Subgroups — C₁₃₁×#13068 #45698952

Order 33536.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#361×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2140×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1814×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1389×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1385×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₁₃₁	—	2	1	no	—
9	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
11	#504×C₁₃₁	—	2	2	no	—
12	#504×C₁₃₁	—	2	2	no	—
13	C₂×M₃₂×C₁₃₁	—	2	2	no	—
14	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
16	#472×C₁₃₁	—	2	2	no	—
17	#472×C₁₃₁	C₄	1	4	yes	complemented
18	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
20	#427×C₁₃₁	C₂²	1	1	yes	characteristic
21	#421×C₁₃₁	—	2	1	no	—
22	C₂×#100×C₁₃₁	—	2	1	no	—
23	#361×C₁₃₁	—	2	2	no	—
24	#357×C₁₃₁	—	2	2	no	—
25	#344×C₁₃₁	—	2	1	no	—
26	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃₁	—	2	1	no	—
28	AGL(1,ℤ/8)×C₁₃₁	—	2	4	no	—
29	AGL(1,ℤ/8)×C₁₃₁	—	4	2	no	—
30	C₂×SD₁₆×C₁₃₁	—	4	1	no	—
31	C₂×D₈×C₁₃₁	—	2	1	no	—
32	C₂×D₈×C₁₃₁	—	2	1	no	—
33	C₂×D₈×C₁₃₁	—	2	1	no	—
34	C₂×D₈×C₁₃₁	—	2	1	no	—
35	C₂×D₈×C₁₃₁	—	2	1	no	—
36	C₂×D₈×C₁₃₁	—	4	1	no	—
37	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
38	#131×C₁₃₁	—	2	1	no	—
39	#131×C₁₃₁	—	2	1	no	—
40	#131×C₁₃₁	C₂×C₄	1	2	yes	—
41	#131×C₁₃₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₁₃₁	—	2	1	no	—
43	C₂×M₁₆×C₁₃₁	—	2	1	no	—
44	C₂×M₁₆×C₁₃₁	—	2	1	no	—
45	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	—	2	1	no	—
47	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
48	#117×C₁₃₁	—	2	1	no	—
49	M₃₂×C₁₃₁	—	2	4	no	—
50	M₃₂×C₁₃₁	—	2	1	no	—
51	M₃₂×C₁₃₁	—	2	1	no	—
52	M₃₂×C₁₃₁	—	2	1	no	—
53	M₃₂×C₁₃₁	D₄	1	2	yes	—
54	C₂×C₁₆×C₁₃₁	—	2	4	no	—
55	C₂×C₁₆×C₁₃₁	—	2	1	no	—
56	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
57	#108×C₁₃₁	—	2	1	no	—
58	#108×C₁₃₁	—	2	1	no	—
59	#108×C₁₃₁	—	2	1	no	—
60	#108×C₁₃₁	C₂×C₄	1	2	yes	—
61	#107×C₁₃₁	—	2	1	no	—
62	#106×C₁₃₁	—	2	1	no	—
63	#104×C₁₃₁	—	4	2	no	—
64	#100×C₁₃₁	—	2	4	no	—
65	C₂⁴×C₁₃₁	—	4	1	no	—
66	#41×C₁₃₁	—	2	1	no	—
67	#41×C₁₃₁	—	2	1	no	—
68	#41×C₁₃₁	C₂²⋊C₄	1	2	yes	—
69	#41×C₁₃₁	D₈	1	2	yes	—
70	C₂×Q₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
71	C₂×D₄×C₁₃₁	—	2	1	no	—
72	C₂×D₄×C₁₃₁	—	2	2	no	—
73	C₂×D₄×C₁₃₁	—	4	1	no	—
74	C₂×D₄×C₁₃₁	—	4	1	no	—
75	C₂×D₄×C₁₃₁	—	4	1	no	—
76	C₂×D₄×C₁₃₁	—	4	1	no	—
77	C₂×D₄×C₁₃₁	—	4	1	no	—
78	C₂×D₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₁	—	2	1	no	—
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	SD₁₆×C₁₃₁	—	4	4	no	—
82	D₈×C₁₃₁	—	2	2	no	—
83	D₈×C₁₃₁	—	2	2	no	—
84	D₈×C₁₃₁	—	4	4	no	—
85	D₈×C₁₃₁	—	4	1	no	—
86	D₈×C₁₃₁	—	4	1	no	—
87	M₁₆×C₁₃₁	—	2	1	no	—
88	M₁₆×C₁₃₁	—	2	2	no	—
89	M₁₆×C₁₃₁	—	2	2	no	—
90	M₁₆×C₁₃₁	—	2	2	no	—
91	M₁₆×C₁₃₁	—	4	1	no	—
92	M₁₆×C₁₃₁	—	4	1	no	—
93	M₁₆×C₁₃₁	—	4	1	no	—
94	M₁₆×C₁₃₁	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₁₃₁	—	2	1	no	—
96	C₂×C₈×C₁₃₁	—	2	2	no	—
97	C₂×C₈×C₁₃₁	—	2	2	no	—
98	C₂×C₈×C₁₃₁	—	4	1	no	—
99	C₂×C₈×C₁₃₁	—	4	1	no	—
