Subgroups — C₇₁×#60647 #31881519

Order 18176.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₇₁	C₂	1	4	yes	maximal, complemented
2	C₇₁×#3498	C₂	1	2	yes	maximal, complemented
3	C₇₁×#3155	C₂	1	12	yes	maximal, complemented
4	C₇₁×#3151	C₂	1	6	yes	maximal, complemented
5	C₇₁×#3147	C₂	1	3	yes	maximal, complemented
6	C₂×#493×C₇₁	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₇₁	C₂²	1	4	yes	—
9	#576×C₇₁	—	2	32	no	—
10	C₂×AGL(1,ℤ/8)×C₇₁	C₂²	1	24	yes	complemented
11	C₂×#135×C₇₁	C₂²	1	12	yes	complemented
12	C₂²×D₈×C₇₁	C₂²	1	12	yes	complemented
13	C₂×#131×C₇₁	C₂²	1	2	yes	—
14	#550×C₇₁	—	2	4	no	—
15	D₄²×C₇₁	—	2	12	no	—
16	#535×C₇₁	C₂²	1	12	yes	complemented
17	#532×C₇₁	C₂²	1	6	yes	complemented
18	C₂×#127×C₇₁	C₂²	1	6	yes	complemented
19	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
20	#496×C₇₁	C₂²	1	24	yes	complemented
21	#495×C₇₁	C₂²	1	12	yes	complemented
22	#494×C₇₁	C₂²	1	1	yes	characteristic, complemented
23	#493×C₇₁	C₂²	1	3	yes	complemented
24	#493×C₇₁	C₂²	1	6	yes	complemented
25	#493×C₇₁	C₂²	1	12	yes	complemented
26	#492×C₇₁	C₂²	1	6	yes	complemented
27	#405×C₇₁	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₇₁	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	complemented
30	H(5,𝔽₂)×C₇₁	—	2	32	no	—
31	C₂×#41×C₇₁	—	2	4	no	—
32	C₂×#41×C₇₁	C₂³	1	12	yes	—
33	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
34	C₂²×D₄×C₇₁	—	2	12	no	—
35	C₂²×D₄×C₇₁	—	2	12	no	—
36	C₂²×D₄×C₇₁	C₂³	1	12	yes	—
37	AGL(1,ℤ/8)×C₇₁	—	2	96	no	—
38	#135×C₇₁	—	2	48	no	—
39	C₂×Q₁₆×C₇₁	C₂³	1	4	yes	—
40	C₂×SD₁₆×C₇₁	C₂³	1	24	yes	—
41	C₂×D₈×C₇₁	—	2	48	no	—
42	C₂×D₈×C₇₁	C₂³	1	12	yes	—
43	C₂×D₈×C₇₁	C₂³	1	24	yes	—
44	#131×C₇₁	D₄	1	16	yes	complemented
45	C₂×M₁₆×C₇₁	C₂³	1	6	yes	—
46	C₂²×C₈×C₇₁	C₂³	1	6	yes	—
47	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
48	#127×C₇₁	—	2	12	no	—
49	#127×C₇₁	C₂³	1	6	yes	—
50	#127×C₇₁	C₂³	1	12	yes	—
51	#124×C₇₁	C₂³	1	12	yes	—
52	#121×C₇₁	—	2	12	no	—
53	#121×C₇₁	—	2	24	no	—
54	#120×C₇₁	—	2	24	no	—
55	C₄×Q₈×C₇₁	D₄	1	2	yes	—
56	C₄×D₄×C₇₁	—	2	12	no	—
57	C₄×D₄×C₇₁	D₄	1	6	yes	—
58	#117×C₇₁	C₂³	1	3	yes	—
59	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
60	#97×C₇₁	C₂³	1	6	yes	—
61	C₄×C₈×C₇₁	C₂³	1	6	yes	—
62	C₄×C₈×C₇₁	C₂³	1	3	yes	—
63	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
64	C₂⁴×C₇₁	—	2	12	no	—
65	#41×C₇₁	—	2	48	no	—
66	#41×C₇₁	—	4	16	no	—
67	#41×C₇₁	C₂×D₄	1	8	yes	—
68	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₂×Q₈×C₇₁	C₂⁴	1	4	yes	—
70	C₂×D₄×C₇₁	—	2	12	no	—
71	C₂×D₄×C₇₁	—	2	48	no	—
72	C₂×D₄×C₇₁	—	2	12	no	—
73	C₂×D₄×C₇₁	—	2	12	no	—
74	C₂×D₄×C₇₁	—	2	24	no	—
75	C₂×D₄×C₇₁	—	4	48	no	—
76	C₂×D₄×C₇₁	—	8	24	no	—
77	C₂×D₄×C₇₁	C₂×D₄	1	3	yes	—
