Subgroups — C₇₁×#2946 #22585508

Order 9088.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₇₁	C₂	1	1	yes	maximal, characteristic
2	#520×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#518×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#442×C₇₁	C₂	1	2	yes	maximal, complemented
5	#440×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₄×SD₁₆×C₇₁	C₂	1	2	yes	maximal, complemented
7	C₄×D₈×C₇₁	C₂	1	2	yes	maximal, complemented
8	#428×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#416×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#102×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₇₁	C₂²	1	1	yes	characteristic
14	AGL(1,ℤ/8)×C₇₁	C₄	1	8	yes	complemented
15	C₂×SD₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×D₈×C₇₁	C₂²	1	2	yes	—
17	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₇₁	C₂²	1	2	yes	complemented
20	C₄×D₄×C₇₁	—	2	1	no	—
21	C₄×D₄×C₇₁	—	2	1	no	—
22	C₄×D₄×C₇₁	—	2	2	no	—
23	C₄×D₄×C₇₁	C₂²	1	2	yes	—
24	C₄×D₄×C₇₁	C₂²	1	2	yes	—
25	C₄×D₄×C₇₁	C₂²	1	2	yes	complemented
26	#117×C₇₁	—	2	1	no	—
27	#117×C₇₁	C₂²	1	1	yes	characteristic
28	#117×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
32	#107×C₇₁	C₂²	1	2	yes	complemented
33	#106×C₇₁	C₂²	1	2	yes	complemented
34	#103×C₇₁	C₂²	1	1	yes	characteristic, complemented
35	#103×C₇₁	C₂²	1	1	yes	characteristic, complemented
36	#102×C₇₁	C₂²	1	2	yes	—
37	#102×C₇₁	C₂²	1	2	yes	—
38	#102×C₇₁	C₂²	1	1	yes	characteristic, complemented
39	#102×C₇₁	C₂²	1	1	yes	characteristic, complemented
40	#97×C₇₁	C₂²	1	2	yes	complemented
41	C₄×C₈×C₇₁	C₂²	1	2	yes	complemented
42	C₂⁴×C₇₁	—	2	1	no	—
43	#41×C₇₁	—	2	2	no	—
44	#41×C₇₁	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₇₁	—	2	1	no	—
47	C₂×D₄×C₇₁	—	2	1	no	—
48	C₂×D₄×C₇₁	—	2	2	no	—
49	C₂×D₄×C₇₁	C₂×C₄	1	4	yes	complemented
50	C₂×D₄×C₇₁	C₂³	1	2	yes	—
51	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	—	2	1	no	—
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	—	2	2	no	—
55	C₂²×C₄×C₇₁	—	2	2	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
58	SD₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
59	D₈×C₇₁	C₂×C₄	1	8	yes	complemented
60	M₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
61	C₂×C₈×C₇₁	—	2	2	no	—
62	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₇₁	—	2	1	no	—
66	C₄⋊C₄×C₇₁	—	2	2	no	—
67	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₇₁	—	2	1	no	—
73	C₂²⋊C₄×C₇₁	—	2	2	no	—
74	C₂²⋊C₄×C₇₁	—	2	2	no	—
75	C₂²⋊C₄×C₇₁	—	2	2	no	—
76	C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
77	C₄²×C₇₁	—	2	2	no	—
78	C₄²×C₇₁	—	2	1	no	—
79	C₄²×C₇₁	C₂³	1	2	yes	—
80	C₂³×C₇₁	—	2	1	no	—
81	C₂³×C₇₁	—	2	2	no	—
82	C₂³×C₇₁	—	2	4	no	—
83	C₂³×C₇₁	—	8	1	no	—
84	C₂³×C₇₁	—	8	1	no	—
85	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₇₁	—	2	1	no	—
87	Q₈×C₇₁	C₂²×C₄	1	2	yes	—
88	D₄×C₇₁	—	2	4	no	—
89	D₄×C₇₁	—	2	4	no	—
90	D₄×C₇₁	—	2	2	no	—
91	D₄×C₇₁	—	2	1	no	—
92	D₄×C₇₁	C₂²×C₄	1	2	yes	—
93	D₄×C₇₁	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	—	2	4	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	2	no	—
99	C₂×C₄×C₇₁	—	8	2	no	—
100	C₂×C₄×C₇₁	—	8	2	no	—
101	C₂×C₄×C₇₁	—	8	2	no	—
