Subgroups — C₁₃₁×#8227 #45694111

Order 33536.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1745×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#330×C₁₃₁	C₂	1	2	yes	maximal
5	#1295×C₁₃₁	C₂	1	2	yes	maximal
6	C₂×#125×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×#97×C₁₃₁	—	2	1	no	—
8	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
9	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
11	#389×C₁₃₁	C₄	1	1	yes	characteristic
12	#389×C₁₃₁	C₄	1	1	yes	characteristic
13	#387×C₁₃₁	—	2	1	no	—
14	#387×C₁₃₁	—	2	1	no	—
15	#382×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
20	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
21	#336×C₁₃₁	—	2	1	no	—
22	#336×C₁₃₁	—	2	1	no	—
23	#330×C₁₃₁	C₄	1	8	yes	complemented
24	C₂²×C₈×C₁₃₁	—	2	2	no	—
25	C₂²×C₈×C₁₃₁	—	2	2	no	—
26	#125×C₁₃₁	—	2	2	no	—
27	#125×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₁	—	2	1	no	—
37	C₂×C₄²×C₁₃₁	—	2	1	no	—
38	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
45	#97×C₁₃₁	—	2	4	no	—
46	#97×C₁₃₁	—	2	2	no	—
47	#97×C₁₃₁	—	2	2	no	—
48	#97×C₁₃₁	C₂×C₄	1	4	yes	—
49	#97×C₁₃₁	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₁₃₁	—	2	4	no	—
51	#95×C₁₃₁	—	2	1	no	—
52	#95×C₁₃₁	—	2	1	no	—
53	#95×C₁₃₁	—	2	1	no	—
54	#95×C₁₃₁	—	2	1	no	—
55	#95×C₁₃₁	—	2	1	no	—
56	#95×C₁₃₁	—	2	1	no	—
57	C₂²×C₄×C₁₃₁	—	2	1	no	—
58	C₂²×C₄×C₁₃₁	—	2	1	no	—
59	C₂²×C₄×C₁₃₁	—	2	1	no	—
60	C₂²×C₄×C₁₃₁	—	2	1	no	—
61	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
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	—	2	4	no	—
72	C₂×C₈×C₁₃₁	—	4	2	no	—
73	C₂×C₈×C₁₃₁	—	4	2	no	—
74	C₄⋊C₄×C₁₃₁	—	2	1	no	—
75	C₄⋊C₄×C₁₃₁	—	2	1	no	—
76	C₄⋊C₄×C₁₃₁	—	2	4	no	—
77	C₄⋊C₄×C₁₃₁	—	2	4	no	—
78	C₄⋊C₄×C₁₃₁	—	4	1	no	—
79	C₄⋊C₄×C₁₃₁	—	4	1	no	—
80	C₄⋊C₄×C₁₃₁	—	4	2	no	—
81	C₄⋊C₄×C₁₃₁	—	4	2	no	—
82	C₄⋊C₄×C₁₃₁	C₄²	1	2	yes	—
83	C₄⋊C₄×C₁₃₁	C₄²	1	2	yes	—
84	C₄²×C₁₃₁	—	2	2	no	—
85	C₄²×C₁₃₁	—	2	2	no	—
86	C₄²×C₁₃₁	—	2	2	no	—
87	C₄²×C₁₃₁	—	4	1	no	—
88	C₄²×C₁₃₁	—	4	1	no	—
89	C₄²×C₁₃₁	—	4	1	no	—
90	C₄²×C₁₃₁	—	4	2	no	—
91	C₄²×C₁₃₁	—	4	2	no	—
92	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃₁	#41	1	2	yes	—
94	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	—	2	1	no	—
96	C₂×C₄×C₁₃₁	—	2	1	no	—
97	C₂×C₄×C₁₃₁	—	2	1	no	—
98	C₂×C₄×C₁₃₁	—	2	1	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	2	1	no	—
101	C₂×C₄×C₁₃₁	—	2	1	no	—
102	C₂×C₄×C₁₃₁	—	2	1	no	—
103	C₂×C₄×C₁₃₁	—	2	1	no	—
104	C₂×C₄×C₁₃₁	—	2	1	no	—
105	C₂×C₄×C₁₃₁	—	2	1	no	—
106	C₂×C₄×C₁₃₁	—	2	1	no	—
107	C₂×C₄×C₁₃₁	—	2	2	no	—
108	C₂×C₄×C₁₃₁	—	2	2	no	—
109	C₂×C₄×C₁₃₁	—	2	1	no	—
110	C₂×C₄×C₁₃₁	—	2	1	no	—
111	C₂×C₄×C₁₃₁	—	2	1	no	—
112	C₂×C₄×C₁₃₁	—	4	1	no	—
113	C₂×C₄×C₁₃₁	—	4	1	no	—
114	C₂×C₄×C₁₃₁	—	4	1	no	—
115	C₂×C₄×C₁₃₁	—	4	1	no	—
116	C₂×C₄×C₁₃₁	—	4	1	no	—
117	C₂×C₄×C₁₃₁	—	4	1	no	—
118	C₂×C₄×C₁₃₁	—	4	1	no	—
119	C₂×C₄×C₁₃₁	—	4	1	no	—
120	C₂×C₄×C₁₃₁	—	4	1	no	—
121	C₂×C₄×C₁₃₁	—	4	1	no	—
122	C₂×C₄×C₁₃₁	#101	1	2	yes	—
123	C₂×C₄×C₁₃₁	#104	1	2	yes	—
124	C₂×C₄×C₁₃₁	#104	1	2	yes	—
125	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
127	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
129	C₈×C₁₃₁	—	2	8	no	—
130	C₈×C₁₃₁	—	4	4	no	—
131	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₃₁	C₂×#101	1	1	yes	characteristic
133	C₂²×C₁₃₁	#413	1	1	yes	characteristic
134	C₂²×C₁₃₁	C₂×#104	1	1	yes	characteristic
135	C₂²×C₁₃₁	C₂×#104	1	1	yes	characteristic
