Subgroups — C₇₁×#12557 #31833429

Order 18176.

630 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
6	C₂×#117×C₇₁	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₇₁	—	2	1	no	—
8	C₂×M₃₂×C₇₁	—	2	2	no	—
9	C₂×M₃₂×C₇₁	—	2	2	no	—
10	C₂×M₃₂×C₇₁	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
12	#427×C₇₁	—	2	2	no	—
13	#427×C₇₁	C₄	1	4	yes	complemented
14	C₂×#107×C₇₁	—	2	1	no	—
15	C₂×#107×C₇₁	—	2	2	no	—
16	C₂×#107×C₇₁	C₄	1	4	yes	complemented
17	C₂×#107×C₇₁	C₂²	1	2	yes	—
18	C₂×#107×C₇₁	C₂²	1	2	yes	—
19	C₂×#106×C₇₁	—	2	1	no	—
20	C₂³×C₄×C₇₁	—	4	1	no	—
21	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	D₄	1	2	yes	—
27	C₂²×C₈×C₇₁	D₄	1	2	yes	—
28	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
29	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	#117×C₇₁	—	2	4	no	—
31	#117×C₇₁	—	4	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
36	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
37	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
38	C₂×C₄²×C₇₁	—	4	1	no	—
39	M₃₂×C₇₁	—	2	8	no	—
40	C₂×C₁₆×C₇₁	—	2	2	no	—
41	C₂×C₁₆×C₇₁	—	2	2	no	—
42	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
43	#107×C₇₁	—	2	4	no	—
44	#107×C₇₁	—	2	2	no	—
45	#107×C₇₁	—	2	2	no	—
46	#107×C₇₁	C₂×C₄	1	4	yes	—
47	#107×C₇₁	C₂×C₄	1	4	yes	complemented
48	#106×C₇₁	—	2	4	no	—
49	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	—	4	4	no	—
51	C₂²×C₄×C₇₁	—	4	2	no	—
52	C₂²×C₄×C₇₁	—	4	2	no	—
53	C₂²×C₄×C₇₁	—	8	4	no	—
54	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₄²	1	4	yes	—
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
75	C₄⋊C₄×C₇₁	—	2	4	no	—
76	C₄⋊C₄×C₇₁	—	2	4	no	—
77	C₄⋊C₄×C₇₁	—	4	4	no	—
78	C₄⋊C₄×C₇₁	—	4	2	no	—
79	C₄⋊C₄×C₇₁	—	4	2	no	—
80	C₂²⋊C₄×C₇₁	—	4	4	no	—
81	C₄²×C₇₁	—	4	4	no	—
82	C₁₆×C₇₁	—	2	4	no	—
83	C₂³×C₇₁	—	2	2	no	—
84	C₂³×C₇₁	—	2	2	no	—
85	C₂³×C₇₁	#102	1	2	yes	—
86	C₂³×C₇₁	#102	1	2	yes	—
87	C₂³×C₇₁	#107	1	2	yes	—
88	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇₁	—	4	4	no	—
90	C₂×C₄×C₇₁	—	4	4	no	—
91	C₂×C₄×C₇₁	—	8	8	no	—
92	C₂×C₄×C₇₁	—	8	2	no	—
93	C₂×C₄×C₇₁	—	8	4	no	—
94	C₂×C₄×C₇₁	—	8	2	no	—
95	C₂×C₄×C₇₁	#95	1	4	yes	—
96	C₂×C₄×C₇₁	#102	1	2	yes	—
97	C₂×C₄×C₇₁	#102	1	2	yes	—
98	C₂×C₄×C₇₁	#103	1	4	yes	—
99	C₂×C₄×C₇₁	#103	1	4	yes	—
100	C₂×C₄×C₇₁	#106	1	4	yes	—
101	C₂×C₄×C₇₁	#107	1	2	yes	—
102	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
108	C₈×C₇₁	#95	1	4	yes	—
109	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
110	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₇₁	—	2	4	no	—
114	C₂²×C₇₁	—	2	2	no	—
115	C₂²×C₇₁	—	2	2	no	—
116	C₂²×C₇₁	—	2	4	no	—
117	C₂²×C₇₁	#340	1	4	yes	—
118	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
120	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
121	C₂²×C₇₁	#509	1	2	yes	—
122	C₂²×C₇₁	#510	1	2	yes	—
123	C₄×C₇₁	—	8	4	no	—
124	C₄×C₇₁	—	8	4	no	—
125	C₄×C₇₁	#340	1	4	yes	—
126	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
127	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
128	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
129	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
130	#12557	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₇₁	—	2	4	no	—
132	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
133	C₂×C₇₁	#2139	1	2	yes	—
134	C₂×C₇₁	#2140	1	1	yes	characteristic
135	C₂×C₇₁	#2141	1	1	yes	characteristic
136	C₂×C₇₁	#2159	1	2	yes	—
137	C₂²×M₃₂	C₂×C₇₁	1	1	yes	characteristic
138	C₂×#427	C₂×C₇₁	1	1	yes	characteristic
139	C₂²×#107	C₂×C₇₁	1	1	yes	characteristic
140	#1380	C₂×C₇₁	1	4	yes	complemented
141	C₇₁	#12557	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
143	C₂×#117	—	2	1	no	—
144	C₂²×C₄⋊C₄	—	2	1	no	—
145	C₂×M₃₂	—	2	2	no	—
146	C₂×M₃₂	—	2	2	no	—
147	C₂×M₃₂	C₄×C₇₁	1	4	yes	complemented
148	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
149	#427	—	2	2	no	—
150	#427	C₄×C₇₁	1	4	yes	complemented
151	C₂×#107	—	2	1	no	—
152	C₂×#107	—	2	2	no	—
153	C₂×#107	C₄×C₇₁	1	4	yes	complemented
154	C₂×#107	C₂²×C₇₁	1	2	yes	—
155	C₂×#107	C₂²×C₇₁	1	2	yes	—
156	C₂×#106	—	2	1	no	—
