Subgroups — C₁₃₁×#8145 #45694029

Order 33536.

678 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#322×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1273×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
9	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₃₁	—	2	4	no	—
11	C₂×C₄×C₈×C₁₃₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#336×C₁₃₁	C₄	1	2	yes	—
17	#336×C₁₃₁	C₄	1	2	yes	—
18	#322×C₁₃₁	—	2	4	no	—
19	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	—	2	4	no	—
23	C₂²×C₈×C₁₃₁	—	2	4	no	—
24	C₂²×C₈×C₁₃₁	—	2	4	no	—
25	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	—	2	2	no	—
30	C₂×C₄²×C₁₃₁	—	2	2	no	—
31	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
36	C₂²⋊C₈×C₁₃₁	C₈	1	8	yes	complemented
37	C₄×C₈×C₁₃₁	—	2	4	no	—
38	C₄×C₈×C₁₃₁	—	2	2	no	—
39	C₄×C₈×C₁₃₁	—	2	2	no	—
40	C₄×C₈×C₁₃₁	—	2	2	no	—
41	C₄×C₈×C₁₃₁	—	2	2	no	—
42	C₄×C₈×C₁₃₁	—	2	2	no	—
43	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
44	C₂⁴×C₁₃₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	—	2	1	no	—
46	C₂²×C₄×C₁₃₁	—	2	1	no	—
47	C₂²×C₄×C₁₃₁	—	2	2	no	—
48	C₂²×C₄×C₁₃₁	—	2	2	no	—
49	C₂²×C₄×C₁₃₁	—	2	2	no	—
50	C₂²×C₄×C₁₃₁	—	2	2	no	—
51	C₂²×C₄×C₁₃₁	—	2	2	no	—
52	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₁₃₁	M₁₆	1	4	yes	—
57	C₂²×C₄×C₁₃₁	M₁₆	1	4	yes	—
58	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	—	2	16	no	—
62	C₂×C₈×C₁₃₁	—	2	2	no	—
63	C₂×C₈×C₁₃₁	—	2	4	no	—
64	C₂×C₈×C₁₃₁	—	2	2	no	—
65	C₂×C₈×C₁₃₁	—	2	2	no	—
66	C₂×C₈×C₁₃₁	—	2	2	no	—
67	C₂×C₈×C₁₃₁	—	2	4	no	—
68	C₂×C₈×C₁₃₁	—	2	4	no	—
69	C₂×C₈×C₁₃₁	—	2	4	no	—
70	C₂×C₈×C₁₃₁	—	2	4	no	—
71	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
72	C₂×C₈×C₁₃₁	C₂×C₈	1	8	yes	complemented
73	C₂×C₈×C₁₃₁	M₁₆	1	4	yes	—
74	C₄²×C₁₃₁	—	2	1	no	—
75	C₄²×C₁₃₁	—	2	1	no	—
76	C₄²×C₁₃₁	—	2	2	no	—
77	C₄²×C₁₃₁	—	2	2	no	—
78	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃₁	#41	1	2	yes	—
80	C₂³×C₁₃₁	—	2	1	no	—
81	C₂³×C₁₃₁	—	2	1	no	—
82	C₂³×C₁₃₁	—	2	1	no	—
83	C₂³×C₁₃₁	—	2	1	no	—
84	C₂³×C₁₃₁	C₄×C₈	1	2	yes	—
85	C₂³×C₁₃₁	#97	1	2	yes	—
86	C₂³×C₁₃₁	#97	1	2	yes	—
87	C₂³×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	—	2	1	no	—
90	C₂×C₄×C₁₃₁	—	2	1	no	—
91	C₂×C₄×C₁₃₁	—	2	1	no	—
92	C₂×C₄×C₁₃₁	—	2	1	no	—
93	C₂×C₄×C₁₃₁	—	2	1	no	—
94	C₂×C₄×C₁₃₁	—	2	4	no	—
95	C₂×C₄×C₁₃₁	—	2	4	no	—
96	C₂×C₄×C₁₃₁	—	2	8	no	—
97	C₂×C₄×C₁₃₁	—	2	8	no	—
98	C₂×C₄×C₁₃₁	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₁₃₁	#97	1	2	yes	—
100	C₂×C₄×C₁₃₁	#97	1	2	yes	—
101	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃₁	#131	1	2	yes	—
110	C₂×C₄×C₁₃₁	#131	1	2	yes	—
111	C₈×C₁₃₁	—	2	4	no	—
112	C₈×C₁₃₁	—	2	4	no	—
113	C₈×C₁₃₁	—	2	8	no	—
114	C₈×C₁₃₁	—	4	4	no	—
115	C₂²×C₁₃₁	—	2	2	no	—
116	C₂²×C₁₃₁	—	2	2	no	—
117	C₂²×C₁₃₁	—	2	2	no	—
118	C₂²×C₁₃₁	—	2	2	no	—
119	C₂²×C₁₃₁	—	2	2	no	—
120	C₂²×C₁₃₁	—	2	2	no	—
121	C₂²×C₁₃₁	#322	1	4	yes	complemented
122	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
124	C₂²×C₁₃₁	C₂×#97	1	1	yes	characteristic
125	C₂²×C₁₃₁	C₂×#97	1	1	yes	characteristic
126	C₂²×C₁₃₁	C₄×M₁₆	1	1	yes	characteristic
127	C₂²×C₁₃₁	#405	1	1	yes	characteristic
128	C₂²×C₁₃₁	#405	1	1	yes	characteristic
129	C₄×C₁₃₁	—	2	2	no	—
130	C₄×C₁₃₁	—	2	2	no	—
131	C₄×C₁₃₁	—	2	2	no	—
132	C₄×C₁₃₁	#407	1	2	yes	—
133	C₄×C₁₃₁	#432	1	2	yes	—
134	C₄×C₁₃₁	C₈×D₄	1	4	yes	—
135	C₄×C₁₃₁	#435	1	4	yes	—
136	C₂×C₁₃₁	—	2	4	no	—
137	C₂×C₁₃₁	C₂×#322	1	1	yes	characteristic
138	C₂×C₁₃₁	C₈×M₁₆	1	1	yes	characteristic
139	C₂×C₁₃₁	#1451	1	1	yes	characteristic
140	C₂×C₁₃₁	#1452	1	1	yes	characteristic
141	C₂×C₁₃₁	#1749	1	1	yes	characteristic
142	C₂×C₁₃₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₁₃₁	#1752	1	1	yes	characteristic
144	#8145	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₁	#8145	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×C₄×C₈	C₂×C₁₃₁	1	1	yes	characteristic
