Subgroups — C₁₃₁×#21282 #45707166

Order 33536.

862 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2319×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1970×C₁₃₁	C₂	1	2	yes	maximal
4	#1942×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1936×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1762×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1726×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
11	C₄×D₈×C₁₃₁	C₄	1	8	yes	complemented
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
15	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	#390×C₁₃₁	C₂²	1	4	yes	—
21	#389×C₁₃₁	C₂²	1	4	yes	—
22	#388×C₁₃₁	—	2	4	no	—
23	#381×C₁₃₁	—	2	2	no	—
24	#380×C₁₃₁	—	2	2	no	—
25	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
26	#376×C₁₃₁	C₂²	1	1	yes	characteristic
27	#340×C₁₃₁	C₂²	1	4	yes	—
28	#340×C₁₃₁	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃₁	—	2	2	no	—
31	C₂×D₈×C₁₃₁	—	2	4	no	—
32	C₂×D₈×C₁₃₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₃₁	—	2	4	no	—
37	C₄×D₄×C₁₃₁	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
44	C₂×C₄²×C₁₃₁	—	2	2	no	—
45	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
47	#107×C₁₃₁	—	2	4	no	—
48	#107×C₁₃₁	C₂×C₄	1	4	yes	—
49	#102×C₁₃₁	—	2	8	no	—
50	#102×C₁₃₁	C₂×C₄	1	8	yes	—
51	#97×C₁₃₁	—	2	4	no	—
52	#97×C₁₃₁	—	2	2	no	—
53	C₄×C₈×C₁₃₁	—	2	4	no	—
54	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
55	#95×C₁₃₁	—	2	4	no	—
56	#95×C₁₃₁	—	2	2	no	—
57	#95×C₁₃₁	D₄	1	2	yes	—
58	#95×C₁₃₁	D₄	1	1	yes	characteristic
59	#95×C₁₃₁	D₄	1	1	yes	characteristic
60	C₂⁴×C₁₃₁	—	2	2	no	—
61	C₂×D₄×C₁₃₁	—	2	4	no	—
62	C₂×D₄×C₁₃₁	—	2	4	no	—
63	C₂×D₄×C₁₃₁	—	2	2	no	—
64	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₁	—	2	2	no	—
66	C₂²×C₄×C₁₃₁	—	2	8	no	—
67	C₂²×C₄×C₁₃₁	—	2	4	no	—
68	C₂²×C₄×C₁₃₁	—	8	4	no	—
69	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
74	D₈×C₁₃₁	—	2	8	no	—
75	C₂×C₈×C₁₃₁	—	2	2	no	—
76	C₂×C₈×C₁₃₁	—	2	2	no	—
77	C₂×C₈×C₁₃₁	—	2	2	no	—
78	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃₁	#41	1	4	yes	—
81	C₂×C₈×C₁₃₁	#41	1	4	yes	—
82	C₄⋊C₄×C₁₃₁	—	2	2	no	—
83	C₄⋊C₄×C₁₃₁	—	2	4	no	—
84	C₄⋊C₄×C₁₃₁	—	2	4	no	—
85	C₄⋊C₄×C₁₃₁	—	8	4	no	—
86	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
88	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
89	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
90	C₄²×C₁₃₁	—	2	2	no	—
91	C₄²×C₁₃₁	—	2	2	no	—
92	C₄²×C₁₃₁	—	8	2	no	—
93	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃₁	#41	1	2	yes	—
95	C₂³×C₁₃₁	—	2	4	no	—
96	C₂³×C₁₃₁	—	4	4	no	—
97	C₂³×C₁₃₁	—	4	2	no	—
98	C₂³×C₁₃₁	—	4	4	no	—
99	C₂³×C₁₃₁	—	4	2	no	—
100	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
101	D₄×C₁₃₁	—	2	4	no	—
102	D₄×C₁₃₁	—	2	4	no	—
103	D₄×C₁₃₁	—	4	4	no	—
104	C₂×C₄×C₁₃₁	—	2	1	no	—
105	C₂×C₄×C₁₃₁	—	2	2	no	—
106	C₂×C₄×C₁₃₁	—	2	1	no	—
107	C₂×C₄×C₁₃₁	—	2	4	no	—
108	C₂×C₄×C₁₃₁	—	2	2	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	2	no	—
112	C₂×C₄×C₁₃₁	—	2	2	no	—
113	C₂×C₄×C₁₃₁	—	2	2	no	—
114	C₂×C₄×C₁₃₁	—	4	4	no	—
115	C₂×C₄×C₁₃₁	—	4	4	no	—
116	C₂×C₄×C₁₃₁	—	4	4	no	—
117	C₂×C₄×C₁₃₁	—	4	4	no	—
118	C₂×C₄×C₁₃₁	—	8	8	no	—
119	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
125	C₈×C₁₃₁	—	2	2	no	—
126	C₈×C₁₃₁	—	2	4	no	—
127	C₈×C₁₃₁	#117	1	4	yes	—
128	C₂²×C₁₃₁	—	4	4	no	—
129	C₂²×C₁₃₁	—	4	4	no	—
130	C₂²×C₁₃₁	—	4	4	no	—
131	C₂²×C₁₃₁	—	8	4	no	—
132	C₂²×C₁₃₁	—	8	4	no	—
133	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₁₃₁	#532	1	2	yes	—
135	C₂²×C₁₃₁	#576	1	2	yes	—
136	C₂²×C₁₃₁	#576	1	1	yes	characteristic
137	C₂²×C₁₃₁	#578	1	1	yes	characteristic
138	C₄×C₁₃₁	—	2	2	no	—
139	C₄×C₁₃₁	—	2	2	no	—
140	C₄×C₁₃₁	—	4	2	no	—
141	C₄×C₁₃₁	—	4	2	no	—
142	C₄×C₁₃₁	—	4	4	no	—
143	C₄×C₁₃₁	—	8	4	no	—
144	C₄×C₁₃₁	C₂×#117	1	1	yes	characteristic
145	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
146	C₄×C₁₃₁	#529	1	2	yes	—
147	C₂×C₁₃₁	—	8	4	no	—
148	C₂×C₁₃₁	#2293	1	1	yes	characteristic
