Subgroups — C₇₁×#3007 #22585569

Order 9088.

754 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#137×C₇₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#452×C₇₁	C₂	1	4	yes	maximal, complemented
5	#451×C₇₁	C₂	1	2	yes	maximal, complemented
6	#450×C₇₁	C₂	1	2	yes	maximal, complemented
7	#417×C₇₁	C₂	1	2	yes	maximal, complemented
8	#407×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₇₁	—	2	8	no	—
10	C₂×#41×C₇₁	—	2	1	no	—
11	C₂×#41×C₇₁	—	2	1	no	—
12	C₂×#41×C₇₁	C₂²	1	2	yes	—
13	C₂×#41×C₇₁	C₂²	1	2	yes	—
14	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₄×C₇₁	—	2	1	no	—
17	C₂²×D₄×C₇₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₇₁	—	2	1	no	—
19	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
20	#137×C₇₁	—	2	8	no	—
21	C₂×Q₁₆×C₇₁	C₂²	1	4	yes	complemented
22	C₂×SD₁₆×C₇₁	C₂²	1	4	yes	complemented
23	C₂×M₁₆×C₇₁	C₂²	1	2	yes	complemented
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	2	yes	complemented
28	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
31	#103×C₇₁	C₂²	1	4	yes	complemented
32	#102×C₇₁	C₂²	1	4	yes	complemented
33	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	complemented
35	C₂⁴×C₇₁	—	2	1	no	—
36	C₂⁴×C₇₁	D₄	1	1	yes	characteristic
37	#41×C₇₁	—	2	8	no	—
38	#41×C₇₁	—	2	8	no	—
39	#41×C₇₁	—	2	4	no	—
40	#41×C₇₁	—	2	4	no	—
41	#41×C₇₁	—	2	4	no	—
42	#41×C₇₁	D₄	1	8	yes	—
43	C₂×Q₈×C₇₁	—	2	4	no	—
44	C₂×Q₈×C₇₁	—	2	1	no	—
45	C₂×Q₈×C₇₁	—	2	1	no	—
46	C₂×Q₈×C₇₁	—	2	4	no	—
47	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
49	C₂×D₄×C₇₁	—	2	8	no	—
50	C₂×D₄×C₇₁	—	2	2	no	—
51	C₂×D₄×C₇₁	—	2	2	no	—
52	C₂×D₄×C₇₁	—	2	4	no	—
53	C₂×D₄×C₇₁	—	2	1	no	—
54	C₂×D₄×C₇₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₇₁	—	2	8	no	—
56	C₂²×C₄×C₇₁	—	2	1	no	—
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	2	4	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	8	no	—
69	SD₁₆×C₇₁	—	2	8	no	—
70	M₁₆×C₇₁	—	2	4	no	—
71	C₂×C₈×C₇₁	C₂³	1	4	yes	—
72	C₄⋊C₄×C₇₁	—	2	2	no	—
73	C₄⋊C₄×C₇₁	—	2	2	no	—
74	C₄⋊C₄×C₇₁	—	2	1	no	—
75	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₇₁	—	2	2	no	—
77	C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₂³×C₇₁	—	2	4	no	—
79	C₂³×C₇₁	—	2	2	no	—
80	C₂³×C₇₁	—	2	2	no	—
81	C₂³×C₇₁	—	2	2	no	—
82	C₂³×C₇₁	—	4	1	no	—
83	C₂³×C₇₁	—	4	2	no	—
84	C₂³×C₇₁	—	4	2	no	—
85	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₇₁	—	2	4	no	—
89	Q₈×C₇₁	—	2	2	no	—
90	Q₈×C₇₁	—	2	4	no	—
91	Q₈×C₇₁	—	4	2	no	—
92	Q₈×C₇₁	C₂×D₄	1	4	yes	—
93	D₄×C₇₁	—	2	8	no	—
94	D₄×C₇₁	—	2	4	no	—
95	D₄×C₇₁	—	2	8	no	—
96	D₄×C₇₁	—	2	2	no	—
97	D₄×C₇₁	C₂×D₄	1	4	yes	—
98	C₂×C₄×C₇₁	—	2	4	no	—
99	C₂×C₄×C₇₁	—	2	8	no	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	2	2	no	—
102	C₂×C₄×C₇₁	—	2	2	no	—
103	C₂×C₄×C₇₁	—	2	8	no	—
