Subgroups — C₁₃₁×#20814 #45706698

Order 33536.

906 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1843×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1832×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1797×C₁₃₁	C₂	1	4	yes	maximal
7	#1768×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²×C₁₃₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
10	#432×C₁₃₁	—	2	4	no	—
11	#431×C₁₃₁	—	2	4	no	—
12	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	8	yes	—
13	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
19	#376×C₁₃₁	C₄	1	2	yes	—
20	#376×C₁₃₁	C₄	1	2	yes	—
21	#376×C₁₃₁	C₂²	1	2	yes	—
22	#376×C₁₃₁	C₂²	1	1	yes	characteristic
23	#376×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×#95×C₁₃₁	C₄	1	2	yes	—
25	C₂×#95×C₁₃₁	C₄	1	2	yes	—
26	#336×C₁₃₁	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃₁	—	2	8	no	—
34	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₁	D₄	1	4	yes	—
38	C₂×C₄²×C₁₃₁	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
43	C₄⋊C₈×C₁₃₁	—	2	4	no	—
44	C₄⋊C₈×C₁₃₁	—	2	4	no	—
45	C₄⋊C₈×C₁₃₁	—	2	16	no	—
46	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
47	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
48	#97×C₁₃₁	—	2	4	no	—
49	C₄×C₈×C₁₃₁	—	2	4	no	—
50	#95×C₁₃₁	C₂×C₄	1	2	yes	—
51	#95×C₁₃₁	C₂×C₄	1	2	yes	—
52	#95×C₁₃₁	C₂×C₄	1	4	yes	—
53	#95×C₁₃₁	C₂×C₄	1	4	yes	—
54	#95×C₁₃₁	C₂×C₄	1	2	yes	—
55	#95×C₁₃₁	C₂×C₄	1	2	yes	—
56	C₂⁴×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	—	2	4	no	—
58	C₂²×C₄×C₁₃₁	—	2	4	no	—
59	C₂²×C₄×C₁₃₁	—	2	8	no	—
60	C₂²×C₄×C₁₃₁	—	2	8	no	—
61	C₂²×C₄×C₁₃₁	—	4	2	no	—
62	C₂²×C₄×C₁₃₁	—	4	4	no	—
63	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	—	2	4	no	—
76	C₂×C₈×C₁₃₁	—	2	4	no	—
77	C₂×C₈×C₁₃₁	—	2	8	no	—
78	C₂×C₈×C₁₃₁	—	2	8	no	—
79	C₄²×C₁₃₁	—	2	8	no	—
80	C₄²×C₁₃₁	—	2	4	no	—
81	C₄²×C₁₃₁	C₄⋊C₄	1	4	yes	—
82	C₄²×C₁₃₁	C₄⋊C₄	1	4	yes	—
83	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
84	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
85	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
86	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
87	C₄²×C₁₃₁	#41	1	8	yes	—
88	C₂³×C₁₃₁	—	2	2	no	—
89	C₂³×C₁₃₁	—	2	2	no	—
90	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₁₃₁	#131	1	4	yes	—
92	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	4	no	—
97	C₂×C₄×C₁₃₁	—	2	2	no	—
98	C₂×C₄×C₁₃₁	—	2	2	no	—
99	C₂×C₄×C₁₃₁	—	2	4	no	—
100	C₂×C₄×C₁₃₁	—	2	8	no	—
101	C₂×C₄×C₁₃₁	—	2	4	no	—
102	C₂×C₄×C₁₃₁	—	2	4	no	—
103	C₂×C₄×C₁₃₁	—	2	4	no	—
104	C₂×C₄×C₁₃₁	—	2	4	no	—
105	C₂×C₄×C₁₃₁	—	2	4	no	—
106	C₂×C₄×C₁₃₁	—	4	4	no	—
107	C₂×C₄×C₁₃₁	—	4	16	no	—
108	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃₁	#117	1	8	yes	—
114	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
117	C₈×C₁₃₁	—	4	4	no	—
118	C₈×C₁₃₁	—	4	4	no	—
119	C₂²×C₁₃₁	—	2	2	no	—
120	C₂²×C₁₃₁	—	2	8	no	—
121	C₂²×C₁₃₁	—	2	2	no	—
122	C₂²×C₁₃₁	#424	1	4	yes	—
123	C₂²×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₁₃₁	C₂×#117	1	2	yes	—
125	C₂²×C₁₃₁	C₂×#131	1	2	yes	—
126	C₂²×C₁₃₁	#568	1	2	yes	—
127	C₄×C₁₃₁	—	2	2	no	—
128	C₄×C₁₃₁	—	2	2	no	—
129	C₄×C₁₃₁	—	2	2	no	—
130	C₄×C₁₃₁	—	2	2	no	—
131	C₄×C₁₃₁	—	2	8	no	—
132	C₄×C₁₃₁	—	2	2	no	—
133	C₄×C₁₃₁	—	4	4	no	—
134	C₄×C₁₃₁	#532	1	2	yes	—
135	C₄×C₁₃₁	#533	1	2	yes	—
136	C₂×C₁₃₁	—	2	4	no	—
137	C₂×C₁₃₁	#2285	1	1	yes	characteristic
138	C₂×C₁₃₁	C₂×#424	1	1	yes	characteristic
139	C₂×C₁₃₁	#2905	1	1	yes	characteristic
