
46	C₂²×C₄×C₇₁	—	4	1	no	—
47	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
54	M₁₆×C₇₁	—	4	1	no	—
55	M₁₆×C₇₁	—	4	1	no	—
56	M₁₆×C₇₁	—	4	2	no	—
57	C₂×C₈×C₇₁	—	4	1	no	—
58	C₂×C₈×C₇₁	—	4	1	no	—
59	C₂×C₈×C₇₁	—	4	2	no	—
60	C₂×C₈×C₇₁	C₂³	1	2	yes	—
61	C₂×C₈×C₇₁	C₂³	1	2	yes	—
62	C₂×C₈×C₇₁	C₂³	1	2	yes	—
63	C₂×C₈×C₇₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
69	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
70	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
71	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
72	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
73	C₄²×C₇₁	C₂³	1	2	yes	—
74	C₂³×C₇₁	—	2	2	no	—
75	C₂³×C₇₁	—	4	1	no	—
76	C₂³×C₇₁	—	4	1	no	—
77	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₇₁	C₂²×C₄	1	2	yes	—
80	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₇₁	—	4	1	no	—
82	D₄×C₇₁	—	2	4	no	—
83	D₄×C₇₁	—	4	2	no	—
84	D₄×C₇₁	—	4	1	no	—
85	D₄×C₇₁	—	4	2	no	—
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	—	2	1	no	—
88	C₂×C₄×C₇₁	—	2	2	no	—
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	1	no	—
94	C₂×C₄×C₇₁	—	4	1	no	—
95	C₂×C₄×C₇₁	—	4	1	no	—
96	C₂×C₄×C₇₁	—	4	1	no	—
97	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
104	C₈×C₇₁	—	2	2	no	—
105	C₈×C₇₁	—	2	2	no	—
106	C₈×C₇₁	—	2	2	no	—
107	C₈×C₇₁	—	2	2	no	—
108	C₂²×C₇₁	—	2	1	no	—
109	C₂²×C₇₁	—	2	1	no	—
110	C₂²×C₇₁	—	2	1	no	—
111	C₂²×C₇₁	—	2	1	no	—
112	C₂²×C₇₁	—	2	2	no	—
113	C₂²×C₇₁	—	2	2	no	—
114	C₂²×C₇₁	—	2	1	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	—	2	1	no	—
117	C₂²×C₇₁	—	2	2	no	—
118	C₂²×C₇₁	—	2	1	no	—
119	C₂²×C₇₁	—	4	4	no	—
120	C₂²×C₇₁	#131	1	2	yes	—
121	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	2	1	no	—
124	C₄×C₇₁	—	2	1	no	—
125	C₄×C₇₁	—	2	2	no	—
126	C₄×C₇₁	—	2	2	no	—
127	C₄×C₇₁	—	2	2	no	—
128	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
129	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₇₁	—	2	2	no	—
131	C₂×C₇₁	—	4	2	no	—
132	C₂×C₇₁	—	4	1	no	—
133	C₂×C₇₁	—	4	1	no	—
134	C₂×C₇₁	#518	1	1	yes	characteristic
135	C₂×C₇₁	C₂×#131	1	1	yes	characteristic
136	C₂×C₇₁	#568	1	1	yes	characteristic
137	#2976	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₇₁	#2976	1	1	yes	characteristic, direct product factor, Sylow
139	#525	C₂×C₇₁	1	1	yes	characteristic
140	#436	C₂×C₇₁	1	2	yes	complemented
141	#435	C₂×C₇₁	1	2	yes	complemented
142	#435	C₂×C₇₁	1	2	yes	complemented
143	C₈×D₄	C₂×C₇₁	1	2	yes	complemented
144	#433	C₂×C₇₁	1	2	yes	complemented
145	#408	C₂×C₇₁	1	2	yes	complemented
146	#407	C₂×C₇₁	1	1	yes	characteristic, complemented
147	C₂×C₂²⋊C₈	C₂×C₇₁	1	1	yes	characteristic, complemented
148	C₂×#41	C₂²×C₇₁	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
150	C₂×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
151	C₂×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
152	C₂×M₁₆	C₂²×C₇₁	1	2	yes	complemented
153	C₂²×C₈	C₂²×C₇₁	1	1	yes	characteristic
154	C₂²×C₈	C₂²×C₇₁	1	1	yes	characteristic
155	C₂²×C₈	C₂²×C₇₁	1	2	yes	complemented
156	#123	C₄×C₇₁	1	2	yes	—
157	C₂²⋊Q₈	C₄×C₇₁	1	1	yes	characteristic
158	C₂²⋊Q₈	C₄×C₇₁	1	1	yes	characteristic
159	#121	C₄×C₇₁	1	1	yes	characteristic
160	#121	C₄×C₇₁	1	1	yes	characteristic
161	#120	C₄×C₇₁	1	2	yes	—
162	C₄×D₄	C₂²×C₇₁	1	2	yes	—
163	C₄×D₄	C₂²×C₇₁	1	2	yes	—
164	#117	C₂²×C₇₁	1	1	yes	characteristic
165	C₄⋊C₈	C₂²×C₇₁	1	2	yes	complemented
166	C₄⋊C₈	C₂²×C₇₁	1	2	yes	complemented
167	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
168	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
169	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