102	C₂×C₄×C₇₁	—	8	1	no	—
103	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#41	1	2	yes	—
108	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₇₁	—	2	1	no	—
110	C₈×C₇₁	—	2	1	no	—
111	C₈×C₇₁	C₂²×C₄	1	4	yes	—
112	C₂²×C₇₁	—	2	1	no	—
113	C₂²×C₇₁	—	2	1	no	—
114	C₂²×C₇₁	—	2	2	no	—
115	C₂²×C₇₁	—	2	4	no	—
116	C₂²×C₇₁	—	4	1	no	—
117	C₂²×C₇₁	—	4	2	no	—
118	C₂²×C₇₁	—	4	2	no	—
119	C₂²×C₇₁	—	4	1	no	—
120	C₂²×C₇₁	—	8	4	no	—
121	C₂²×C₇₁	C₄×D₄	1	2	yes	—
122	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₇₁	—	2	2	no	—
124	C₄×C₇₁	—	2	2	no	—
125	C₄×C₇₁	—	2	2	no	—
126	C₄×C₇₁	—	4	1	no	—
127	C₄×C₇₁	—	4	2	no	—
128	C₄×C₇₁	—	4	1	no	—
129	C₄×C₇₁	C₄×D₄	1	2	yes	—
130	C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
132	C₂×C₇₁	—	2	2	no	—
133	C₂×C₇₁	—	4	2	no	—
134	C₂×C₇₁	—	4	4	no	—
135	C₂×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₇₁	#576	1	1	yes	characteristic
137	C₂×C₇₁	#577	1	1	yes	characteristic
138	#2946	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₇₁	#2946	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×AGL(1,ℤ/8)	C₂×C₇₁	1	1	yes	characteristic
141	#520	C₂×C₇₁	1	1	yes	characteristic, complemented
142	#518	C₂×C₇₁	1	1	yes	characteristic, complemented
143	#442	C₂×C₇₁	1	2	yes	complemented
144	#440	C₂×C₇₁	1	2	yes	complemented
145	C₄×SD₁₆	C₂×C₇₁	1	2	yes	complemented
146	C₄×D₈	C₂×C₇₁	1	2	yes	complemented
147	#428	C₂×C₇₁	1	1	yes	characteristic, complemented
148	#416	C₂×C₇₁	1	1	yes	characteristic, complemented
149	C₂×#102	C₂×C₇₁	1	1	yes	characteristic, complemented
150	#405	C₂×C₇₁	1	1	yes	characteristic, complemented
151	C₂×#41	C₂²×C₇₁	1	1	yes	characteristic
152	C₂²×D₄	C₂²×C₇₁	1	1	yes	characteristic
153	AGL(1,ℤ/8)	C₄×C₇₁	1	8	yes	complemented
154	C₂×SD₁₆	C₂²×C₇₁	1	2	yes	—
155	C₂×D₈	C₂²×C₇₁	1	2	yes	—
156	C₂×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₇₁	1	1	yes	characteristic
158	C₄×Q₈	C₂²×C₇₁	1	2	yes	complemented
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	C₂²×C₇₁	1	2	yes	—
163	C₄×D₄	C₂²×C₇₁	1	2	yes	—
164	C₄×D₄	C₂²×C₇₁	1	2	yes	complemented
165	#117	—	2	1	no	—
166	#117	C₂²×C₇₁	1	1	yes	characteristic
167	#117	C₂²×C₇₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	#107	C₂²×C₇₁	1	2	yes	complemented
172	#106	C₂²×C₇₁	1	2	yes	complemented
173	#103	C₂²×C₇₁	1	1	yes	characteristic, complemented
174	#103	C₂²×C₇₁	1	1	yes	characteristic, complemented
175	#102	C₂²×C₇₁	1	2	yes	—
176	#102	C₂²×C₇₁	1	2	yes	—
177	#102	C₂²×C₇₁	1	1	yes	characteristic, complemented
178	#102	C₂²×C₇₁	1	1	yes	characteristic, complemented
179	#97	C₂²×C₇₁	1	2	yes	complemented
180	C₄×C₈	C₂²×C₇₁	1	2	yes	complemented
181	C₂⁴	—	2	1	no	—
182	#41	—	2	2	no	—
183	#41	C₂×C₄×C₇₁	1	4	yes	complemented
184	C₂×Q₈	C₂³×C₇₁	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×C₄×C₇₁	1	4	yes	complemented
189	C₂×D₄	C₂³×C₇₁	1	2	yes	—
190	C₂×D₄	C₂³×C₇₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
197	SD₁₆	C₂×C₄×C₇₁	1	8	yes	complemented
198	D₈	C₂×C₄×C₇₁	1	8	yes	complemented
199	M₁₆	C₂×C₄×C₇₁	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂³×C₇₁	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₇₁	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₇₁	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