140	C₂×C₁₃₁	#2924	1	4	yes	—
141	#20814	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₁	#20814	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#432	C₂×C₁₃₁	1	1	yes	characteristic
144	C₂×#431	C₂×C₁₃₁	1	1	yes	characteristic
145	C₂×#376	C₂×C₁₃₁	1	1	yes	characteristic
146	#1843	C₂×C₁₃₁	1	2	yes	complemented
147	#1832	C₂×C₁₃₁	1	2	yes	complemented
148	#1797	C₂×C₁₃₁	1	4	yes	—
149	#1768	C₂×C₁₃₁	1	4	yes	complemented
150	C₂²×C₄²	C₂²×C₁₃₁	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
152	#432	—	2	4	no	—
153	#431	—	2	4	no	—
154	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	8	yes	—
155	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
158	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
159	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
161	#376	C₄×C₁₃₁	1	2	yes	—
162	#376	C₄×C₁₃₁	1	2	yes	—
163	#376	C₂²×C₁₃₁	1	1	yes	characteristic
164	#376	C₂²×C₁₃₁	1	1	yes	characteristic
165	#376	C₂²×C₁₃₁	1	2	yes	—
166	C₂×#95	C₄×C₁₃₁	1	2	yes	—
167	C₂×#95	C₄×C₁₃₁	1	2	yes	—
168	#336	C₂²×C₁₃₁	1	8	yes	—
169	C₂³×C₄	C₂×C₄×C₁₃₁	1	4	yes	—
170	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
174	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
175	C₂×C₄²	—	2	8	no	—
176	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₁₃₁	1	4	yes	—
180	C₂×C₄²	Q₈×C₁₃₁	1	4	yes	—
181	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	16	no	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	#95	C₂×C₄×C₁₃₁	1	2	yes	—
193	#95	C₂×C₄×C₁₃₁	1	2	yes	—
194	#95	C₂×C₄×C₁₃₁	1	4	yes	—
195	#95	C₂×C₄×C₁₃₁	1	4	yes	—
196	#95	C₂×C₄×C₁₃₁	1	2	yes	—
197	#95	C₂×C₄×C₁₃₁	1	2	yes	—
198	C₂⁴	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
206	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	4	no	—
223	C₄²	C₄⋊C₄×C₁₃₁	1	4	yes	—
224	C₄²	C₄⋊C₄×C₁₃₁	1	4	yes	—
225	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
226	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
227	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
228	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
229	C₄²	#41×C₁₃₁	1	8	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
233	C₂³	#131×C₁₃₁	1	4	yes	—
234	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
251	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
255	C₂×C₄	#117×C₁₃₁	1	8	yes	—
256	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	4	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	#424×C₁₃₁	1	4	yes	—
265	C₂²	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
266	C₂²	C₂×#117×C₁₃₁	1	2	yes	—
267	C₂²	C₂×#131×C₁₃₁	1	2	yes	—
268	C₂²	#568×C₁₃₁	1	2	yes	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	8	no	—
275	C₄	—	4	4	no	—
276	C₄	#532×C₁₃₁	1	2	yes	—
277	C₄	#533×C₁₃₁	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	#2285×C₁₃₁	1	1	yes	characteristic
280	C₂	C₂×#424×C₁₃₁	1	1	yes	characteristic
281	C₂	#2905×C₁₃₁	1	1	yes	characteristic
282	C₂	#2924×C₁₃₁	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #1843×C₁₃₁	C₂	semidirect
5. #1832×C₁₃₁	C₂	semidirect
7. #1768×C₁₃₁	C₂	semidirect
141. #20814	C₁₃₁	direct (N × H)
142. C₁₃₁	#20814	direct (N × H)
146. #1843	C₂×C₁₃₁	semidirect
147. #1832	C₂×C₁₃₁	semidirect
149. #1768	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₃₁	2
C₂×#431×C₁₃₁	2
C₂×#376×C₁₃₁	2
#1843×C₁₃₁	2
#1832×C₁₃₁	2
#1797×C₁₃₁	2
#1768×C₁₃₁	2
#20814	131

Subgroups — C131×#20814 #45706698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20814 #45706698