78	C₂×D₄×C₇₁	C₂⁴	1	12	yes	—
79	C₂²×C₄×C₇₁	—	2	12	no	—
80	C₂²×C₄×C₇₁	C₂×D₄	1	6	yes	—
81	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
82	Q₁₆×C₇₁	—	2	8	no	—
83	SD₁₆×C₇₁	—	2	48	no	—
84	D₈×C₇₁	—	2	24	no	—
85	D₈×C₇₁	—	2	48	no	—
86	M₁₆×C₇₁	C₂×D₄	1	24	yes	complemented
87	C₂×C₈×C₇₁	C₂×D₄	1	24	yes	complemented
88	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
89	C₂×C₈×C₇₁	C₂⁴	1	6	yes	—
90	C₄⋊C₄×C₇₁	—	2	4	no	—
91	C₄⋊C₄×C₇₁	C₂×D₄	1	6	yes	—
92	C₂²⋊C₄×C₇₁	—	2	24	no	—
93	C₂²⋊C₄×C₇₁	C₂×D₄	1	6	yes	—
94	C₄²×C₇₁	C₂×D₄	1	6	yes	—
95	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
96	C₄²×C₇₁	C₂⁴	1	3	yes	—
97	C₂³×C₇₁	—	2	12	no	—
98	C₂³×C₇₁	—	4	24	no	—
99	C₂³×C₇₁	—	8	24	no	—
100	C₂³×C₇₁	C₂²×D₄	1	3	yes	—
101	Q₈×C₇₁	—	2	8	no	—
102	Q₈×C₇₁	#127	1	4	yes	—
103	D₄×C₇₁	—	2	24	no	—
104	D₄×C₇₁	—	4	24	no	—
105	D₄×C₇₁	—	4	24	no	—
106	D₄×C₇₁	—	4	24	no	—
107	D₄×C₇₁	—	8	24	no	—
108	D₄×C₇₁	#127	1	12	yes	—
109	C₂×C₄×C₇₁	—	2	12	no	—
110	C₂×C₄×C₇₁	—	2	4	no	—
111	C₂×C₄×C₇₁	—	4	24	no	—
112	C₂×C₄×C₇₁	C₂²×D₄	1	6	yes	—
113	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₇₁	C₂²×D₄	1	12	yes	—
115	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
117	C₈×C₇₁	C₂²×D₄	1	4	yes	—
118	C₈×C₇₁	C₂²×D₄	1	12	yes	—
119	C₂²×C₇₁	—	2	6	no	—
120	C₂²×C₇₁	—	4	12	no	—
121	C₂²×C₇₁	—	4	24	no	—
122	C₂²×C₇₁	—	16	24	no	—
123	C₂²×C₇₁	C₂×#127	1	6	yes	—
124	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₇₁	—	2	6	no	—
126	C₄×C₇₁	—	4	4	no	—
127	C₄×C₇₁	C₂×#127	1	6	yes	—
128	C₄×C₇₁	#576	1	4	yes	complemented
129	C₄×C₇₁	C₂³×D₄	1	2	yes	—
130	#60647	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₇₁	—	2	6	no	—
132	C₂×C₇₁	—	8	12	no	—
133	C₂×C₇₁	C₂²×#127	1	1	yes	characteristic
134	C₂×C₇₁	C₂×#576	1	2	yes	—
135	C₂×#576	C₂×C₇₁	1	4	yes	complemented
136	#3498	C₂×C₇₁	1	2	yes	complemented
137	#3155	C₂×C₇₁	1	12	yes	complemented
138	#3151	C₂×C₇₁	1	6	yes	complemented
139	#3147	C₂×C₇₁	1	3	yes	complemented
140	C₂×#493	C₂×C₇₁	1	3	yes	complemented
141	C₄×#131	C₂×C₇₁	1	1	yes	characteristic, complemented
142	C₇₁	#60647	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×H(5,𝔽₂)	C₂²×C₇₁	1	4	yes	—
144	#576	—	2	32	no	—
145	C₂×AGL(1,ℤ/8)	C₂²×C₇₁	1	24	yes	complemented
146	C₂×#135	C₂²×C₇₁	1	12	yes	complemented
147	C₂²×D₈	C₂²×C₇₁	1	12	yes	complemented
148	C₂×#131	C₂²×C₇₁	1	2	yes	—
149	#550	—	2	4	no	—
150	D₄²	—	2	12	no	—
151	#535	C₂²×C₇₁	1	12	yes	complemented
152	#532	C₂²×C₇₁	1	6	yes	complemented
153	C₂×#127	C₂²×C₇₁	1	6	yes	complemented
154	C₄×#41	C₂²×C₇₁	1	1	yes	characteristic
155	#496	C₂²×C₇₁	1	24	yes	complemented
156	#495	C₂²×C₇₁	1	12	yes	complemented
157	#494	C₂²×C₇₁	1	1	yes	characteristic, complemented
158	#493	C₂²×C₇₁	1	3	yes	complemented
