Subgroups — C₁₃₁×#12811 #45698695

Order 33536.

510 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	C₂×C₄×D₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#357×C₁₃₁	C₂	1	2	yes	maximal
4	#1845×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1380×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₁	—	2	1	no	—
8	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
9	C₄×D₈×C₁₃₁	—	2	2	no	—
10	C₄×D₈×C₁₃₁	C₄	1	4	yes	complemented
11	C₂×#107×C₁₃₁	—	2	1	no	—
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×#102×C₁₃₁	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#384×C₁₃₁	—	2	1	no	—
17	C₄⋊C₁₆×C₁₃₁	—	2	2	no	—
18	#357×C₁₃₁	—	2	4	no	—
19	C₂²×D₄×C₁₃₁	—	2	1	no	—
20	C₂³×C₄×C₁₃₁	—	4	1	no	—
21	C₂×D₈×C₁₃₁	—	2	1	no	—
22	C₂×D₈×C₁₃₁	—	2	1	no	—
23	C₂×D₈×C₁₃₁	—	2	1	no	—
24	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×D₈×C₁₃₁	D₄	1	2	yes	—
26	C₂×D₈×C₁₃₁	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₁₃₁	—	2	4	no	—
30	C₄×D₄×C₁₃₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
35	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₃₁	—	2	2	no	—
38	C₂×C₁₆×C₁₃₁	—	2	2	no	—
39	C₂×C₁₆×C₁₃₁	—	2	2	no	—
40	#107×C₁₃₁	—	2	1	no	—
41	#107×C₁₃₁	—	2	2	no	—
42	#107×C₁₃₁	—	2	2	no	—
43	#107×C₁₃₁	—	4	2	no	—
44	#107×C₁₃₁	C₂×C₄	1	2	yes	—
45	#102×C₁₃₁	—	2	4	no	—
46	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
48	C₂⁴×C₁₃₁	—	4	1	no	—
49	C₂×D₄×C₁₃₁	—	2	2	no	—
50	C₂×D₄×C₁₃₁	—	2	2	no	—
51	C₂×D₄×C₁₃₁	—	2	2	no	—
52	C₂×D₄×C₁₃₁	—	4	1	no	—
53	C₂×D₄×C₁₃₁	—	4	1	no	—
54	C₂×D₄×C₁₃₁	—	4	1	no	—
55	C₂²×C₄×C₁₃₁	—	4	1	no	—
56	C₂²×C₄×C₁₃₁	—	4	4	no	—
57	C₂²×C₄×C₁₃₁	—	4	2	no	—
58	C₂²×C₄×C₁₃₁	—	8	2	no	—
59	C₂²×C₄×C₁₃₁	—	8	2	no	—
60	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	D₈×C₁₃₁	—	2	2	no	—
64	D₈×C₁₃₁	—	2	2	no	—
65	D₈×C₁₃₁	—	2	2	no	—
66	D₈×C₁₃₁	C₄⋊C₄	1	4	yes	complemented
67	C₂×C₈×C₁₃₁	—	2	1	no	—
68	C₂×C₈×C₁₃₁	—	2	1	no	—
69	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₁	—	2	2	no	—
77	C₄⋊C₄×C₁₃₁	—	4	1	no	—
78	C₄⋊C₄×C₁₃₁	—	4	2	no	—
79	C₄⋊C₄×C₁₃₁	—	4	2	no	—
80	C₄⋊C₄×C₁₃₁	—	8	2	no	—
81	C₄⋊C₄×C₁₃₁	—	8	2	no	—
82	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
83	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
85	C₁₆×C₁₃₁	—	4	2	no	—
86	C₂³×C₁₃₁	—	4	2	no	—
87	C₂³×C₁₃₁	—	4	2	no	—
88	C₂³×C₁₃₁	—	4	2	no	—
89	C₂³×C₁₃₁	—	8	1	no	—
90	C₂³×C₁₃₁	—	8	1	no	—
91	C₂³×C₁₃₁	—	8	1	no	—
92	C₂³×C₁₃₁	—	8	1	no	—
93	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
94	D₄×C₁₃₁	—	2	4	no	—
95	D₄×C₁₃₁	—	4	2	no	—
96	D₄×C₁₃₁	—	4	2	no	—
97	D₄×C₁₃₁	—	4	2	no	—
98	C₂×C₄×C₁₃₁	—	2	1	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	4	2	no	—
101	C₂×C₄×C₁₃₁	—	8	1	no	—
102	C₂×C₄×C₁₃₁	—	8	1	no	—
103	C₂×C₄×C₁₃₁	—	8	8	no	—
104	C₂×C₄×C₁₃₁	—	8	2	no	—
105	C₂×C₄×C₁₃₁	—	8	2	no	—
106	C₂×C₄×C₁₃₁	—	8	2	no	—