207	C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
208	C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
209	C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂³×C₇₁	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×C₄×C₇₁	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	1	no	—
231	D₄	C₂²×C₄×C₇₁	1	2	yes	—
232	D₄	C₂²×C₄×C₇₁	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₂²×C₄×C₇₁	1	2	yes	—
243	C₂×C₄	C₂×D₄×C₇₁	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₇₁	1	2	yes	—
245	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
246	C₂×C₄	#41×C₇₁	1	2	yes	—
247	C₂×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂²×C₄×C₇₁	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₄×D₄×C₇₁	1	2	yes	—
261	C₂²	C₂²×D₄×C₇₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	C₄×D₄×C₇₁	1	2	yes	—
269	C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
270	C₄	C₂×#41×C₇₁	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
275	C₂	#576×C₇₁	1	1	yes	characteristic
276	C₂	#577×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₇₁
prime radical	C₂×AGL(1,ℤ/8)×C₇₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ #117×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #520×C₇₁	C₂	semidirect
3. #518×C₇₁	C₂	semidirect
4. #442×C₇₁	C₂	semidirect
5. #440×C₇₁	C₂	semidirect
6. C₄×SD₁₆×C₇₁	C₂	semidirect
7. C₄×D₈×C₇₁	C₂	semidirect
8. #428×C₇₁	C₂	semidirect
9. #416×C₇₁	C₂	semidirect
10. C₂×#102×C₇₁	C₂	semidirect
11. #405×C₇₁	C₂	semidirect
14. AGL(1,ℤ/8)×C₇₁	C₄	semidirect
19. C₄×Q₈×C₇₁	C₂²	semidirect
25. C₄×D₄×C₇₁	C₂²	semidirect
32. #107×C₇₁	C₂²	semidirect
33. #106×C₇₁	C₂²	semidirect
34. #103×C₇₁	C₂²	semidirect
35. #103×C₇₁	C₂²	semidirect
38. #102×C₇₁	C₂²	semidirect
39. #102×C₇₁	C₂²	semidirect
40. #97×C₇₁	C₂²	semidirect
41. C₄×C₈×C₇₁	C₂²	semidirect
44. #41×C₇₁	C₂×C₄	semidirect
49. C₂×D₄×C₇₁	C₂×C₄	semidirect
58. SD₁₆×C₇₁	C₂×C₄	semidirect
59. D₈×C₇₁	C₂×C₄	semidirect
60. M₁₆×C₇₁	C₂×C₄	semidirect
138. #2946	C₇₁	direct (N × H)
139. C₇₁	#2946	direct (N × H)
141. #520	C₂×C₇₁	semidirect
142. #518	C₂×C₇₁	semidirect
143. #442	C₂×C₇₁	semidirect
144. #440	C₂×C₇₁	semidirect
145. C₄×SD₁₆	C₂×C₇₁	semidirect
146. C₄×D₈	C₂×C₇₁	semidirect
147. #428	C₂×C₇₁	semidirect
148. #416	C₂×C₇₁	semidirect
149. C₂×#102	C₂×C₇₁	semidirect
150. #405	C₂×C₇₁	semidirect
153. AGL(1,ℤ/8)	C₄×C₇₁	semidirect
158. C₄×Q₈	C₂²×C₇₁	semidirect
164. C₄×D₄	C₂²×C₇₁	semidirect
171. #107	C₂²×C₇₁	semidirect
172. #106	C₂²×C₇₁	semidirect
173. #103	C₂²×C₇₁	semidirect
174. #103	C₂²×C₇₁	semidirect
177. #102	C₂²×C₇₁	semidirect
178. #102	C₂²×C₇₁	semidirect
179. #97	C₂²×C₇₁	semidirect
180. C₄×C₈	C₂²×C₇₁	semidirect
183. #41	C₂×C₄×C₇₁	semidirect
188. C₂×D₄	C₂×C₄×C₇₁	semidirect
197. SD₁₆	C₂×C₄×C₇₁	semidirect
198. D₈	C₂×C₄×C₇₁	semidirect
199. M₁₆	C₂×C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₇₁	2
#520×C₇₁	2
#518×C₇₁	2
#442×C₇₁	2
#440×C₇₁	2
C₄×SD₁₆×C₇₁	2
C₄×D₈×C₇₁	2
#428×C₇₁	2
#416×C₇₁	2
C₂×#102×C₇₁	2
#405×C₇₁	2
#2946	71

Subgroups — C71×#2946 #22585508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2946 #22585508