149	C₂×C₁₃₁	#2948	1	1	yes	characteristic
150	C₂×C₁₃₁	#2951	1	1	yes	characteristic
151	C₂×C₁₃₁	#3175	1	2	yes	—
152	C₂×C₁₃₁	#3176	1	2	yes	—
153	#21282	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₁	#21282	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×C₄×D₈	C₂×C₁₃₁	1	1	yes	characteristic
156	#2319	C₂×C₁₃₁	1	2	yes	complemented
157	#1970	C₂×C₁₃₁	1	2	yes	—
158	#1942	C₂×C₁₃₁	1	2	yes	complemented
159	#1936	C₂×C₁₃₁	1	4	yes	complemented
160	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
161	#1762	C₂×C₁₃₁	1	2	yes	complemented
162	#1726	C₂×C₁₃₁	1	1	yes	characteristic, complemented
163	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
165	C₄×D₈	C₄×C₁₃₁	1	8	yes	complemented
166	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
167	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
168	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
169	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
170	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
171	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
174	#390	C₂²×C₁₃₁	1	4	yes	—
175	#389	C₂²×C₁₃₁	1	4	yes	—
176	#388	—	2	4	no	—
177	#381	—	2	2	no	—
178	#380	—	2	2	no	—
179	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
180	#376	C₂²×C₁₃₁	1	1	yes	characteristic
181	#340	C₂²×C₁₃₁	1	2	yes	—
182	#340	C₂²×C₁₃₁	1	4	yes	—
183	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	C₂×C₄×C₁₃₁	1	4	yes	—
187	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
190	C₄×D₄	—	2	4	no	—
191	C₄×D₄	C₂×C₄×C₁₃₁	1	8	yes	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₂²⋊C₄	—	2	4	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	C₂×C₄×C₁₃₁	1	4	yes	—
203	#102	—	2	8	no	—
204	#102	C₂×C₄×C₁₃₁	1	8	yes	—
205	#97	—	2	4	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	D₄×C₁₃₁	1	1	yes	characteristic
212	#95	D₄×C₁₃₁	1	1	yes	characteristic
213	#95	D₄×C₁₃₁	1	2	yes	—
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
228	D₈	—	2	8	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
234	C₂×C₈	#41×C₁₃₁	1	4	yes	—
235	C₂×C₈	#41×C₁₃₁	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	8	4	no	—
240	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	8	4	no	—
243	C₂²⋊C₄	—	8	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	8	2	no	—
247	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
248	C₄²	#41×C₁₃₁	1	2	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
275	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
277	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
278	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	#117×C₁₃₁	1	4	yes	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
288	C₂²	#532×C₁₃₁	1	2	yes	—
289	C₂²	#576×C₁₃₁	1	2	yes	—
290	C₂²	#576×C₁₃₁	1	1	yes	characteristic
291	C₂²	#578×C₁₃₁	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×#117×C₁₃₁	1	1	yes	characteristic
299	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
300	C₄	#529×C₁₃₁	1	2	yes	—
301	C₂	—	8	4	no	—
302	C₂	#2293×C₁₃₁	1	1	yes	characteristic
303	C₂	#2948×C₁₃₁	1	1	yes	characteristic
304	C₂	#2951×C₁₃₁	1	1	yes	characteristic
305	C₂	#3175×C₁₃₁	1	2	yes	—
306	C₂	#3176×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2319×C₁₃₁	C₂	semidirect
4. #1942×C₁₃₁	C₂	semidirect
5. #1936×C₁₃₁	C₂	semidirect
6. C₄×#107×C₁₃₁	C₂	semidirect
7. #1762×C₁₃₁	C₂	semidirect
8. #1726×C₁₃₁	C₂	semidirect
11. C₄×D₈×C₁₃₁	C₄	semidirect
153. #21282	C₁₃₁	direct (N × H)
154. C₁₃₁	#21282	direct (N × H)
156. #2319	C₂×C₁₃₁	semidirect
158. #1942	C₂×C₁₃₁	semidirect
159. #1936	C₂×C₁₃₁	semidirect
160. C₄×#107	C₂×C₁₃₁	semidirect
161. #1762	C₂×C₁₃₁	semidirect
162. #1726	C₂×C₁₃₁	semidirect
165. C₄×D₈	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₁₃₁	2
#2319×C₁₃₁	2
#1970×C₁₃₁	2
#1942×C₁₃₁	2
#1936×C₁₃₁	2
C₄×#107×C₁₃₁	2
#1762×C₁₃₁	2
#1726×C₁₃₁	2
#21282	131

Subgroups — C131×#21282 #45707166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#21282 #45707166