Subgroups — C₁₃₁×#11459 #45697343

Order 33536.

386 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2162×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1322×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1314×C₁₃₁	C₂	1	2	yes	maximal
5	C₂²×C₄²×C₁₃₁	C₄	1	1	yes	characteristic
6	#431×C₁₃₁	—	2	2	no	—
7	#431×C₁₃₁	—	2	2	no	—
8	C₂×C₄⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₃₁	—	2	2	no	—
15	C₂²⋊C₁₆×C₁₃₁	C₂²	1	4	yes	—
16	C₄×C₁₆×C₁₃₁	—	2	2	no	—
17	C₂³×C₄×C₁₃₁	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃₁	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃₁	—	2	1	no	—
25	C₂×C₄²×C₁₃₁	—	2	1	no	—
26	C₂×C₄²×C₁₃₁	—	2	1	no	—
27	C₂×C₄²×C₁₃₁	—	2	1	no	—
28	C₂×C₄²×C₁₃₁	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₃₁	—	2	4	no	—
33	C₄⋊C₈×C₁₃₁	—	2	2	no	—
34	C₄⋊C₈×C₁₃₁	—	2	2	no	—
35	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
36	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
37	C₄×C₈×C₁₃₁	—	2	1	no	—
38	C₄×C₈×C₁₃₁	—	2	1	no	—
39	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
40	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
41	C₂⁴×C₁₃₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	—	2	1	no	—
43	C₂²×C₄×C₁₃₁	—	2	1	no	—
44	C₂²×C₄×C₁₃₁	—	2	2	no	—
45	C₂²×C₄×C₁₃₁	—	2	1	no	—
46	C₂²×C₄×C₁₃₁	—	2	1	no	—
47	C₂²×C₄×C₁₃₁	—	2	1	no	—
48	C₂²×C₄×C₁₃₁	—	2	1	no	—
49	C₂²×C₄×C₁₃₁	—	4	2	no	—
50	C₂²×C₄×C₁₃₁	—	4	1	no	—
51	C₂²×C₄×C₁₃₁	—	4	1	no	—
52	C₂²×C₄×C₁₃₁	C₁₆	1	4	yes	complemented
53	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	—	2	1	no	—
60	C₂×C₈×C₁₃₁	—	2	1	no	—
61	C₂×C₈×C₁₃₁	—	4	1	no	—
62	C₂×C₈×C₁₃₁	—	4	1	no	—
63	C₂×C₈×C₁₃₁	—	4	1	no	—
64	C₂×C₈×C₁₃₁	—	4	1	no	—
65	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
67	C₄²×C₁₃₁	—	2	1	no	—
68	C₄²×C₁₃₁	—	2	1	no	—
69	C₄²×C₁₃₁	—	2	1	no	—
70	C₄²×C₁₃₁	—	2	1	no	—
71	C₄²×C₁₃₁	—	2	1	no	—
72	C₄²×C₁₃₁	—	2	1	no	—
73	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
76	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
77	C₁₆×C₁₃₁	—	4	4	no	—
78	C₂³×C₁₃₁	—	2	1	no	—
79	C₂³×C₁₃₁	—	2	1	no	—
80	C₂³×C₁₃₁	—	4	1	no	—
81	C₂³×C₁₃₁	—	4	1	no	—
82	C₂³×C₁₃₁	C₂×C₁₆	1	2	yes	—
83	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	—	2	1	no	—
85	C₂×C₄×C₁₃₁	—	2	1	no	—
86	C₂×C₄×C₁₃₁	—	2	4	no	—
87	C₂×C₄×C₁₃₁	—	2	4	no	—
88	C₂×C₄×C₁₃₁	—	2	2	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	—	4	2	no	—
94	C₂×C₄×C₁₃₁	—	4	2	no	—
95	C₂×C₄×C₁₃₁	—	4	2	no	—
96	C₂×C₄×C₁₃₁	—	4	1	no	—
97	C₂×C₄×C₁₃₁	—	4	1	no	—
98	C₂×C₄×C₁₃₁	—	4	1	no	—
99	C₂×C₄×C₁₃₁	—	4	1	no	—
100	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₂×C₁₆	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₂×C₁₆	1	2	yes	—
106	C₂×C₄×C₁₃₁	M₃₂	1	2	yes	—
107	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
109	C₈×C₁₃₁	—	2	2	no	—
110	C₈×C₁₃₁	—	2	2	no	—
111	C₈×C₁₃₁	—	4	1	no	—
112	C₈×C₁₃₁	—	4	1	no	—
113	C₂²×C₁₃₁	—	2	1	no	—
114	C₂²×C₁₃₁	—	2	1	no	—
115	C₂²×C₁₃₁	—	2	2	no	—
116	C₂²×C₁₃₁	—	4	2	no	—
117	C₂²×C₁₃₁	—	4	2	no	—
118	C₂²×C₁₃₁	—	4	1	no	—
119	C₂²×C₁₃₁	—	4	1	no	—
120	C₂²×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₁₃₁	C₂²×C₁₆	1	1	yes	characteristic