104	C₂×C₄×C₇₁	—	2	2	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	4	2	no	—
110	C₂×C₄×C₇₁	—	4	2	no	—
111	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₄	1	4	yes	—
115	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
116	C₈×C₇₁	—	2	4	no	—
117	C₂²×C₇₁	—	2	1	no	—
118	C₂²×C₇₁	—	2	1	no	—
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	—	2	1	no	—
121	C₂²×C₇₁	—	2	1	no	—
122	C₂²×C₇₁	—	2	4	no	—
123	C₂²×C₇₁	—	2	1	no	—
124	C₂²×C₇₁	—	2	1	no	—
125	C₂²×C₇₁	—	4	4	no	—
126	C₂²×C₇₁	—	4	8	no	—
127	C₂²×C₇₁	—	4	2	no	—
128	C₂²×C₇₁	—	4	2	no	—
129	C₂²×C₇₁	#120	1	2	yes	—
130	C₂²×C₇₁	#137	1	2	yes	complemented
131	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	—	2	4	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	4	2	no	—
136	C₄×C₇₁	—	4	2	no	—
137	C₄×C₇₁	#120	1	2	yes	—
138	C₄×C₇₁	C₂²×D₄	1	2	yes	—
139	C₂×C₇₁	—	2	2	no	—
140	C₂×C₇₁	—	2	2	no	—
141	C₂×C₇₁	—	4	1	no	—
142	C₂×C₇₁	—	4	4	no	—
143	C₂×C₇₁	C₂×#120	1	1	yes	characteristic
144	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
145	C₂×C₇₁	#575	1	1	yes	characteristic
146	#3007	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₇₁	#3007	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×#41	C₂×C₇₁	1	1	yes	characteristic
149	C₂×#137	C₂×C₇₁	1	2	yes	complemented
150	C₂×C₂²⋊Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
151	#452	C₂×C₇₁	1	4	yes	complemented
152	#451	C₂×C₇₁	1	2	yes	complemented
153	#450	C₂×C₇₁	1	2	yes	complemented
154	#417	C₂×C₇₁	1	2	yes	complemented
155	#407	C₂×C₇₁	1	1	yes	characteristic, complemented
156	C₂×#41	—	2	8	no	—
157	C₂×#41	—	2	1	no	—
158	C₂×#41	—	2	1	no	—
159	C₂×#41	C₂²×C₇₁	1	2	yes	—
160	C₂×#41	C₂²×C₇₁	1	2	yes	—
161	C₂²×Q₈	C₂²×C₇₁	1	1	yes	characteristic
162	C₂²×Q₈	C₂²×C₇₁	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	C₂²×C₇₁	1	1	yes	characteristic
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
167	#137	—	2	8	no	—
168	C₂×Q₁₆	C₂²×C₇₁	1	4	yes	complemented
169	C₂×SD₁₆	C₂²×C₇₁	1	4	yes	complemented
170	C₂×M₁₆	C₂²×C₇₁	1	2	yes	complemented
171	C₂²⋊Q₈	—	2	1	no	—
172	C₂²⋊Q₈	—	2	1	no	—
173	C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
174	C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	complemented
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic, complemented
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	#103	C₂²×C₇₁	1	4	yes	complemented
179	#102	C₂²×C₇₁	1	4	yes	complemented
180	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
181	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	complemented
182	C₂⁴	—	2	1	no	—
183	C₂⁴	D₄×C₇₁	1	1	yes	characteristic
184	#41	—	2	8	no	—
185	#41	—	2	8	no	—
186	#41	—	2	4	no	—
187	#41	—	2	4	no	—
188	#41	—	2	4	no	—
189	#41	D₄×C₇₁	1	8	yes	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₂³×C₇₁	1	2	yes	—