147	C₄×C₂²⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
148	C₄×C₂²⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
149	C₂×#322	C₂×C₁₃₁	1	2	yes	complemented
150	#1273	C₂×C₁₃₁	1	2	yes	complemented
151	C₂³×C₈	C₄×C₁₃₁	1	2	yes	—
152	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
154	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
155	C₂×C₄×C₈	—	2	4	no	—
156	C₂×C₄×C₈	C₄×C₁₃₁	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
161	#336	C₄×C₁₃₁	1	2	yes	—
162	#336	C₄×C₁₃₁	1	2	yes	—
163	#322	—	2	4	no	—
164	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
171	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
172	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
173	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
180	C₂²⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	C₈×C₁₃₁	1	8	yes	complemented
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
189	C₂⁴	C₄²×C₁₃₁	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₈×C₁₃₁	1	4	yes	—
201	C₂²×C₄	M₁₆×C₁₃₁	1	4	yes	—
202	C₂²×C₄	M₁₆×C₁₃₁	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
206	C₂×C₈	—	2	16	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
217	C₂×C₈	C₂×C₈×C₁₃₁	1	8	yes	complemented
218	C₂×C₈	M₁₆×C₁₃₁	1	4	yes	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
224	C₄²	#41×C₁₃₁	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₄×C₈×C₁₃₁	1	2	yes	—
230	C₂³	#97×C₁₃₁	1	2	yes	—
231	C₂³	#97×C₁₃₁	1	2	yes	—
232	C₂³	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	C₄×C₈×C₁₃₁	1	2	yes	—
244	C₂×C₄	#97×C₁₃₁	1	2	yes	—
245	C₂×C₄	#97×C₁₃₁	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
247	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
249	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
250	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
251	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
252	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
253	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
254	C₂×C₄	#131×C₁₃₁	1	2	yes	—
255	C₂×C₄	#131×C₁₃₁	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	2	4	no	—
258	C₈	—	2	8	no	—
259	C₈	—	4	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	#322×C₁₃₁	1	4	yes	complemented
267	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
268	C₂²	C₂×C₄×C₈×C₁₃₁	1	1	yes	characteristic
269	C₂²	C₂×#97×C₁₃₁	1	1	yes	characteristic
270	C₂²	C₂×#97×C₁₃₁	1	1	yes	characteristic
271	C₂²	C₄×M₁₆×C₁₃₁	1	1	yes	characteristic
272	C₂²	#405×C₁₃₁	1	1	yes	characteristic
273	C₂²	#405×C₁₃₁	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	#407×C₁₃₁	1	2	yes	—
278	C₄	#432×C₁₃₁	1	2	yes	—
279	C₄	C₈×D₄×C₁₃₁	1	4	yes	—
280	C₄	#435×C₁₃₁	1	4	yes	—
281	C₂	—	2	4	no	—
282	C₂	C₂×#322×C₁₃₁	1	1	yes	characteristic
283	C₂	C₈×M₁₆×C₁₃₁	1	1	yes	characteristic
284	C₂	#1451×C₁₃₁	1	1	yes	characteristic
285	C₂	#1452×C₁₃₁	1	1	yes	characteristic
286	C₂	#1749×C₁₃₁	1	1	yes	characteristic
287	C₂	C₈×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
288	C₂	#1752×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₁₃₁	C₂³
commutator	C₂²	C₂×C₄×C₈×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂³×C₁₃₁	C₂×C₄²
prime radical	C₂⁴×C₁₃₁	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#322×C₁₃₁	C₂	semidirect
5. #1273×C₁₃₁	C₂	semidirect
36. C₂²⋊C₈×C₁₃₁	C₈	semidirect
72. C₂×C₈×C₁₃₁	C₂×C₈	semidirect
121. C₂²×C₁₃₁	#322	semidirect
144. #8145	C₁₃₁	direct (N × H)
145. C₁₃₁	#8145	direct (N × H)
149. C₂×#322	C₂×C₁₃₁	semidirect
150. #1273	C₂×C₁₃₁	semidirect
181. C₂²⋊C₈	C₈×C₁₃₁	semidirect
217. C₂×C₈	C₂×C₈×C₁₃₁	semidirect
266. C₂²	#322×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃₁	2
C₄×C₂²⋊C₈×C₁₃₁	2
C₂×#322×C₁₃₁	2
#1273×C₁₃₁	2
#8145	131

Subgroups — C131×#8145 #45694029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8145 #45694029