122	C₂²×C₁₃₁	C₂×M₃₂	1	1	yes	characteristic
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	2	1	no	—
125	C₄×C₁₃₁	—	2	1	no	—
126	C₄×C₁₃₁	—	4	1	no	—
127	C₄×C₁₃₁	—	4	1	no	—
128	C₄×C₁₃₁	C₂²⋊C₁₆	1	2	yes	—
129	C₄×C₁₃₁	C₂²⋊C₁₆	1	2	yes	—
130	C₄×C₁₃₁	#349	1	2	yes	—
131	C₄×C₁₃₁	#410	1	1	yes	characteristic
132	C₄×C₁₃₁	#413	1	1	yes	characteristic
133	C₂×C₁₃₁	—	2	1	no	—
134	C₂×C₁₃₁	—	2	1	no	—
135	C₂×C₁₃₁	—	4	2	no	—
136	C₂×C₁₃₁	#1458	1	1	yes	characteristic
137	C₂×C₁₃₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
138	C₂×C₁₃₁	C₂×#349	1	1	yes	characteristic
139	#11459	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₁	#11459	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#431	C₂×C₁₃₁	1	1	yes	characteristic
142	#2162	C₂×C₁₃₁	1	2	yes	complemented
143	#1322	C₂×C₁₃₁	1	2	yes	complemented
144	#1314	C₂×C₁₃₁	1	2	yes	—
145	C₂²×C₄²	C₄×C₁₃₁	1	1	yes	characteristic
146	#431	—	2	2	no	—
147	#431	—	2	2	no	—
148	C₂×C₄⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
154	C₄⋊C₁₆	—	2	2	no	—
155	C₂²⋊C₁₆	C₂²×C₁₃₁	1	4	yes	—
156	C₄×C₁₆	—	2	2	no	—
157	C₂³×C₄	C₈×C₁₃₁	1	1	yes	characteristic
158	C₂³×C₄	C₈×C₁₃₁	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₈×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄²	C₈×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
172	C₂×C₁₆	—	2	4	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
180	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
181	C₂⁴	C₂×C₈×C₁₃₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	C₁₆×C₁₃₁	1	4	yes	complemented
193	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
214	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
215	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
216	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
217	C₁₆	—	4	4	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	1	no	—
222	C₂³	C₂×C₁₆×C₁₃₁	1	2	yes	—
223	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₁₆×C₁₃₁	1	2	yes	—
245	C₂×C₄	C₂×C₁₆×C₁₃₁	1	2	yes	—
246	C₂×C₄	M₃₂×C₁₃₁	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
261	C₂²	C₂²×C₁₆×C₁₃₁	1	1	yes	characteristic
262	C₂²	C₂×M₃₂×C₁₃₁	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂²⋊C₁₆×C₁₃₁	1	2	yes	—
269	C₄	C₂²⋊C₁₆×C₁₃₁	1	2	yes	—
270	C₄	#349×C₁₃₁	1	2	yes	—
271	C₄	#410×C₁₃₁	1	1	yes	characteristic
272	C₄	#413×C₁₃₁	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	4	2	no	—
276	C₂	#1458×C₁₃₁	1	1	yes	characteristic
277	C₂	C₂×C₂²⋊C₁₆×C₁₃₁	1	1	yes	characteristic
278	C₂	C₂×#349×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₃₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2162×C₁₃₁	C₂	semidirect
3. #1322×C₁₃₁	C₂	semidirect
52. C₂²×C₄×C₁₃₁	C₁₆	semidirect
139. #11459	C₁₃₁	direct (N × H)
140. C₁₃₁	#11459	direct (N × H)
142. #2162	C₂×C₁₃₁	semidirect
143. #1322	C₂×C₁₃₁	semidirect
192. C₂²×C₄	C₁₆×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₃₁	2
#2162×C₁₃₁	2
#1322×C₁₃₁	2
#1314×C₁₃₁	2
#11459	131

Subgroups — C131×#11459 #45697343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#11459 #45697343