Subgroups — C₁₃₁×#8957 #45694841

Order 33536.

586 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2972×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1479×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1460×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1276×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×#124×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₁₃₁	—	2	4	no	—
10	#431×C₁₃₁	C₂²	1	2	yes	—
11	#431×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₃₁	—	2	2	no	—
13	#391×C₁₃₁	C₄	1	1	yes	characteristic
14	#390×C₁₃₁	C₄	1	1	yes	characteristic
15	#388×C₁₃₁	—	2	2	no	—
16	#382×C₁₃₁	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
18	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
19	Q₈⋊C₈×C₁₃₁	—	2	4	no	—
20	#323×C₁₃₁	—	2	2	no	—
21	C₂²×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₁	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	—	2	4	no	—
25	#124×C₁₃₁	—	2	2	no	—
26	#124×C₁₃₁	C₈	1	4	yes	complemented
27	C₄×Q₈×C₁₃₁	—	2	2	no	—
28	C₄×Q₈×C₁₃₁	—	2	2	no	—
29	C₄×D₄×C₁₃₁	—	2	2	no	—
30	C₄×D₄×C₁₃₁	—	2	2	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×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃₁	—	2	1	no	—
38	C₂×C₄²×C₁₃₁	—	2	1	no	—
39	C₂×C₄²×C₁₃₁	—	2	1	no	—
40	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₁₃₁	—	2	2	no	—
44	C₄⋊C₈×C₁₃₁	D₄	1	4	yes	—
45	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
46	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
47	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
48	C₄×C₈×C₁₃₁	—	2	2	no	—
49	C₄×C₈×C₁₃₁	—	2	2	no	—
50	#95×C₁₃₁	—	2	2	no	—
51	#95×C₁₃₁	—	2	2	no	—
52	C₂⁴×C₁₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂×Q₈×C₁₃₁	—	2	1	no	—
54	C₂×Q₈×C₁₃₁	—	2	2	no	—
55	C₂×Q₈×C₁₃₁	—	2	2	no	—
56	C₂×Q₈×C₁₃₁	C₂×C₈	1	2	yes	complemented
57	C₂×D₄×C₁₃₁	—	2	2	no	—
58	C₂×D₄×C₁₃₁	—	2	2	no	—
59	C₂×D₄×C₁₃₁	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₃₁	M₁₆	1	2	yes	—
61	C₂²×C₄×C₁₃₁	—	2	1	no	—
62	C₂²×C₄×C₁₃₁	—	2	1	no	—
63	C₂²×C₄×C₁₃₁	—	2	1	no	—
64	C₂²×C₄×C₁₃₁	—	2	2	no	—
65	C₂²×C₄×C₁₃₁	—	2	2	no	—
66	C₂²×C₄×C₁₃₁	—	2	1	no	—
67	C₂²×C₄×C₁₃₁	—	4	4	no	—
68	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃₁	—	2	4	no	—
73	C₂×C₈×C₁₃₁	—	4	2	no	—
74	C₂×C₈×C₁₃₁	—	8	4	no	—
75	C₄⋊C₄×C₁₃₁	—	2	2	no	—
76	C₄⋊C₄×C₁₃₁	—	2	2	no	—
77	C₄⋊C₄×C₁₃₁	—	4	2	no	—
78	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
79	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
80	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
81	C₄²×C₁₃₁	—	2	1	no	—
82	C₄²×C₁₃₁	—	2	2	no	—
83	C₄²×C₁₃₁	—	4	2	no	—
84	C₄²×C₁₃₁	—	4	2	no	—
85	C₄²×C₁₃₁	—	4	1	no	—
86	C₄²×C₁₃₁	—	4	1	no	—
87	C₄²×C₁₃₁	C₂×C₈	1	2	yes	complemented
88	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃₁	#41	1	2	yes	—
90	C₂³×C₁₃₁	—	2	2	no	—
91	C₂³×C₁₃₁	—	2	2	no	—
92	C₂³×C₁₃₁	—	4	2	no	—
93	C₂³×C₁₃₁	—	4	2	no	—
94	C₂³×C₁₃₁	C₂²⋊C₈	1	4	yes	—
95	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₁₃₁	—	2	4	no	—
97	Q₈×C₁₃₁	—	4	2	no	—
98	D₄×C₁₃₁	—	2	4	no	—
99	D₄×C₁₃₁	—	2	4	no	—
100	C₂×C₄×C₁₃₁	—	2	2	no	—
101	C₂×C₄×C₁₃₁	—	2	2	no	—
102	C₂×C₄×C₁₃₁	—	2	2	no	—
103	C₂×C₄×C₁₃₁	—	2	2	no	—
104	C₂×C₄×C₁₃₁	—	2	2	no	—
105	C₂×C₄×C₁₃₁	—	4	1	no	—
106	C₂×C₄×C₁₃₁	—	4	1	no	—
107	C₂×C₄×C₁₃₁	—	4	1	no	—
108	C₂×C₄×C₁₃₁	—	4	1	no	—
109	C₂×C₄×C₁₃₁	—	4	1	no	—
110	C₂×C₄×C₁₃₁	—	4	1	no	—
111	C₂×C₄×C₁₃₁	—	4	1	no	—
112	C₂×C₄×C₁₃₁	—	4	4	no	—
113	C₂×C₄×C₁₃₁	—	4	2	no	—
114	C₂×C₄×C₁₃₁	—	4	2	no	—
115	C₂×C₄×C₁₃₁	—	4	1	no	—
116	C₂×C₄×C₁₃₁	—	8	2	no	—
117	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	#121	1	2	yes	—
120	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃₁	#131	1	2	yes	—
124	C₈×C₁₃₁	—	4	4	no	—
125	C₈×C₁₃₁	—	8	2	no	—
126	C₂²×C₁₃₁	—	2	1	no	—
127	C₂²×C₁₃₁	—	2	1	no	—
128	C₂²×C₁₃₁	—	2	4	no	—
129	C₂²×C₁₃₁	—	4	4	no	—
130	C₂²×C₁₃₁	—	4	2	no	—