159	#493	C₂²×C₇₁	1	6	yes	complemented
160	#493	C₂²×C₇₁	1	12	yes	complemented
161	#492	C₂²×C₇₁	1	6	yes	complemented
162	#405	C₂²×C₇₁	1	6	yes	complemented
163	C₄×M₁₆	C₂²×C₇₁	1	3	yes	complemented
164	C₂×C₄×C₈	C₂²×C₇₁	1	3	yes	complemented
165	H(5,𝔽₂)	—	2	32	no	—
166	C₂×#41	—	2	4	no	—
167	C₂×#41	C₂³×C₇₁	1	12	yes	—
168	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
169	C₂²×D₄	—	2	12	no	—
170	C₂²×D₄	—	2	12	no	—
171	C₂²×D₄	C₂³×C₇₁	1	12	yes	—
172	AGL(1,ℤ/8)	—	2	96	no	—
173	#135	—	2	48	no	—
174	C₂×Q₁₆	C₂³×C₇₁	1	4	yes	—
175	C₂×SD₁₆	C₂³×C₇₁	1	24	yes	—
176	C₂×D₈	—	2	48	no	—
177	C₂×D₈	C₂³×C₇₁	1	12	yes	—
178	C₂×D₈	C₂³×C₇₁	1	24	yes	—
179	#131	D₄×C₇₁	1	16	yes	complemented
180	C₂×M₁₆	C₂³×C₇₁	1	6	yes	—
181	C₂²×C₈	C₂³×C₇₁	1	6	yes	—
182	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
183	#127	—	2	12	no	—
184	#127	C₂³×C₇₁	1	6	yes	—
185	#127	C₂³×C₇₁	1	12	yes	—
186	#124	C₂³×C₇₁	1	12	yes	—
187	#121	—	2	12	no	—
188	#121	—	2	24	no	—
189	#120	—	2	24	no	—
190	C₄×Q₈	D₄×C₇₁	1	2	yes	—
191	C₄×D₄	—	2	12	no	—
192	C₄×D₄	D₄×C₇₁	1	6	yes	—
193	#117	C₂³×C₇₁	1	3	yes	—
194	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
195	#97	C₂³×C₇₁	1	6	yes	—
196	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₇₁	1	3	yes	—
198	C₄×C₈	C₂³×C₇₁	1	6	yes	—
199	C₂⁴	—	2	12	no	—
200	#41	—	2	48	no	—
201	#41	—	4	16	no	—
202	#41	C₂×D₄×C₇₁	1	8	yes	—
203	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
204	C₂×Q₈	C₂⁴×C₇₁	1	4	yes	—
205	C₂×D₄	—	2	12	no	—
206	C₂×D₄	—	2	48	no	—
207	C₂×D₄	—	2	12	no	—
208	C₂×D₄	—	2	12	no	—
209	C₂×D₄	—	2	24	no	—
210	C₂×D₄	—	4	48	no	—
211	C₂×D₄	—	8	24	no	—
212	C₂×D₄	C₂×D₄×C₇₁	1	3	yes	—
213	C₂×D₄	C₂⁴×C₇₁	1	12	yes	—
214	C₂²×C₄	—	2	12	no	—
215	C₂²×C₄	C₂×D₄×C₇₁	1	6	yes	—
216	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
217	Q₁₆	—	2	8	no	—
218	SD₁₆	—	2	48	no	—
219	D₈	—	2	24	no	—
220	D₈	—	2	48	no	—
221	M₁₆	C₂×D₄×C₇₁	1	24	yes	complemented
222	C₂×C₈	C₂×D₄×C₇₁	1	24	yes	complemented
223	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
224	C₂×C₈	C₂⁴×C₇₁	1	6	yes	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	C₂×D₄×C₇₁	1	6	yes	—
227	C₂²⋊C₄	—	2	24	no	—
228	C₂²⋊C₄	C₂×D₄×C₇₁	1	6	yes	—
229	C₄²	C₂×D₄×C₇₁	1	6	yes	—
230	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
231	C₄²	C₂⁴×C₇₁	1	3	yes	—
232	C₂³	—	2	12	no	—
233	C₂³	—	4	24	no	—
234	C₂³	—	8	24	no	—
235	C₂³	C₂²×D₄×C₇₁	1	3	yes	—
236	Q₈	—	2	8	no	—
237	Q₈	#127×C₇₁	1	4	yes	—
238	D₄	—	2	24	no	—
239	D₄	—	4	24	no	—
240	D₄	—	4	24	no	—
241	D₄	—	4	24	no	—
242	D₄	—	8	24	no	—
243	D₄	#127×C₇₁	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	24	no	—
247	C₂×C₄	C₂²×D₄×C₇₁	1	6	yes	—
248	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