100	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
105	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₁	—	4	1	no	—
107	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
108	C₄²×C₁₃₁	—	4	1	no	—
109	C₁₆×C₁₃₁	—	2	2	no	—
110	C₁₆×C₁₃₁	—	2	2	no	—
111	C₂³×C₁₃₁	—	2	1	no	—
112	C₂³×C₁₃₁	—	4	2	no	—
113	C₂³×C₁₃₁	—	8	1	no	—
114	C₂³×C₁₃₁	—	8	1	no	—
115	C₂³×C₁₃₁	—	16	2	no	—
116	C₂³×C₁₃₁	#135	1	1	yes	characteristic
117	Q₈×C₁₃₁	—	2	1	no	—
118	Q₈×C₁₃₁	#102	1	2	yes	—
119	D₄×C₁₃₁	—	2	2	no	—
120	D₄×C₁₃₁	—	2	2	no	—
121	D₄×C₁₃₁	—	2	1	no	—
122	D₄×C₁₃₁	—	4	2	no	—
123	D₄×C₁₃₁	—	4	2	no	—
124	D₄×C₁₃₁	—	8	1	no	—
125	D₄×C₁₃₁	—	8	1	no	—
126	D₄×C₁₃₁	#102	1	2	yes	—
127	C₂×C₄×C₁₃₁	—	2	1	no	—
128	C₂×C₄×C₁₃₁	—	2	2	no	—
129	C₂×C₄×C₁₃₁	—	2	2	no	—
130	C₂×C₄×C₁₃₁	—	8	1	no	—
131	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
137	C₈×C₁₃₁	—	2	2	no	—
138	C₈×C₁₃₁	—	4	2	no	—
139	C₈×C₁₃₁	—	8	1	no	—
140	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
141	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
142	C₈×C₁₃₁	#120	1	1	yes	characteristic
143	C₈×C₁₃₁	#123	1	1	yes	characteristic
144	C₂²×C₁₃₁	—	2	2	no	—
145	C₂²×C₁₃₁	—	4	1	no	—
146	C₂²×C₁₃₁	—	8	1	no	—
147	C₂²×C₁₃₁	—	8	2	no	—
148	C₂²×C₁₃₁	—	16	2	no	—
149	C₂²×C₁₃₁	—	16	2	no	—
150	C₂²×C₁₃₁	—	16	2	no	—
151	C₂²×C₁₃₁	#416	1	1	yes	characteristic
152	C₂²×C₁₃₁	#465	1	1	yes	characteristic
153	C₂²×C₁₃₁	#466	1	1	yes	characteristic
154	C₄×C₁₃₁	—	2	2	no	—
155	C₄×C₁₃₁	—	16	1	no	—
156	C₄×C₁₃₁	#386	1	1	yes	characteristic
157	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
158	C₄×C₁₃₁	#465	1	1	yes	characteristic
159	C₄×C₁₃₁	#466	1	1	yes	characteristic
160	C₂×C₁₃₁	—	2	1	no	—
161	C₂×C₁₃₁	—	2	1	no	—
162	C₂×C₁₃₁	—	2	1	no	—
163	C₂×C₁₃₁	—	4	2	no	—
164	C₂×C₁₃₁	—	16	2	no	—
165	C₂×C₁₃₁	#1893	1	1	yes	characteristic
166	#13068	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₁₃₁	#13068	1	1	yes	characteristic, direct product factor, Sylow
168	#3407	C₂×C₁₃₁	1	1	yes	characteristic, complemented
169	C₂×#472	C₂×C₁₃₁	1	1	yes	characteristic
170	C₂×#361	C₂×C₁₃₁	1	1	yes	characteristic, complemented
171	#2140	C₂×C₁₃₁	1	1	yes	characteristic, complemented
172	#1814	C₂×C₁₃₁	1	1	yes	characteristic, complemented
173	#1389	C₂×C₁₃₁	1	1	yes	characteristic, complemented
174	#1385	C₂×C₁₃₁	1	1	yes	characteristic, complemented
175	C₂×AGL(1,ℤ/8)	—	2	1	no	—
176	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
177	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
178	#504	—	2	2	no	—
179	#504	—	2	2	no	—
180	C₂×M₃₂	—	2	2	no	—
181	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
183	#472	—	2	2	no	—
184	#472	C₄×C₁₃₁	1	4	yes	complemented
185	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
186	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
187	#427	C₂²×C₁₃₁	1	1	yes	characteristic
188	#421	—	2	1	no	—
189	C₂×#100	—	2	1	no	—
190	#361	—	2	2	no	—
191	#357	—	2	2	no	—
192	#344	—	2	1	no	—
193	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
194	C₂²×D₄	—	2	1	no	—
195	AGL(1,ℤ/8)	—	2	4	no	—