131	C₂²×C₁₃₁	—	4	2	no	—
132	C₂²×C₁₃₁	—	8	2	no	—
133	C₂²×C₁₃₁	#386	1	1	yes	characteristic
134	C₂²×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₃₁	#408	1	1	yes	characteristic
136	C₄×C₁₃₁	—	2	2	no	—
137	C₄×C₁₃₁	—	2	1	no	—
138	C₄×C₁₃₁	—	4	2	no	—
139	C₄×C₁₃₁	—	4	2	no	—
140	C₄×C₁₃₁	—	4	2	no	—
141	C₄×C₁₃₁	—	4	2	no	—
142	C₄×C₁₃₁	—	8	1	no	—
143	C₄×C₁₃₁	C₈×D₄	1	2	yes	—
144	C₄×C₁₃₁	#435	1	2	yes	—
145	C₄×C₁₃₁	#455	1	2	yes	—
146	C₂×C₁₃₁	—	2	1	no	—
147	C₂×C₁₃₁	—	2	1	no	—
148	C₂×C₁₃₁	—	4	4	no	—
149	C₂×C₁₃₁	#1869	1	1	yes	characteristic
150	C₂×C₁₃₁	#1884	1	1	yes	characteristic
151	C₂×C₁₃₁	#1885	1	1	yes	characteristic
152	#8957	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₃₁	#8957	1	1	yes	characteristic, direct product factor, Sylow
154	#2972	C₂×C₁₃₁	1	1	yes	characteristic
155	C₄×#124	C₂×C₁₃₁	1	1	yes	characteristic
156	#1479	C₂×C₁₃₁	1	2	yes	complemented
157	#1460	C₂×C₁₃₁	1	1	yes	characteristic
158	#1276	C₂×C₁₃₁	1	2	yes	complemented
159	C₂×#124	C₄×C₁₃₁	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
162	C₈×D₄	—	2	4	no	—
163	#431	C₂²×C₁₃₁	1	2	yes	—
164	#431	C₂²×C₁₃₁	1	2	yes	—
165	C₂×C₂²⋊C₈	—	2	2	no	—
166	#391	C₄×C₁₃₁	1	1	yes	characteristic
167	#390	C₄×C₁₃₁	1	1	yes	characteristic
168	#388	—	2	2	no	—
169	#382	C₄×C₁₃₁	1	1	yes	characteristic
170	C₄×C₂²⋊C₄	—	2	2	no	—
171	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
172	Q₈⋊C₈	—	2	4	no	—
173	#323	—	2	2	no	—
174	C₂²×Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
175	C₂²×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₁₃₁	1	2	yes	—
177	C₂²×C₈	—	2	4	no	—
178	#124	—	2	2	no	—
179	#124	C₈×C₁₃₁	1	4	yes	complemented
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	D₄×C₁₃₁	1	4	yes	—
198	C₂²⋊C₈	—	2	4	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊C₄×C₁₃₁	1	2	yes	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂×C₈×C₁₃₁	1	2	yes	complemented
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂×C₈×C₁₃₁	1	2	yes	—
213	C₂×D₄	M₁₆×C₁₃₁	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	8	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂×C₈×C₁₃₁	1	2	yes	complemented
241	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
242	C₄²	#41×C₁₃₁	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	C₂²⋊C₈×C₁₃₁	1	4	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
271	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄	#121×C₁₃₁	1	2	yes	—
273	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
276	C₂×C₄	#131×C₁₃₁	1	2	yes	—
277	C₈	—	4	4	no	—
278	C₈	—	8	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#386×C₁₃₁	1	1	yes	characteristic
287	C₂²	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
288	C₂²	#408×C₁₃₁	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	1	no	—
296	C₄	C₈×D₄×C₁₃₁	1	2	yes	—
297	C₄	#435×C₁₃₁	1	2	yes	—
298	C₄	#455×C₁₃₁	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	4	no	—
302	C₂	#1869×C₁₃₁	1	1	yes	characteristic
303	C₂	#1884×C₁₃₁	1	1	yes	characteristic
304	C₂	#1885×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	#120
commutator	C₂×C₄	C₂²×C₈×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#386
prime radical	C₂²×D₄×C₁₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1479×C₁₃₁	C₂	semidirect
5. #1276×C₁₃₁	C₂	semidirect
26. #124×C₁₃₁	C₈	semidirect
56. C₂×Q₈×C₁₃₁	C₂×C₈	semidirect
87. C₄²×C₁₃₁	C₂×C₈	semidirect
152. #8957	C₁₃₁	direct (N × H)
153. C₁₃₁	#8957	direct (N × H)
156. #1479	C₂×C₁₃₁	semidirect
158. #1276	C₂×C₁₃₁	semidirect
179. #124	C₈×C₁₃₁	semidirect
209. C₂×Q₈	C₂×C₈×C₁₃₁	semidirect
240. C₄²	C₂×C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2972×C₁₃₁	2
C₄×#124×C₁₃₁	2
#1479×C₁₃₁	2
#1460×C₁₃₁	2
#1276×C₁₃₁	2
#8957	131

Subgroups — C131×#8957 #45694841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8957 #45694841