136	C₂²×C₁₃₁	#421	1	1	yes	characteristic
137	C₂²×C₁₃₁	#421	1	1	yes	characteristic
138	C₄×C₁₃₁	—	2	1	no	—
139	C₄×C₁₃₁	—	2	1	no	—
140	C₄×C₁₃₁	—	2	2	no	—
141	C₄×C₁₃₁	—	2	2	no	—
142	C₄×C₁₃₁	—	2	2	no	—
143	C₄×C₁₃₁	—	2	2	no	—
144	C₄×C₁₃₁	—	2	2	no	—
145	C₄×C₁₃₁	—	4	2	no	—
146	C₄×C₁₃₁	—	4	2	no	—
147	C₄×C₁₃₁	—	4	1	no	—
148	C₄×C₁₃₁	—	8	1	no	—
149	C₄×C₁₃₁	—	8	1	no	—
150	C₂×C₁₃₁	#1528	1	1	yes	characteristic
151	C₂×C₁₃₁	#1529	1	1	yes	characteristic
152	C₂×C₁₃₁	#1531	1	1	yes	characteristic
153	C₂×C₁₃₁	#1532	1	1	yes	characteristic
154	C₂×C₁₃₁	C₄×#101	1	1	yes	characteristic
155	C₂×C₁₃₁	C₄×#104	1	1	yes	characteristic
156	C₂×C₁₃₁	#1759	1	1	yes	characteristic
157	#8227	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₁	#8227	1	1	yes	characteristic, direct product factor, Sylow
159	C₄×#125	C₂×C₁₃₁	1	1	yes	characteristic
160	#1745	C₂×C₁₃₁	1	1	yes	characteristic
161	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic
162	C₂×#330	C₂×C₁₃₁	1	2	yes	—
163	#1295	C₂×C₁₃₁	1	2	yes	—
164	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
165	C₂×#97	—	2	1	no	—
166	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
167	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
168	C₂×C₄×C₈	—	2	1	no	—
169	#389	C₄×C₁₃₁	1	1	yes	characteristic
170	#389	C₄×C₁₃₁	1	1	yes	characteristic
171	#387	—	2	1	no	—
172	#387	—	2	1	no	—
173	#382	C₂²×C₁₃₁	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	C₄×C₁₃₁	1	1	yes	characteristic
177	C₄×C₄⋊C₄	C₄×C₁₃₁	1	1	yes	characteristic
178	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
179	#336	—	2	1	no	—
180	#336	—	2	1	no	—
181	#330	C₄×C₁₃₁	1	8	yes	complemented
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	#125	—	2	2	no	—
185	#125	C₂×C₄×C₁₃₁	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
203	#97	—	2	4	no	—
204	#97	—	2	2	no	—
205	#97	—	2	2	no	—
206	#97	C₂×C₄×C₁₃₁	1	4	yes	—
207	#97	C₂×C₄×C₁₃₁	1	4	yes	—
208	C₄×C₈	—	2	4	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₄²×C₁₃₁	1	2	yes	—
241	C₄⋊C₄	C₄²×C₁₃₁	1	2	yes	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
251	C₄²	#41×C₁₃₁	1	2	yes	—
252	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	#101×C₁₃₁	1	2	yes	—
281	C₂×C₄	#104×C₁₃₁	1	2	yes	—
282	C₂×C₄	#104×C₁₃₁	1	2	yes	—
283	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
284	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
287	C₈	—	2	8	no	—
288	C₈	—	4	4	no	—
289	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
290	C₂²	C₂×#101×C₁₃₁	1	1	yes	characteristic
291	C₂²	#413×C₁₃₁	1	1	yes	characteristic
292	C₂²	C₂×#104×C₁₃₁	1	1	yes	characteristic
293	C₂²	C₂×#104×C₁₃₁	1	1	yes	characteristic
294	C₂²	#421×C₁₃₁	1	1	yes	characteristic
295	C₂²	#421×C₁₃₁	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#1528×C₁₃₁	1	1	yes	characteristic
309	C₂	#1529×C₁₃₁	1	1	yes	characteristic
310	C₂	#1531×C₁₃₁	1	1	yes	characteristic
311	C₂	#1532×C₁₃₁	1	1	yes	characteristic
312	C₂	C₄×#101×C₁₃₁	1	1	yes	characteristic
313	C₂	C₄×#104×C₁₃₁	1	1	yes	characteristic
314	C₂	#1759×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
23. #330×C₁₃₁	C₄	semidirect
157. #8227	C₁₃₁	direct (N × H)
158. C₁₃₁	#8227	direct (N × H)
181. #330	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₁₃₁	2
#1745×C₁₃₁	2
C₄×#97×C₁₃₁	2
C₂×#330×C₁₃₁	2
#1295×C₁₃₁	2
#8227	131

Subgroups — C131×#8227 #45694111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8227 #45694111