196	AGL(1,ℤ/8)	—	4	2	no	—
197	C₂×SD₁₆	—	4	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	4	1	no	—
204	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
205	#131	—	2	1	no	—
206	#131	—	2	1	no	—
207	#131	C₂×C₄×C₁₃₁	1	2	yes	—
208	#131	D₄×C₁₃₁	1	2	yes	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
215	#117	—	2	1	no	—
216	M₃₂	—	2	4	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	D₄×C₁₃₁	1	2	yes	—
221	C₂×C₁₆	—	2	4	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
224	#108	—	2	1	no	—
225	#108	—	2	1	no	—
226	#108	—	2	1	no	—
227	#108	C₂×C₄×C₁₃₁	1	2	yes	—
228	#107	—	2	1	no	—
229	#106	—	2	1	no	—
230	#104	—	4	2	no	—
231	#100	—	2	4	no	—
232	C₂⁴	—	4	1	no	—
233	#41	—	2	1	no	—
234	#41	—	2	1	no	—
235	#41	C₂²⋊C₄×C₁₃₁	1	2	yes	—
236	#41	D₈×C₁₃₁	1	2	yes	—
237	C₂×Q₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
248	SD₁₆	—	4	4	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	4	no	—
252	D₈	—	4	1	no	—
253	D₈	—	4	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	C₂²⋊C₄×C₁₃₁	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
271	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
272	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	16	2	no	—
283	C₂³	#135×C₁₃₁	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	#102×C₁₃₁	1	2	yes	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	1	no	—
292	D₄	—	8	1	no	—
293	D₄	#102×C₁₃₁	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
308	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
309	C₈	#120×C₁₃₁	1	1	yes	characteristic
310	C₈	#123×C₁₃₁	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	—	16	2	no	—
317	C₂²	—	16	2	no	—
318	C₂²	#416×C₁₃₁	1	1	yes	characteristic
319	C₂²	#465×C₁₃₁	1	1	yes	characteristic
320	C₂²	#466×C₁₃₁	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	16	1	no	—
323	C₄	#386×C₁₃₁	1	1	yes	characteristic
324	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
325	C₄	#465×C₁₃₁	1	1	yes	characteristic
326	C₄	#466×C₁₃₁	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	2	no	—
331	C₂	—	16	2	no	—
332	C₂	#1893×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₁	#1893
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂×#472×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×#131×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3407×C₁₃₁	C₂	semidirect
3. C₂×#361×C₁₃₁	C₂	semidirect
4. #2140×C₁₃₁	C₂	semidirect
5. #1814×C₁₃₁	C₂	semidirect
6. #1389×C₁₃₁	C₂	semidirect
7. #1385×C₁₃₁	C₂	semidirect
17. #472×C₁₃₁	C₄	semidirect
166. #13068	C₁₃₁	direct (N × H)
167. C₁₃₁	#13068	direct (N × H)
168. #3407	C₂×C₁₃₁	semidirect
170. C₂×#361	C₂×C₁₃₁	semidirect
171. #2140	C₂×C₁₃₁	semidirect
172. #1814	C₂×C₁₃₁	semidirect
173. #1389	C₂×C₁₃₁	semidirect
174. #1385	C₂×C₁₃₁	semidirect
184. #472	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3407×C₁₃₁	2
C₂×#472×C₁₃₁	2
C₂×#361×C₁₃₁	2
#2140×C₁₃₁	2
#1814×C₁₃₁	2
#1389×C₁₃₁	2
#1385×C₁₃₁	2
#13068	131

Subgroups — C131×#13068 #45698952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13068 #45698952