157	C₂³×C₄	—	4	1	no	—
158	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₇₁	1	2	yes	—
160	C₂²×C₈	C₂×C₄×C₇₁	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₇₁	1	2	yes	—
162	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₇₁	1	2	yes	—
164	C₂²×C₈	D₄×C₇₁	1	2	yes	—
165	C₂²×C₈	Q₈×C₇₁	1	2	yes	—
166	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
167	#117	—	2	4	no	—
168	#117	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₄⋊C₄	—	4	2	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₄²	—	4	1	no	—
176	M₃₂	—	2	8	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	C₂×C₄×C₇₁	1	4	yes	complemented
180	#107	—	2	4	no	—
181	#107	—	2	2	no	—
182	#107	—	2	2	no	—
183	#107	C₂×C₄×C₇₁	1	4	yes	—
184	#107	C₂×C₄×C₇₁	1	4	yes	complemented
185	#106	—	2	4	no	—
186	C₂⁴	D₈×C₇₁	1	1	yes	characteristic
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	8	4	no	—
191	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
193	C₂²×C₄	C₄⋊C₄×C₇₁	1	2	yes	—
194	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
195	C₂²×C₄	SD₁₆×C₇₁	1	2	yes	—
196	C₂²×C₄	SD₁₆×C₇₁	1	2	yes	—
197	C₂²×C₄	Q₁₆×C₇₁	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
199	C₂×C₈	C₄²×C₇₁	1	4	yes	—
200	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
201	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
202	C₂×C₈	C₂²⋊C₄×C₇₁	1	4	yes	—
203	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
204	C₂×C₈	C₄⋊C₄×C₇₁	1	4	yes	—
205	C₂×C₈	C₄⋊C₄×C₇₁	1	4	yes	—
206	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
211	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	4	no	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	#102×C₇₁	1	2	yes	—
223	C₂³	#102×C₇₁	1	2	yes	—
224	C₂³	#107×C₇₁	1	2	yes	—
225	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	8	8	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#95×C₇₁	1	4	yes	—
233	C₂×C₄	#102×C₇₁	1	2	yes	—
234	C₂×C₄	#102×C₇₁	1	2	yes	—
235	C₂×C₄	#103×C₇₁	1	4	yes	—
236	C₂×C₄	#103×C₇₁	1	4	yes	—
237	C₂×C₄	#106×C₇₁	1	4	yes	—
238	C₂×C₄	#107×C₇₁	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
242	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
244	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
245	C₈	#95×C₇₁	1	4	yes	—
246	C₈	C₂×C₄²×C₇₁	1	1	yes	characteristic
247	C₈	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
248	C₈	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
249	C₈	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#340×C₇₁	1	4	yes	—
255	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
256	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
257	C₂²	C₂×#107×C₇₁	1	1	yes	characteristic
258	C₂²	#509×C₇₁	1	2	yes	—
259	C₂²	#510×C₇₁	1	2	yes	—
260	C₄	—	8	4	no	—
261	C₄	—	8	4	no	—
262	C₄	#340×C₇₁	1	4	yes	—
263	C₄	C₂×#95×C₇₁	1	1	yes	characteristic
264	C₄	C₂×#103×C₇₁	1	1	yes	characteristic
265	C₄	C₂×#103×C₇₁	1	1	yes	characteristic
266	C₄	C₂×#106×C₇₁	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	C₂×#340×C₇₁	1	1	yes	characteristic
269	C₂	C₇₁×#2139	1	2	yes	—
270	C₂	C₇₁×#2140	1	1	yes	characteristic
271	C₂	C₇₁×#2141	1	1	yes	characteristic
272	C₂	C₇₁×#2159	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂×D₈
commutator	C₈	C₂×C₄²×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₂⁴×C₇₁	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ C₂³×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇₁×#1380	C₂	semidirect
10. C₂×M₃₂×C₇₁	C₄	semidirect
13. #427×C₇₁	C₄	semidirect
16. C₂×#107×C₇₁	C₄	semidirect
42. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
47. #107×C₇₁	C₂×C₄	semidirect
130. #12557	C₇₁	direct (N × H)
140. #1380	C₂×C₇₁	semidirect
141. C₇₁	#12557	direct (N × H)
147. C₂×M₃₂	C₄×C₇₁	semidirect
150. #427	C₄×C₇₁	semidirect
153. C₂×#107	C₄×C₇₁	semidirect
179. C₂×C₁₆	C₂×C₄×C₇₁	semidirect
184. #107	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
C₂²×M₃₂×C₇₁	2
C₂×#427×C₇₁	2
C₂²×#107×C₇₁	2
C₇₁×#1380	2
#12557	71

Subgroups — C71×#12557 #31833429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12557 #31833429