249	C₂×C₄	C₂²×D₄×C₇₁	1	12	yes	—
250	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
251	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
252	C₈	C₂²×D₄×C₇₁	1	4	yes	—
253	C₈	C₂²×D₄×C₇₁	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	4	24	no	—
256	C₂²	—	4	12	no	—
257	C₂²	—	16	24	no	—
258	C₂²	C₂×#127×C₇₁	1	6	yes	—
259	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	4	4	no	—
262	C₄	C₂×#127×C₇₁	1	6	yes	—
263	C₄	#576×C₇₁	1	4	yes	complemented
264	C₄	C₂³×D₄×C₇₁	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	8	12	no	—
267	C₂	C₂²×#127×C₇₁	1	1	yes	characteristic
268	C₂	C₂×#576×C₇₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄×#41×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#576×C₇₁	C₂	semidirect
2. C₇₁×#3498	C₂	semidirect
3. C₇₁×#3155	C₂	semidirect
4. C₇₁×#3151	C₂	semidirect
5. C₇₁×#3147	C₂	semidirect
6. C₂×#493×C₇₁	C₂	semidirect
7. C₄×#131×C₇₁	C₂	semidirect
10. C₂×AGL(1,ℤ/8)×C₇₁	C₂²	semidirect
11. C₂×#135×C₇₁	C₂²	semidirect
12. C₂²×D₈×C₇₁	C₂²	semidirect
16. #535×C₇₁	C₂²	semidirect
17. #532×C₇₁	C₂²	semidirect
18. C₂×#127×C₇₁	C₂²	semidirect
20. #496×C₇₁	C₂²	semidirect
21. #495×C₇₁	C₂²	semidirect
22. #494×C₇₁	C₂²	semidirect
23. #493×C₇₁	C₂²	semidirect
24. #493×C₇₁	C₂²	semidirect
25. #493×C₇₁	C₂²	semidirect
26. #492×C₇₁	C₂²	semidirect
27. #405×C₇₁	C₂²	semidirect
28. C₄×M₁₆×C₇₁	C₂²	semidirect
29. C₂×C₄×C₈×C₇₁	C₂²	semidirect
44. #131×C₇₁	D₄	semidirect
86. M₁₆×C₇₁	C₂×D₄	semidirect
87. C₂×C₈×C₇₁	C₂×D₄	semidirect
128. C₄×C₇₁	#576	semidirect
130. #60647	C₇₁	direct (N × H)
135. C₂×#576	C₂×C₇₁	semidirect
136. #3498	C₂×C₇₁	semidirect
137. #3155	C₂×C₇₁	semidirect
138. #3151	C₂×C₇₁	semidirect
139. #3147	C₂×C₇₁	semidirect
140. C₂×#493	C₂×C₇₁	semidirect
141. C₄×#131	C₂×C₇₁	semidirect
142. C₇₁	#60647	direct (N × H)
145. C₂×AGL(1,ℤ/8)	C₂²×C₇₁	semidirect
146. C₂×#135	C₂²×C₇₁	semidirect
147. C₂²×D₈	C₂²×C₇₁	semidirect
151. #535	C₂²×C₇₁	semidirect
152. #532	C₂²×C₇₁	semidirect
153. C₂×#127	C₂²×C₇₁	semidirect
155. #496	C₂²×C₇₁	semidirect
156. #495	C₂²×C₇₁	semidirect
157. #494	C₂²×C₇₁	semidirect
158. #493	C₂²×C₇₁	semidirect
159. #493	C₂²×C₇₁	semidirect
160. #493	C₂²×C₇₁	semidirect
161. #492	C₂²×C₇₁	semidirect
162. #405	C₂²×C₇₁	semidirect
163. C₄×M₁₆	C₂²×C₇₁	semidirect
164. C₂×C₄×C₈	C₂²×C₇₁	semidirect
179. #131	D₄×C₇₁	semidirect
221. M₁₆	C₂×D₄×C₇₁	semidirect
222. C₂×C₈	C₂×D₄×C₇₁	semidirect
263. C₄	#576×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₇₁	2
C₇₁×#3498	2
C₇₁×#3155	2
C₇₁×#3151	2
C₇₁×#3147	2
C₂×#493×C₇₁	2
C₄×#131×C₇₁	2
#60647	71

Subgroups — C71×#60647 #31881519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#60647 #31881519