195	C₂×Q₈	C₂³×C₇₁	1	2	yes	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	C₂³×C₇₁	1	2	yes	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
210	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
211	C₂²×C₄	D₄×C₇₁	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	SD₁₆	—	2	8	no	—
217	M₁₆	—	2	4	no	—
218	C₂×C₈	C₂³×C₇₁	1	4	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
233	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
234	C₂³	C₂×D₄×C₇₁	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	4	no	—
238	Q₈	—	4	2	no	—
239	Q₈	C₂×D₄×C₇₁	1	4	yes	—
240	D₄	—	2	8	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	8	no	—
243	D₄	—	2	2	no	—
244	D₄	C₂×D₄×C₇₁	1	4	yes	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₇₁	1	4	yes	—
262	C₂×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
263	C₈	—	2	4	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	#120×C₇₁	1	2	yes	—
277	C₂²	#137×C₇₁	1	2	yes	complemented
278	C₂²	C₂²×D₄×C₇₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	4	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	#120×C₇₁	1	2	yes	—
285	C₄	C₂²×D₄×C₇₁	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	2	2	no	—
288	C₂	—	4	1	no	—
289	C₂	—	4	4	no	—
290	C₂	C₂×#120×C₇₁	1	1	yes	characteristic
291	C₂	C₂×#137×C₇₁	1	1	yes	characteristic
292	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	C₂²×#41×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×#137×C₇₁	C₂	semidirect
3. C₂×C₂²⋊Q₈×C₇₁	C₂	semidirect
4. #452×C₇₁	C₂	semidirect
5. #451×C₇₁	C₂	semidirect
6. #450×C₇₁	C₂	semidirect
7. #417×C₇₁	C₂	semidirect
8. #407×C₇₁	C₂	semidirect
15. C₂²×Q₈×C₇₁	C₂²	semidirect
21. C₂×Q₁₆×C₇₁	C₂²	semidirect
22. C₂×SD₁₆×C₇₁	C₂²	semidirect
23. C₂×M₁₆×C₇₁	C₂²	semidirect
27. C₂²⋊Q₈×C₇₁	C₂²	semidirect
29. C₂×C₄⋊C₄×C₇₁	C₂²	semidirect
31. #103×C₇₁	C₂²	semidirect
32. #102×C₇₁	C₂²	semidirect
34. C₂²⋊C₈×C₇₁	C₂²	semidirect
130. C₂²×C₇₁	#137	semidirect
146. #3007	C₇₁	direct (N × H)
147. C₇₁	#3007	direct (N × H)
149. C₂×#137	C₂×C₇₁	semidirect
150. C₂×C₂²⋊Q₈	C₂×C₇₁	semidirect
151. #452	C₂×C₇₁	semidirect
152. #451	C₂×C₇₁	semidirect
153. #450	C₂×C₇₁	semidirect
154. #417	C₂×C₇₁	semidirect
155. #407	C₂×C₇₁	semidirect
162. C₂²×Q₈	C₂²×C₇₁	semidirect
168. C₂×Q₁₆	C₂²×C₇₁	semidirect
169. C₂×SD₁₆	C₂²×C₇₁	semidirect
170. C₂×M₁₆	C₂²×C₇₁	semidirect
174. C₂²⋊Q₈	C₂²×C₇₁	semidirect
176. C₂×C₄⋊C₄	C₂²×C₇₁	semidirect
178. #103	C₂²×C₇₁	semidirect
179. #102	C₂²×C₇₁	semidirect
181. C₂²⋊C₈	C₂²×C₇₁	semidirect
277. 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
C₂²×#41×C₇₁	2
C₂×#137×C₇₁	2
C₂×C₂²⋊Q₈×C₇₁	2
#452×C₇₁	2
#451×C₇₁	2
#450×C₇₁	2
#417×C₇₁	2
#407×C₇₁	2
#3007	71

Subgroups — C71×#3007 #22585569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#3007 #22585569