107	C₂×C₄×C₁₃₁	#102	1	2	yes	—
108	C₂×C₄×C₁₃₁	#102	1	2	yes	—
109	C₂×C₄×C₁₃₁	D₁₆	1	2	yes	—
110	C₂×C₄×C₁₃₁	SD₃₂	1	2	yes	—
111	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
117	C₈×C₁₃₁	—	2	2	no	—
118	C₈×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₁₃₁	#117	1	1	yes	characteristic
120	C₈×C₁₃₁	#121	1	1	yes	characteristic
121	C₈×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₁₃₁	—	4	4	no	—
123	C₂²×C₁₃₁	—	8	2	no	—
124	C₂²×C₁₃₁	—	8	2	no	—
125	C₂²×C₁₃₁	—	8	2	no	—
126	C₂²×C₁₃₁	—	8	2	no	—
127	C₂²×C₁₃₁	—	8	2	no	—
128	C₂²×C₁₃₁	—	8	2	no	—
129	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₃₁	#459	1	1	yes	characteristic
131	C₂²×C₁₃₁	#474	1	1	yes	characteristic
132	C₂²×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
133	C₂²×C₁₃₁	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×C₁₃₁	#509	1	1	yes	characteristic
135	C₂²×C₁₃₁	#510	1	1	yes	characteristic
136	C₄×C₁₃₁	—	2	2	no	—
137	C₄×C₁₃₁	—	8	2	no	—
138	C₄×C₁₃₁	—	16	2	no	—
139	C₄×C₁₃₁	#357	1	4	yes	complemented
140	C₄×C₁₃₁	#380	1	1	yes	characteristic
141	C₄×C₁₃₁	#417	1	1	yes	characteristic
142	C₄×C₁₃₁	#461	1	1	yes	characteristic
143	C₄×C₁₃₁	#476	1	1	yes	characteristic
144	C₂×C₁₃₁	—	8	4	no	—
145	C₂×C₁₃₁	#1802	1	1	yes	characteristic
146	C₂×C₁₃₁	C₂×#357	1	1	yes	characteristic
147	C₂×C₁₃₁	#2139	1	1	yes	characteristic
148	C₂×C₁₃₁	#2206	1	1	yes	characteristic
149	C₂×C₁₃₁	#2208	1	1	yes	characteristic
150	C₂×C₁₃₁	#2224	1	1	yes	characteristic
151	C₂×C₁₃₁	#2226	1	1	yes	characteristic
152	#12811	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₃₁	#12811	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×C₄×D₈	C₂×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
156	C₂×#357	C₂×C₁₃₁	1	2	yes	—
157	#1845	C₂×C₁₃₁	1	1	yes	characteristic, complemented
158	#1380	C₂×C₁₃₁	1	2	yes	complemented
159	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
160	C₂×C₄×D₄	—	2	1	no	—
161	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
162	C₄×D₈	—	2	2	no	—
163	C₄×D₈	C₄×C₁₃₁	1	4	yes	complemented
164	C₂×#107	—	2	1	no	—
165	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
166	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
167	C₂×#102	—	2	1	no	—
168	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
169	#384	—	2	1	no	—
170	C₄⋊C₁₆	—	2	2	no	—
171	#357	—	2	4	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
178	C₂×D₈	D₄×C₁₃₁	1	2	yes	—
179	C₂×D₈	Q₈×C₁₃₁	1	2	yes	—
180	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	2	2	no	—
193	#107	—	2	1	no	—
194	#107	—	2	2	no	—
195	#107	—	2	2	no	—
196	#107	—	4	2	no	—
197	#107	C₂×C₄×C₁₃₁	1	2	yes	—
198	#102	—	2	4	no	—
199	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
200	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
201	C₂⁴	—	4	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	8	2	no	—