170	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
171	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	complemented
172	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	complemented
173	#97	C₂²×C₇₁	1	2	yes	complemented
174	C₄×C₈	C₂²×C₇₁	1	2	yes	complemented
175	C₂⁴	C₂×C₄×C₇₁	1	1	yes	characteristic
176	#41	—	4	1	no	—
177	#41	—	4	1	no	—
178	C₂×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
179	C₂×D₄	C₂×C₄×C₇₁	1	2	yes	—
180	C₂×D₄	C₂×C₄×C₇₁	1	2	yes	—
181	C₂×D₄	C₂×C₄×C₇₁	1	2	yes	—
182	C₂×D₄	C₂×C₄×C₇₁	1	1	yes	characteristic
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	1	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	C₂³×C₇₁	1	2	yes	—
199	C₂×C₈	C₂³×C₇₁	1	2	yes	—
200	C₂×C₈	C₂³×C₇₁	1	2	yes	—
201	C₂×C₈	C₂³×C₇₁	1	2	yes	—
202	C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
203	C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
204	C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
205	C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
206	C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
207	C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
208	C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
209	C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
210	C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
211	C₄²	C₂³×C₇₁	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂²×C₄×C₇₁	1	1	yes	characteristic
216	C₂³	C₂²×C₄×C₇₁	1	1	yes	characteristic
217	C₂³	C₂²×C₄×C₇₁	1	2	yes	—
218	C₂³	C₂²×C₄×C₇₁	1	1	yes	characteristic
219	Q₈	—	4	1	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	1	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	C₂²×C₄×C₇₁	1	2	yes	—
236	C₂×C₄	C₂²×C₄×C₇₁	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₄×C₇₁	1	2	yes	—
240	C₂×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
241	C₂×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	#131×C₇₁	1	2	yes	—
259	C₂²	C₂³×C₄×C₇₁	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
267	C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	4	2	no	—
272	C₂	#518×C₇₁	1	1	yes	characteristic
273	C₂	C₂×#131×C₇₁	1	1	yes	characteristic
274	C₂	#568×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₇₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂³×C₄
prime radical	#525×C₇₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #436×C₇₁	C₂	semidirect
3. #435×C₇₁	C₂	semidirect
4. #435×C₇₁	C₂	semidirect
5. C₈×D₄×C₇₁	C₂	semidirect
6. #433×C₇₁	C₂	semidirect
7. #408×C₇₁	C₂	semidirect
8. #407×C₇₁	C₂	semidirect
9. C₂×C₂²⋊C₈×C₇₁	C₂	semidirect
14. C₂×M₁₆×C₇₁	C₂²	semidirect
17. C₂²×C₈×C₇₁	C₂²	semidirect
27. C₄⋊C₈×C₇₁	C₂²	semidirect
28. C₄⋊C₈×C₇₁	C₂²	semidirect
33. C₂²⋊C₈×C₇₁	C₂²	semidirect
34. C₂²⋊C₈×C₇₁	C₂²	semidirect
35. #97×C₇₁	C₂²	semidirect
36. C₄×C₈×C₇₁	C₂²	semidirect
137. #2976	C₇₁	direct (N × H)
138. C₇₁	#2976	direct (N × H)
140. #436	C₂×C₇₁	semidirect
141. #435	C₂×C₇₁	semidirect
142. #435	C₂×C₇₁	semidirect
143. C₈×D₄	C₂×C₇₁	semidirect
144. #433	C₂×C₇₁	semidirect
145. #408	C₂×C₇₁	semidirect
146. #407	C₂×C₇₁	semidirect
147. C₂×C₂²⋊C₈	C₂×C₇₁	semidirect
152. C₂×M₁₆	C₂²×C₇₁	semidirect
155. C₂²×C₈	C₂²×C₇₁	semidirect
165. C₄⋊C₈	C₂²×C₇₁	semidirect
166. C₄⋊C₈	C₂²×C₇₁	semidirect
171. C₂²⋊C₈	C₂²×C₇₁	semidirect
172. C₂²⋊C₈	C₂²×C₇₁	semidirect
173. #97	C₂²×C₇₁	semidirect
174. C₄×C₈	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
#436×C₇₁	2
#435×C₇₁	2
C₈×D₄×C₇₁	2
#433×C₇₁	2
#408×C₇₁	2
#407×C₇₁	2
C₂×C₂²⋊C₈×C₇₁	2
#2976	71

Subgroups — C71×#2976 #22585538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2976 #22585538