212	C₂²×C₄	—	8	2	no	—
213	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
214	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
216	D₈	—	2	2	no	—
217	D₈	—	2	2	no	—
218	D₈	—	2	2	no	—
219	D₈	C₄⋊C₄×C₁₃₁	1	4	yes	complemented
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
226	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
227	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
228	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	8	2	no	—
234	C₄⋊C₄	—	8	2	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
237	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
238	C₁₆	—	4	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	—	8	1	no	—
246	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	#102×C₁₃₁	1	2	yes	—
261	C₂×C₄	#102×C₁₃₁	1	2	yes	—
262	C₂×C₄	D₁₆×C₁₃₁	1	2	yes	—
263	C₂×C₄	SD₃₂×C₁₃₁	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
265	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
268	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
269	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
272	C₈	#117×C₁₃₁	1	1	yes	characteristic
273	C₈	#121×C₁₃₁	1	1	yes	characteristic
274	C₈	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
275	C₂²	—	4	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
283	C₂²	#459×C₁₃₁	1	1	yes	characteristic
284	C₂²	#474×C₁₃₁	1	1	yes	characteristic
285	C₂²	C₂×D₁₆×C₁₃₁	1	1	yes	characteristic
286	C₂²	C₂×SD₃₂×C₁₃₁	1	1	yes	characteristic
287	C₂²	#509×C₁₃₁	1	1	yes	characteristic
288	C₂²	#510×C₁₃₁	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	16	2	no	—
292	C₄	#357×C₁₃₁	1	4	yes	complemented
293	C₄	#380×C₁₃₁	1	1	yes	characteristic
294	C₄	#417×C₁₃₁	1	1	yes	characteristic
295	C₄	#461×C₁₃₁	1	1	yes	characteristic
296	C₄	#476×C₁₃₁	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	#1802×C₁₃₁	1	1	yes	characteristic
299	C₂	C₂×#357×C₁₃₁	1	1	yes	characteristic
300	C₂	#2139×C₁₃₁	1	1	yes	characteristic
301	C₂	#2206×C₁₃₁	1	1	yes	characteristic
302	C₂	#2208×C₁₃₁	1	1	yes	characteristic
303	C₂	#2224×C₁₃₁	1	1	yes	characteristic
304	C₂	#2226×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₃₁	C₂	semidirect
4. #1845×C₁₃₁	C₂	semidirect
5. #1380×C₁₃₁	C₂	semidirect
10. C₄×D₈×C₁₃₁	C₄	semidirect
66. D₈×C₁₃₁	C₄⋊C₄	semidirect
139. C₄×C₁₃₁	#357	semidirect
152. #12811	C₁₃₁	direct (N × H)
153. C₁₃₁	#12811	direct (N × H)
155. C₂×C₄⋊C₁₆	C₂×C₁₃₁	semidirect
157. #1845	C₂×C₁₃₁	semidirect
158. #1380	C₂×C₁₃₁	semidirect
163. C₄×D₈	C₄×C₁₃₁	semidirect
219. D₈	C₄⋊C₄×C₁₃₁	semidirect
292. C₄	#357×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
C₂×C₄⋊C₁₆×C₁₃₁	2
C₂×#357×C₁₃₁	2
#1845×C₁₃₁	2
#1380×C₁₃₁	2
#12811	131

Subgroups — C131×#12811 #45698695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#12811 #45698695